text:
\section{Introduction}
\label{sec:intro}

\noindent {\bf Texture classification.} \quad In numerous modern applications (satellite imagery \cite{RAD2000}, geography \cite{f97}, biomedical imagery \cite{Lundahl:1986,aK2001,Rachidi::2008,Richard::2010,B2010}, geophysics \cite{SL87}, art investigation \cite{ABRY:2012:A}, \ldots), the data available for analysis consist of images of homogeneous textures, that need to be characterized.
Texture classification thus consists of classical problem in image processing that received considerable efforts in recent years (cf. e.g.,  \cite{Lundahl:1986,U1995,N02,D2002,C05,vsv99,P84,K99} and references therein). \\*[-.2cm]

\noindent {\bf Scale invariance and Self-similarity.} \quad Amongst the many different ways texture characterizations have been investigated, techniques based on  \emph{scale invariance}, or \emph{fractal}, concepts are considered as promising, notably for the application fields listed above (cf. e.g., \cite{Pesquet-Popescu::2002} for a review).
Scale invariance can be defined as the fact that there exists no specific space-scale in data that play a preferred role in their space dynamic, or equivalently that all space-scales are equally important.
Scale invariance in data implies that they are analyzed with (statistical) models that do not rely on the identification of specific scales (such as Markov models) but instead with models that aim at characterizing a relation amongst scales.
Because \emph{Self-Similarity} is a theoretically well-grounded, and relatively simple instance of scale invariance behaviors, it has often been proposed that Gaussian self-similar fields are relevant models enabling efficient characterization and classification of the textures (cf. e.g, \cite{K99,P84}). \\*[-.2cm]

\noindent {\bf Anisotropy.} \quad However, textures are also often characterized by anisotropy, which may either be deeply tied to self-similarity itself \cite{BE03,BMS07} or exist as an independent property that is superimposed to an isotropic self-similarity.
In both cases, it is a crucial stake in analysis to disentangle self-similarity from anisotropy, to discriminate whether self-similarity and anisotropy are independent properties or if they are stemming from the same constructive mechanism, as well as to be able to estimate accurately the self-similarity parameter $H$, despite anisotropy.
It has already been pointed out that fractal analysis and estimation is very sensitive to anisotropy (cf. e.g., \cite{SL87}).
In the literature, anisotropy is often analyzed from 1D slices extracted from images along different directions \cite{Richard::2010} or by making use of local directional differential estimators {\cite{R92,S94}.  \\*[-.2cm]

\noindent {\bf Goals and contributions.} \quad In this context, elaborating on a preliminary attempt \cite{rcvja11}, the present contribution aims at proposing an efficient and elegant solution to the joint analysis and estimation of self-similarity and anisotropy in 2D fields.
Though the devised procedure aims at, and is designed for, being applied to real-world textures, its performance are assessed by means of Monte Carlo simulation performed on synthetic isotropic and non isotropic Gaussian textures.
While the (discrete) wavelet transform (DWT)  is nowadays a classical tool for image processing, the key originality of the present work is to show that the classical discrete wavelet transform fails at providing a relevant analysis of self-similarity in presence of anisotropy and it is instead here proposed to replace it with the \emph{Hyperbolic Wavelet Transform} (HWT) (defined e.g.,  in \cite{DeVore1998}).
Indeed, the use of different dilation factors on the axes $x$ and $y$ potentially permits to \emph{see} the anisotropy, as opposed to the classical 2D Discrete Wavelet Transform relying on a single and isotropic dilation operator.
The Hyperbolic Wavelet Transform is defined in Section~\ref{sec:HWT}.
Note that the HWT had appear earlier in the literature under different names, such as \emph{Tensor-product wavelet} \cite{Yu96}, \emph{anisotropic wavelet transform} \cite{Ro1999} or \emph{rectangular wavelet transform} \cite{Zav07}, without specific exploration though of its benefits to study anisotropy in textures.
Also, redundant (or overcomplete) wavelet representations (such as M-band, dual tree and Hilbert pair complex wavelets, cf. e.g., \cite{S05} for a enlightening review) may be used to analyze images and textures.
However, while they suffer from a larger computational cost, they have been observed (in preliminary attempts performed by the authors) to yield little, if not no, practical benefit for the study of scale invariance and the estimation of the corresponding parameters.
Overcomplete wavelet representations are thus excluded from the present study.

As representative of 2D model mixing self-similarity and anisotropy, self-similar Gaussian 2D random fields with built-in anisotropy, such as those proposed in e.g., \cite{BE03,BMS07}, are used here.
These processes are defined and illustrated in Section~\ref{sec:OSGRF} and their hyperbolic wavelet analysis is detailed in Section~\ref{sec:HWTOSGRF}.

Estimation procedures for the parameters characterizing self-similarity and anisotropy are defined and their performance assessed in Section~\ref{sec:esta}.
The definition of anisotropy involves a rigid definition of reference (orthogonal) axes, that have no reason a priori to match those of the sensor used to acquire the image and thus to coincide with the horizontal $x$ and vertical $y$ axis with which the image is presented for analysis.
Therefore, the model introduced in Section~\ref{sec:OSGRF} includes a rotation parameter that accounts for this unknown.
An estimation procedure for this rotation is devised and analyzed in Section~\ref{sec:estb}.
Therefore, the parameters characterizing rotation, anisotropy and self-similarity are estimated jointly.

For application purposes, it is crucial to be able to decide whether textures should be modeled by isotropic or anisotropic models.
Therefore, a procedure for testing the null hypothesis that the texture is isotropic is constructed and studied in Section~\ref{sec:test}.
It is based on a non parametric bootstrap procedure performed on the hyperbolic wavelet coefficients (in the spirit of the construction devised in \cite{WENDT:2007:E}) and can thus be applied to each single analyzed texture independently.
Incidentally, the bootstrap procedures also provides us with confidence intervals for the estimates, a very important feature for practical purposes.

To finish with, the analysis procedures proposed here are applied in Section~\ref{sec:othermodel} to a variety of isotropic and anisotropic fields that differ from the precise model used as a reference model (cf. Section~\ref{sec:OSGRF}), hence illustrating the robustness and generality of the tools proposed here.
Notably, it is shown that the proposed analysis enables to clearly distinguish between a truly anisotropic self-similar field from a isotropic self-similar field (with same self-similar parameter) to which directional, hence anisotropic, oscillations have been additively superimposed.
Joint anisotropy and self-similarity is hence clearly disentangled from isotropic self-similarity, with unrelated superimposed anisotropic trend.

\begin{figure}[t]
\begin{minipage}[b]{1.0\linewidth}
\includegraphics[width=.33\linewidth]{fig_modele5.pdf}~\includegraphics[width=.33\linewidth]{fig_modele3.pdf}~\includegraphics[width=.33\linewidth]{fig_modele6.pdf}\\*[.04cm]
\includegraphics[width=.33\linewidth]{fig_modele6_1bis.pdf}~\includegraphics[width=.33\linewidth]{fig_modele6_2bis.pdf}~\includegraphics[width=.33\linewidth]{fig_modele6_3bis.pdf}
\end{minipage
\vspace{-.2cm}
\caption{\label{fig:model} {\bf Sample fields of $X_{\theta_0,\alpha_0,H_0}$.} Top line : $(\theta_0,\alpha_0,H_0)=(0,\alpha_0,0.2)$ with, from left to right $\alpha_0=1$ (isotropic); $\alpha_0=0.7$ and $\alpha_0=0.3$. Bottom line : $(\theta_0,\alpha_0,H_0)=(\theta_0,0.7,0.2)$ and, from left to right $\theta_0=\pi/6$, $\theta_0=\pi/4$ and $\theta_0=\pi/3$.}
\end{figure}

\section{Hyperbolic Wavelet Analysis of anisotropic self-similar random fields}

\subsection{Anisotropic self-similar random fields}
\label{sec:OSGRF}

\subsubsection{Definition}

Because of its generic and representative nature, it has been chosen to work with the class of anisotropic Gaussian self-similar fields, introduced in \cite{BE03,BMS07},  referred to as \emph{Operator Scaling Gaussian Random Field} ({\bf OSGRF}) {which can be defined using the following harmonizable representation}:
\begin{equation}
{X_{f,E_0,H_0}(\underline{x})} =  \int_{\mathbb{R}^2} (e^{i\langle \underline{x}, \, \underline{\xi} \rangle} -1) f(\underline{\xi})^{-(H_0 +1)} d\widehat{W}(\underline{\xi})\,,
\label{eq:modela}
\end{equation}
where $\underline{x}=(x_1,x_2)$,  $\underline{\xi}=(\xi_1,\xi_2)$, $E_0$ is a matrix satisfying Tr$(E_0)=2$,
$f$ a  { $E_0$--homogeneous continuous positive function} (hence   {satisfying the homogeneity relationship
$f(a^{E_0}\underline{\xi})=af(\underline{\xi})$ on $\mathbb{R}^2$)
such that $\int (1\wedge |\underline{\xi}|^2)f(\underline{\xi})^{-2(H_0 +1)} d\underline{\xi}<+\infty$, and  where $d\widehat{W}(\underline{\xi})$ stands for a 2D  {Wiener measure}.}

When $f$ is not a radial function, the Gaussian field is not isotropic.
In this study, it is chosen to use the following 2-parameter (related to anisotropy and rotation) explicit form:
$$ f_{\theta_0,\alpha_0}(\underline{\xi})=(\vert \zeta_1 \vert^{1/\alpha_0} + \vert \zeta_2 \vert^ {1/(2-\alpha_0)}),$$
with $\underline{\zeta}=(\zeta_1,\zeta_2)=R_{\theta_0} \underline{\xi}$ and rotation matrix $\displaystyle  R_{\theta_0}$ defined as:
$$\displaystyle  R_{\theta_0} = \begin{pmatrix}\cos(\theta_0)&-\sin(\theta_0)\\\sin(\theta_0) & \cos(\theta_0)\end{pmatrix}.$$
In this model,  $E_0=\begin{pmatrix}\alpha_0&0\\0&2-\alpha_0\end{pmatrix}$,  $0<\alpha_0<2$.

A rotation parameter $\theta_0$ has been added to  the definition \ref{eq:modela} : It accounts for the fact that the rigid axes according to which anisotropy is defined need not match a priori the sensor axes according to which the image is digitalized  (for real-world data) or numerically produced (for synthetic textures).
In the sequel, {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$ thus refers to the following model, relying on $3$ parameters, $\theta_0,\alpha_0,H_0$, characterizing respectively, rotation, anisotropy and self-similarity
\begin{eqnarray}
X_{\theta_0,\alpha_0,H_0}(\underline{x}) = &  \int_{\mathbb{R}^2} (e^{i\langle \underline{x}, \, \underline{\xi} \rangle} -1) f_{\alpha_0,\theta_0}(\underline{\xi})^{-(H_0 +1)} d\widehat{W}(\underline{\xi}
\label{eq:model}
\end{eqnarray}

\subsubsection{Properties}
With this construction, {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$ has stationary increments.
It possesses a built-in anisotropy characterized by the parameter $ \alpha_0 \in (0, 2)$.
When $\alpha_0=1$, the field is isotropic and the case $1<\alpha_0<2$ correspond to the case $0<\alpha_0<1$ with the axes $(x_1,x_2)$ permuted.

{\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$
satisfies  (where $\overset{\mathcal{L}}{=}  $ denotes equality for all finite dimensional distributions):
\begin{equation}\label{e:selfsimilar}
\{ X_{\alpha_0,H_0,\theta_0}(a^{E}  \underline{x} )\}
\overset{\mathcal{L}}{=} \{ a^{H_0} X_{\alpha_0,H_0,\theta_0}( \underline{x}) \}.
\end{equation}
with $E_0=R_{\theta_0}E R_{-\theta_0}$. It is thus exactly self-similar with parameter $0< H_0 < \makebox{ min }(\alpha_0, 2-\alpha_0) <2 $.

Fig.~\ref{fig:model} displays realizations of $X_{\theta_0,\alpha_0,H_0}(\underline{x})$, obtained from {\sc Matlab} routines written by ourselves and available upon request.
On top row, $(H_0,\theta_0)=(0.2,0)$ are kept fixed while $\alpha_0$ is varied from  $\alpha_0 =1$, $0.7$ and $0.3$  (from left to right).
The practical goal is to estimate correctly $H_0$ despite these different unknown anisotropy strengths $\alpha_0$.
On bottom row, a strongly anisotropic field is shown,  $(\alpha_0=0.3,H_0=0.2)$, with $3$ different rotation angles $\theta_0$
(from left to right  $\theta_0=\pi/6$, $\pi/4$ and $\pi/3$).
The targeted goal is here to estimate correctly $(\alpha_0,H_0)$ despite such unknown rotation.

This three-parameter {\bf OSGRF} stochastic process  provides us with a rich and versatile model for selfsimilar (an)isotropic textures.

\subsubsection{Numerical simulation}
Realizations (or sample fields) of the synthetic processes defined in Eq.~(\ref{eq:model}) are produced numerically following the classical procedure, recalled in e.g., \cite{BMS07}, relying on drawing at random realizations of white-noise $d\widehat{W}(\underline{\xi})$, followed by standard numerical integration procedures.

\begin{figure}[t]
\begin{minipage}[b]{1.0\linewidth}
  \centering
  \centerline{\includegraphics[width=1.0\linewidth]{fig_algobis.pdf}}
\end{minipage}
\vspace{-.6cm}
\caption{\label{fig:algo} {\bf Hyperbolic Wavelet Transform.}
Top line : one step of HWT consists of one step of 2D-DWT (left) (with 1D-DWT performed on each line of HL and each column of LH  subbands (right)).
Bottom line : second step of HWT (left) and locations of the HWT vs DWT (black dots) coefficients in the frequency domain (right).
Black dots correspond to 2D-DWT while for HWT, white dots indicate subband HH, black triangles indicate LH and HL, squares correspond to the approximation coefficients).
The circle symbols correspond to the coefficients $\psi_{j_1,j_2}$ $j_1,j_2\neq 0$  and square symbols  to  the coefficients $\psi_{0,j_2}, \psi_{j_1,0}, \psi_{0,0}$.
}
\end{figure}

\subsection{Hyperbolic Wavelet Transform}
\label{sec:HWT}

The 2D Hyperbolic Wavelet Transform (HWT) differs from the 2D Discrete Wavelet Transform (DWT) insofar as its definition relies on the use of two different dilation factors along the horizontal and vertical axes, as opposed to the 2D-DWT that makes use of a single and same dilation factor along both axes.
This difference turns out to be crucial for the analysis of anisotropy.

The collection of functions constituting the orthogonal basis underlying the HWT is defined as tensor products of univariate wavelets (cf. e.g., \cite{Mallat1998}).
Let $\varphi$ and $\psi$ denote the scaling function and the wavelet of a given one-dimensional multiresolution analysis.
The HWT basis of ${\cal L}^2({\mathbb R}^2)$ is defined as (cf.~\cite{DeVore1998}):
\begin{eqnarray*}
\psi_{j_1,j_2,k_1,k_2}(x_1,x_2)\!\!\!\!\!&=&\!\!\!\!\!
\psi(2^{j_1}x_1-k_1)\psi(2^{j_2}x_2-k_2),\\
\psi_{0,j_2,k_1,k_2}(x_1,x_2)\!\!\!\!\!&=&\!\!\!\!\!
\varphi(x_1-k_1)\psi(2^{j_2}x_2-k_2),\\
\psi_{j_1,0,k_1,k_2}(x_1,x_2)\!\!\!\!\!&=&\!\!\!\!\!
\psi(2^{j_1}x_1-k_1)\varphi(x_2-k_2),\\
\psi_{0,0,k_1,k_2}(x_1,\!\!\!\!\!&x_2&\!\!\!\!\!)= \varphi(x_1-k_1)\varphi(x_2-k_2),
\end{eqnarray*}
for all  $\underline{j}=(j_1,j_2)\in \mathbb{N}^{*2}$ and $\underline{k}=(k_1,k_2) \in \mathbb Z^2$.
The HWT shares a deep relation with Triebel bases, used in mathematic literature, to characterize anisotropic functional spaces (cf.~\cite{Trie06}).

The hyperbolic wavelet coefficients of the process $X$
are defined, $\forall \underline{j}=(j_1,j_2)\in\mathbb{N}^{*2}$,  as :
\begin{equation}
d_X(\underline{j},\underline{k})= 2^{j_1+j_2}\int_{\mathbb{R}^2}\psi_{\underline{j},\underline{k}}(x_1,x_2)X(x_1,x_2)d
x_1 d x_2.
\label{eq:coef}
\end{equation}
Note that a ${\cal L}^1$-normalization is used (instead of the classical ${\cal L}^2$-norm), as it better suits self-similarity analysis (cf. e.g., \cite{BMA93}).
With these notations, fine resolution scales correspond to the limit $2^{j_1}, 2^{j_2} \rightarrow  +\infty$, and index $j$ in the decomposition corresponds to the actual resolution $2^{J-j}$, where $J=\log_2(N)$, for an image of size ($N\times N$).

Such coefficients $d_X(\underline{j},\underline{k})$ can be computed efficiently, using a recursive pyramidal filter bank based algorithm comparable to that underlying the 2D-DWT.
In Fig.~\ref{fig:algo}, the first  two iterations are illustrated, in the Fourier domain.
One iteration of HWT practically consists of  the combination of one iteration of the
2D-DWT algorithm, with 1D-DWT  performed on each line of the vertical details (HL) and 1D DWT performed on each column of the horizontal details (LH).
Because the central frequencies of the dilated scaling function of $\varphi(2^j x)$ and mother-wavelet $\psi(2^j x)$ can be approximated as $f_j=\frac{1}{4}2^{j}$ and $f_j=\frac{3}{4}2^{j}$, respectively, the HWT coefficients $d_X(\underline{j},\underline{k})$ can be located in a (log-) frequency-frequency plane as shown in   Fig.~\ref{fig:algo} bottom right) and thus compared to the location of the 2D-DWT coefficients.

In what follows, a 1D-Daubechies-$3$ multiresolution is used \cite{daubechies92}.

\subsection{Analysis of anisotropic self-similar random fields}
\label{sec:HWTOSGRF}

From Eq. (\ref{eq:coef}),  the HWT coefficients  can be rewritten, $\forall \underline{j}=(j_1,j_2)\in\mathbb{N}^{*2}$ and
$\forall \underline{k}=(k_1,k_2)\in\mathbb{Z}^2$,  as stochastic integrals:
\begin{equation}
\label{def-wc}
d_X(\underline{j},\underline{k})=\int_{\mathbb{R}^2}
\frac{\left(\prod_{\ell=1}^{2}e^{i
2^{-j_\ell}k_\ell\xi_\ell}\overline{\widehat{\psi}(2^{-j_\ell}\xi_\ell)}\right)}
{\left(|\xi_1|^{1/\alpha_0}+|\xi_2|^{1/(2-\alpha_0)}\right)^{H_0+1}}\;d\widehat{W}(\xi_1,\xi_2)
\;.
\end{equation}
{Following the methodology in ~\cite{CV10b,ACJRV2012}, it can be proven that the HWT coefficients are weakly correlated,  i.e.,  $\forall (j_1,j_2,k_1,k_2,k'_1,k'_2)$
\begin{multline}
\left|\mathbb{E}(d_X((j_1,j_2),(k_1,k_2))d_X((j_1,j_2),(k'_1,k'_2)))\right| \\
\leq
\frac{\mathbb{E}(|d_X((j_1,j_2,0,0)|^2)}{1+|k_1-k'_1|+|k_2-k'_2|}\;.
\end{multline}
Using the substitution
$\zeta_1=2^{-j_1}\xi_1$,
$\zeta_2=2^{-j_2}\xi_2$ in the rewriting of definition of the wavelet coefficients (cf. Eq.~(\ref{def-wc})), we have been able to show that  the HWT coefficients typically behave as~\cite{CV10b,ACJRV2012} :
\[
d_X(\underline{j},\underline{k})\simeq 2^{\frac{j_1+j_2}{2}}\int_{\mathbb{R}^2}
\frac{\left(\prod_{\ell=1}^{2}e^{i
k_\ell\zeta_\ell}\overline{\widehat{\psi}(\zeta_\ell)}\right)d\widehat{W}(\zeta_1,\zeta_2)}
{\left(2^{\frac{j_1}{\alpha_0}}|\zeta_1|^{\frac{1}{\alpha_0}}+2^{\frac{j_2}{2-\alpha_0}}|\zeta_2|^{\frac{1}{2-\alpha_0}}\right)^{H_0+1}}
\;,
\]
When $j_1/\alpha_0>j_2/(2-\alpha_0)$, we derive that, for all $\zeta_1,\zeta_2$:
\begin{multline*}
2^{\frac{j_1}{\alpha_0}}|\zeta_1|^{\frac{1}{\alpha_0}} \leq
2^{\frac{j_1}{\alpha_0}}|\zeta_1|^{\frac{1}{\alpha_0}}+2^{\frac{j_2}{2-\alpha_0}}|\zeta_2|^{\frac{1}{2-\alpha_0}} \leq \\
 2^{\frac{j_1}{\alpha_0}}\left(|\zeta_1|^{\frac{1}{\alpha_0}}+|\zeta_2|^{\frac{1}{2-\alpha_0}}\right)
\end{multline*}
and further
\begin{multline*}
2^{\frac{-j_1H_0}{\alpha_0}}
\frac{1}{(|\zeta_1|^{\frac{1}{\alpha_0}}+|\zeta_2|^{\frac{1}{2-\alpha_0}})^{H_0+1}} \leq  \\
\frac{1}{2^{\frac{j_1}{\alpha_0}}|\zeta_1|^{\frac{1}{\alpha_0}}+2^{\frac{j_2}{2-\alpha_0}}|\zeta_2|^{\frac{1}{2-\alpha_0}}}\leq
2^{\frac{-j_1H_0}{\alpha_0}}\frac{1}{\left(|\zeta_1|^{\frac{H_0+1}{\alpha_0}}\right)}
\end{multline*}
which enabled us to obtain the following inequality:
\begin{multline*}
C_1 2^{\frac{j_1+j_2}{2}}2^{-\frac{j_1(H_0+1)}{\alpha_0}}\leq
\mathbb{E}(|d_X(\underline{j},\underline{k})|^2)^{1/2}\leq \\ C_2
2^{\frac{j_1+j_2}{2}}2^{-\frac{j_1(H_0+1)}{\alpha_0}}\; ,
\end{multline*}
\[
\makebox{ with } C_1=\left(\int_{\mathbb{R}^2}
\frac{\prod_{\ell=1}^{2}|\widehat{\psi}(\zeta_\ell)|^2d\xi}{(|\zeta_1|^{\frac{1}{\alpha_0}}+|\zeta_2|^{\frac{1}{2-\alpha_0}})^{H_0+1}}\right)^{1/2}\;,
\]
\[
\makebox{ and } C_2=\left(\int_{\mathbb{R}^2}
\frac{\prod_{\ell=1}^{2}|\widehat{\psi}(\zeta_\ell)|^2 d\xi}{(|\zeta_1|^{\frac{H_0+1}{\alpha_0}})}\right)^{1/2}\;.
\]
}
Combined to similar arguments for the case $j_1/\alpha_0\leq j_2/(2-\alpha_0)$, these calculations enable us to show that the order of magnitude of the expectation of the squared HWT coefficients reads, $\forall (j_1,j_2)$:
\begin{equation}
\label{eq:HWT}
\mathbb{E}(|d_X(\underline{j},\underline{k})|^2)^{1/2}\approx
2^{\frac{j_1+j_2}{2}}2^{-(H_0+1)\max(\frac{j_1}{\alpha_0},\frac{j_2}{2-\alpha_0})}\;.
\end{equation}
Because the processes of interest here are Gaussian, this can straightforwardly be extended to any  $ q \geq  -2 $, cf. \cite{CV10a,CV10b,ACJRV2012}:
\begin{equation}
\label{eq:HWTq}
\mathbb{E}(|d_X(\underline{j},\underline{k})|^q)\approx
2^{q(j_1+j_2)}2^{-q(H_0+1)\max(\frac{j_1}{\alpha_0},\frac{j_2}{2-\alpha_0})}\;.
\end{equation}
These key results constitute the founding ingredient for the estimation procedures defined below.

\section{Parameter Estimation}
\label{sec:est}

The goal is now to define estimation procedures for the three-parameters entering the definition of {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$ and to study their statistical performance.
It is assumed, first, that $\theta_0$ is known and equal to $\theta_0 \equiv 0$ and estimation is devoted to parameters $\alpha_0$ and $H_0$.
In the second part, $\theta_0$ is unknown and needs to be estimated as well.
This is performed by applying the estimation of $\alpha_0$ and $H_0$ to a collection of rotated images.
It will be shown that the correct estimation angle is estimated when the estimation of the anisotropy coefficients reaches its minimum.

\subsection{Self-similarity and anisotropy parameters}
\label{sec:esta}

In this section, $\theta_0$ is assumed to be known and taken equal to $0$ for simplicity.  \\

\begin{figure}[t]
\begin{minipage}[t]{\linewidth}
\includegraphics[width=\linewidth]{figloglognew.pdf}
\end{minipage}
\vspace{-.7cm}
\caption{\label{fig:estim} {\bf Illustrations of the estimation procedures.} For anisotropic, $\alpha_0 = 0.7$  (top row), and isotropic, $\alpha_0 = 1$ (bottom row), {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$, with $\theta_0= 0, H_0=0.7$.
Left column: Structure functions $S(q,j_1,j_2)$. The solid line indicates the direction $\hat{\alpha}$ while the dashed lines corresponds to  $ \alpha = 1$ and hence the sole direction actually reachable with the coefficients of 2D-DWT.
Middle column: Estimation of $H(\alpha)$, based on $\tau(q=2,\alpha=\hat \alpha)$ obtained from a linear regression of $\log_2 S(q,\alpha j,(2-\alpha) j) $ versus $j$ ($\circ$).
Solid lines correspond to the theoretical $\tau(q=2,\alpha_0)$.
Stars (in (b)) correspond to the (biased) estimation of $H$ from $\tau(q=2,\alpha=1)$, i.e. by using only the 2D-DWT coefficients.
Right column: Plots of $\hat H_2(\alpha) = \tau(2,\alpha)/2$ versus $\alpha$.
The black dot shows the location of the maximum of $H_2(\alpha) = \tau(2,\alpha)/2$ thus yielding the estimated $\hat \alpha$ and $\hat H$.
In (f), as expected for an isotropic image, $\hat \alpha = 1$.
The mixed line corresponds to the theoretical values of $ \tau(2,\alpha)/2$ (cf. Eq. \ref{eq:estimalpha}).
 }
\end{figure}

\subsubsection{Estimation procedure}

By analogy to what has classically been done for the analysis of self-similarity, or scale invariance in general, (cf. e.g., \cite{abfrv02}), the space averages at joint scales $(j_1, j_2)$ (also referred to as structure functions) are used as estimators for the ensemble averages appearing in Eq. (\ref{eq:HWTq}) above:
\begin{equation}
\label{sec:tauqalpha}
 S(q,j_1,j_2) = \frac{1}{n_{j_1,j_2}}\sum_{(k_1,k_2)\in\mathbb{Z}^2} \vert d_X(j_1,j_2,k_1,k_2)\vert^q,
\end{equation}
where $n_{j_1,j_2}$ stands for the number of available coefficients jointly at scales $(2^{j_1},2^{j_2})$.

\noindent Let us further define $\tau_j(q,\alpha)$ as a function of the statistical order $q > 0$ and of the anisotropy parameter $\alpha$:
\begin{equation}
\label{sec:tauqalpha}
\tau(q,\alpha)=\liminf_j\frac{-\alpha \log_2 (S(q,\alpha j,(2-\alpha)j)}{j }.
\end{equation}
In essence, Eq. (\ref{sec:tauqalpha}) amounts to assuming a power-law behavior of the structure functions with respect to scales, in the limit of fine scales $2^j \rightarrow +\infty$, along direction $\alpha$:
$$  S(q,\alpha j,(2-\alpha) j)  \simeq S_0(q) 2^{-j {\tau(q,\alpha)\over \alpha}}. $$

\noindent Eq. (\ref{eq:HWTq}) above indicates that, on average, and with the specific choice $(j_1, j_2) = (\alpha j,(2-\alpha)j)$:
$$ S(q,\alpha j,(2-\alpha) j) \approx 2^{j \frac{q}{\alpha}\left(1-(H_0+1)\max(\frac{\alpha}{\alpha_0},\frac{2-\alpha}{2-\alpha_0})  \right) }.  $$

\noindent Comparing these two last relations suggests that ${\tau(q,\alpha)\over \alpha} = \frac{q}{\alpha}\left(1-(H+1)\max(\frac{\alpha}{\alpha_0},\frac{2-\alpha}{2-\alpha_0})  \right)$, so that, for a given fixed $q$, the anisotropy parameter $\alpha_0$ can be estimated as:
\begin{equation}
\label{eq:estimalpha}
\hat \alpha_{0,q}  =  \makebox{argmin}_\alpha \tau(q,\alpha),
\end{equation}
and that the self-similarity parameter $H$ can be estimated as:
\begin{equation}
\label{eq:estimH}
\hat{H}_q  =  \tau(q,\hat{\alpha}_{0,q})/q.
\end{equation}

\subsubsection{Illustrations}
\label{sec:estillust}

The estimation procedure proposed here is sketched in Fig.~\ref{fig:estim}, for $q=2$, for an anisotropic (\ref{fig:estim}a-c) and an isotropic  (\ref{fig:estim}d-e)  {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$.
It can be decomposed into three steps (for a given $q>0$).

{\bf Step 1:} The HWT coefficients $d_X(\underline{j},\underline{k})$ and corresponding structure functions $S(q,j_1,j_2)$ are computed.
Examples are shown in Fig.~\ref{fig:estim} (left column).

{\bf Step 2:} The surface $\log_2 S(q,j_1,j_2)$, seen as a function of the variables $j_1, j_2$ is interpolated  {(by nearest neighbor)} along the line $\alpha j_1+1=(2-\alpha) j_2+1$. Then, a non weighted least-square regression of  $\log_2 S(q,\alpha j,(2-\alpha) j)$ versus $\log_2 2^j = j$ is performed across all available scales, hence yielding an estimate of $\tau(q,\alpha)$, for each $\alpha$ and each $q$, as sketched in  Fig. \ref{fig:estim}-b and \ref{fig:estim}-e).

{\bf Step 3:} The estimated $\tau(q,\alpha)$ are plotted for a given $q$, as a function of $\alpha$, and its maximum yields the estimate $\hat{\alpha}$ of $\alpha_0$ (cf. Fig. \ref{fig:estim}-c and \ref{fig:estim}-f).
The estimation of the self--similarity parameter $H_0$ is further given by $\hat{H_q}=\tau(q,\hat{\alpha})/q$. \\
This procedure calls for the following comments.
First, Step~2 is performed for all accessible $\alpha$s, that is, for all values of $\alpha$, that connect at least two pairs of
 dyadic scales, i.e.,  $(a_1=2^{j_1}, a_2=2^{j_2})$ with integers $(j_1,j_2)=[1, 2, \ldots, J]^2$.
Therefore, the actual resolution of the values of $\alpha$  that can actually be used depends on the size $N \times N$ of the analyzed image, and hence so does the resolution of the estimate of the anisotropy parameter.
This discretized resolution can be observed in Fig. \ref{fig:test}.

Second, the structure functions $S(q,j,j)$ (hence for $\alpha=1$) are computed from HWT coefficients that actually corresponds to those of the 2D-DWT (cf. Fig.~\ref{fig:estim}-b, dashed line).
For isotropic fields, it is found that $\hat \alpha =1 $, and thus that the coefficients of the HWT that need to be actually used for the estimate of $H_0$ are those of the 2D-DWT.
Conversely, for anisotropic fields, basing the estimate of $H$ on $S(q,j,j)$ results into significant biases, as illustrated in Fig.~\ref{fig:estim}-b, dashed line and -e, where the estimate of $H_0$ for $\alpha \equiv 1$ significantly differs from that obtained for $\alpha = \hat \alpha$.
This illustrates the major benefits of replacing the 2D-DWT with the 2D-HWT.

\begin{figure}[t]
\begin{minipage}[t]{\linewidth}
\includegraphics[width=\linewidth]{figMCsize.pdf}\\
\end{minipage}
\vspace{-.6cm}
\caption{\label{fig:perf} {\bf Estimation performance.} As functions of the sample size $N$ (image size $N \times N$), biases (top row) and standard deviation for $\hat H_{q=2} $ (left column) and $\hat \alpha_{q=2}$ (right column), obtained as average of estimation performed on $500$ realizations of  {\bf OSGRF} $X_{0,\alpha_0,H_0}$ with parameters $\alpha_0=1, H_0=0.7, 0.5$  and $0.3$ ($\circ$),
$\alpha_0=0.8, H_0=0.7, 0.5, 0.3$ ($\ast$) and $\alpha_0=0.6, H_0= 0.5, 0.3$ ({\large $\displaystyle \cdot$}).
Bottom row, dashed lines illustrates the expected $1/sqrt{N \times N}$ decrease of the standard deviations.
 }
\end{figure}

\subsubsection{Estimation Performance}

To complement the theoretical study reported above and to further assess the performance of the proposed estimation procedures, Monte-Carlo simulations are now performed further expanding on numerical investigations presented in  \cite{rcvja11}.
Biases and the standard deviations of $\hat \alpha_q$ and $\hat H_q$ are obtained from averages of estimates computed over $500$ independent realizations of {\bf OSGRF} $X_{0,\alpha_0,H_0}$, numerically produced by {\sc Matlab} routines designed by ourselves and available upon request.

Fig.~\ref{fig:perf} reports biases and  standard deviations as a function of (the $\log_2$ of)  the sample size $N$ (image size is $N\times N$), for various parameters $(\alpha_0,H_0)$.
Fig.~\ref{fig:perf} essentially shows that the estimation performance both for $\hat \alpha $ and $\hat H$ does not depend on the actual $H_0$, a result that is highly reminiscent of the $1D$ case (cf. e.g., \cite{va01}).
However, dependences on the anisotropy parameter $\alpha_0$ do exist and are clearly visible on the standard deviation, which unexpectedly decrease with significant departures of isotropy.

Estimates are found to be asymptotically unbiased, as expected from theoretical analysis, and that standard deviations roughly decrease as $1/\sqrt(N\times N) = 1/N$, in agreement with the weak correlation property of the HWT coefficients.
Using other values of $q>0$ (ranging from $1$ to $5$) yields similar conclusions.

To conclude this section, let us put the emphasis on the fact that image sizes are varied from small  ($2^8 \times 2^8$) to (very) large ($2^{13} \times 2^{13}$).
This illustrates that both the synthesis and analysis procedures corresponding to the definition of {\bf OSGRF} and its analysis can be implemented efficiently and benefits from a remarkably low computational cost.

Estimation performance were reported here only for $q=2$, as it was found empirically that the use of other values of $q$ did not improve performance, as can be expected for Gaussian processes.

\begin{figure}[t]
\begin{minipage}[t]{\linewidth}
\includegraphics[width=\linewidth]{figresrot.pdf}
\end{minipage}
\vspace{-.6cm}
\caption{\label{fig:results} {\bf Joint three parameter estimation procedure.} Estimations of $\hat{\alpha}$ (top row) and $\hat{H}$ (bottom row) for three differents fields with $\theta_0=\pi/3$ and
$(\alpha_0,H_0)=(0.7,0.6)$ ((a) and (d));  $(\alpha_0,H_0)=(0.3,0.2)$ ((b) and (e)); $(\alpha_0,H_0)=(0.7,0.2)$ ((c) and (f)).
The dashed line illustrates the expected theoretical behavior of $\hat{\alpha}$ as a function of $\theta$  and the 'circle' with confidence intervals to Monte-Carlo averages.
The estimation of $\hat{\theta}$ corresponds to the  location of the minimum of $\hat{\alpha}(\theta)$ and satisfactorily corresponds to $\theta_0=\pi/3$.
Final estimates for $\alpha_0$ and $H_0$ are obtained as $\hat{\alpha}^*=\hat{\alpha}(\hat{\theta})$ and $\hat{H}^*=\hat{H}(\hat{\theta})$ and thus show satisfactory agreement with the theoretical values, marked by $ \ast $.
Error bars correspond to $\sigma_{\hat H}$ (resp. $\sigma_{\hat \alpha}$).
}
\end{figure}

\subsection{Rotation parameter}
\label{sec:estb}

\subsubsection{Estimation procedure}

Let us now consider the case where, in addition to $H_0$ and $\alpha_0$, the rotation angle $\theta_0$ is unknown.
To estimate jointly the three unknown parameters, it is here proposed to apply the above procedure to estimate $H_0$ and $\alpha_0$ to a collection of rotated version of the original image, with rotation angles $\theta$.
The estimation of the anisotropy direction relies on the following observations, illustrated in Fig.~\ref{fig:results} (top row):
i) The estimate $\hat{\alpha}(\theta)$ is a $\pi$-periodic function~;
ii) it also has the symmetry $\hat{\alpha}(\theta_0+\theta)=\hat{\alpha}(\theta_0-\theta)$~;
iii) when $\theta=\theta_0$, $\hat{\alpha} \simeq \alpha$~;
iv) when  $\theta=\theta_0+\pi/2$, $\hat{\alpha}=2-\alpha$~;
v) and when  $\theta=\theta_0+\pi/4$, $\hat{\alpha}=1$.
Thus, the following joint estimation procedure for ($\theta_0, \alpha_0, H_0$) can be proposed:
\begin{eqnarray}
\hat \theta    & = & \makebox{argmin}_\theta \hat{\alpha}(\theta),\\
\hat{\alpha}^* & = & \hat{\alpha}(\hat{\theta})\\
 \hat{H}^*_q         & = & \hat{H}_q(\hat{\alpha}(\hat{\theta})).
\label{eq:estim2}
\end{eqnarray}
Because the minimum of  $ \hat{\alpha}(\theta)$ is (arbitrarily) picked, this procedure necessarily implies $\hat{\alpha}^* \leq 1$, there is thus a remaining indetermination whether the correct choice is $\hat{\alpha}^*$ or $2- \hat{\alpha}^*$ and therefore of $\pi/2$ in $\theta_0$.
As previously mentioned, this only amounts to exchanging the roles of the axis $x$ and $y$.
{For isotropic fields, $\hat \alpha(\theta)$ fluctuates around $\alpha=1$,  and no clear minimum (or maximum) is visible.
Furthermore,  $\hat \theta_{max}- \hat \theta_{min}$ differs from $\pi/2$.
When $\hat \theta_{max}-\hat \theta_{min}<\pi/4$, the field is thus declared isotropic and we set $\hat \theta =0$. }

To practically perform the rotation of $\theta$ on the image of analysis, a nearest neighbor interpolation is applied.
The procedure is totally automated and no human supervision is needed.

\subsubsection{Illustrations and performance}

To assess the performance of the joint three-parameter estimate procedure, Monte-Carlo numerical simulations are conducted, and biases and standard deviations are computed from average over 100 realizations of {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$, for various choices of  $(\theta_0,\alpha_0,H_0)$, and with $q=2$.

Fig.~\ref{fig:results} shows, top row, that the estimation $\hat{\alpha}(\theta)$ clearly follows a piecewise linear variation along $\theta$ (modeled by the dashed line) and displays clear extrema for ${\theta} \approx \theta_0$ and ${\theta} \approx \theta_0+\pi/2$.
For  ${\theta} \approx \theta_0$,  $\hat{\alpha}(\hat{\theta})$ and $\hat{H}(\hat{\theta})$ (bottom line of Fig. \ref{fig:results}) provide satisfactory estimates of $\alpha_0$ and $H_0$.
For $\theta=\hat{\theta}+\pi/2$, the estimations are $2-\hat{\alpha}$ and $\hat H$.
{Table \ref{table:test} displays the biases, standard deviations and Mean square errors for several isotropic and anisotropic  {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$ fields.}
It can be observed that $\hat H$ shows more bias when a rotation of the original image is performed.
This is likely due to the interpolation procedure that smoothes out data and thus that \emph{distorts} self-similarity and thus scale invariance at the finest scales.
Better estimations for $H$ can be achieved by discarding a few of the finest scales from the linear regression, when image size permits.

\begin{figure}[t]
\begin{minipage}[t]{\linewidth}
\includegraphics[width=\linewidth]{figMCBS.pdf}
\end{minipage}
\vspace{-.6cm}
\caption{\label{fig:BS} {\bf Bootstrap versus Monte Carlo Estimates of standard deviations.}  (a) Estimations of
$\displaystyle {\mathbb E} \sigma_{\mbox{\tiny BS}}(\hat H)/\sigma_{\mbox{\tiny MC}}(\hat H)$
as a function of $H_0$,
 for $q=2$, obtained from $R=100$ bootstraps applied to $100$ independent copies of {\bf OSGRF} $X_{0,\alpha_0,H_0}$ (image size $(2^{10},2^{10})$), with parameters $\alpha_0=1$
 ($\circ$), $\alpha_0=0.8$ ($\ast$)
and $\alpha_0=0.6$ ({\large $\displaystyle \cdot$}).
 (b) Estimations of  $\displaystyle {\mathbb E} \sigma_{\mbox{\tiny BS}}(\hat \alpha)/\sigma_{\mbox{\tiny MC}}(\hat \alpha)$ as a function of $H_0$,
for $\alpha_0=1$ ($\circ$), $\alpha_0=0.8$ ($\ast$)
and $\alpha_0=0.6$ ({\large $\displaystyle \cdot$}).
}
\end{figure}

\section{Bootstrap-based anisotropy test and confidence intervals}
\label{sec:test}

In applications, it is often of crucial importance to be able to test the isotropy assumption (i.e., whether $\alpha_0 = 1$ or not) for each single image independently.
This theoretically requires the knowledge of the distribution of  $\hat \alpha$.
Though it is found empirically Gaussian, the variance of the distribution remains unknown and, as suggested in Section~\ref{sec:esta} and Fig.~\ref{fig:perf}, it depends not only on the sample size $N$ but also on the unknown parameter $ \alpha_0$ itself.
Asymptotic Gaussian expansions for the calculations of the theoretical variance of  $\hat \alpha$, in the spirit of those proposed for fractional Brownian motion in e.g., \cite{va99}, have been observed to perform poorly (not reported here).
Instead, it is proposed to apply non parametric bootstrap procedure in the HWT coefficient domain, in the spirit of the procedures developed and assessed in \cite{WENDT:2007:E,W07,W08,W09}.
This procedure is detailed in the next section while the corresponding bootstrapped based isotropy test is defined and assessed in Section~\ref{sec:BSTest}.

\subsection{Bootstrap resampling schemes}
\label{sec:bs1}
In a nutshell, nonparametric bootstrap makes use of available samples, many times, by a drawing with replacement procedure, to yield an approximation of the  unknown  population distribution.
In turn, this estimated population distribution is used to construct confidence intervals or test (cf., e.g., \cite{Efron82} and \cite{Hall92}).

For the present work, following \cite{WENDT:2007:E}, the resampling procedure is applied in the HWT coefficient domain.
Because HWT coefficients do not consist of independent random variables, but possess a residual correlation, a time-block bootstrap procedure is used:
At each octave $\underline{j}$, block of size $l$ of HWT coefficients are drawn randomly with replacement.
This yields a set of bootstrapped HWT coefficients $d^{\ast}_X(\underline{j},\underline{k})$, from which bootstrap estimates $\hat \alpha^{\ast}$ and $\hat H^{\ast}$ of $\alpha_0$ and $H_0$, respectively, are obtained.

This procedure is repeated $R$ times, and the population distribution of $\hat \alpha$ and $\hat H$ are inferred from the boostrap estimates $\hat \alpha^{\ast,r}$ and $\hat H^{\ast,r}$, $r=1, \ldots, R$, notably variances can be estimated.

\subsection{Bootstrap-based estimates of variance}

It has been found empirically that $l$ need not depend on octave $\underline{j}$ and can be kept small.
As documented in \cite{WENDT:2007:E},  $l$ is set to twice the size of the support of the mother wavelet (e.g., for a Daubechies3 wavelet used here $l=6$), as correlations amongst HWT coefficients is found to remain significant essentially over a space-scale controlled by the size of the wavelet support.

Fig.~\ref{fig:BS} compares the standard deviations of $\hat H$ (left) and $\hat \alpha $ obtained from 100 Monte Carlo simulations for anisotropic fields (of size $2^{10}\times 2^{10}$) against those obtained by  the bootstrap procedure (with $R=100$ for each of the $100$ Monte Carlo simulations).
Fig.~\ref{fig:BS} shows that the ratios $\displaystyle \sigma_{\mbox{\tiny BS}}(\hat H)/\sigma_{\mbox{\tiny MC}}(\hat H)$ and $\displaystyle \sigma_{\mbox{\tiny BS}}(\hat \alpha)/\sigma_{\mbox{\tiny MC}}(\hat \alpha)$ depend neither on  $\alpha_0$ nor on $H_0$ and remain close to $1$, with a slight overestimation  (from $10$ to $20 \%$) for the former and quasi perfect match for the latter.
Equivalent conclusions are drawn from different sample sizes $N$.
These results indicate that the bootstrap estimates of the variances provide valuable approximations of the true variances of $\hat H$ and $\hat \alpha$.
Together with the Gaussian distribution empirical fact, this yields very satisfactory confidence intervals for $\hat H$ and $\hat \alpha$.

\begin{figure}[t]
\begin{minipage}[t]{\linewidth}
\includegraphics[width=\linewidth]{figTest.pdf}
\end{minipage}
\vspace{-.6cm}
\caption{\label{fig:test} {\bf Anisotropy test}. a) Histogram of $\hat\alpha_{\mbox{MC}}$ (light gray) and of $\hat\alpha_{\mbox{BS}}$ (black) for {\bf OSGRF} $X_{0,\alpha_0,H_0}$ (image size $(2^{10},2^{10})$), with $\alpha_0=0.75$ (a) and $\alpha_0=1$ (b).
Right plot (c) shows the rejection level of the test (with a significance level of $90\%$) obtained for $R=100$ bootstraps,  averaged on $100$ realizations of $X_{0,\alpha_0,H_0}$,  with $H_0=0.3$ ($\circ$), $0.5$ ($\vartriangle$) and $0.7$ ($\triangledown$).
}
\end{figure}

\subsection{Test procedure and performance}
\label{sec:BSTest}

\subsubsection{Test procedure}

To test isotropy in a given image, the null and alternative hypothesis respectively read:
\begin{equation}
{\cal H}_0 : \alpha_0-1=0, \; \mbox{ and } {\cal H}_A : \alpha_0-1 \neq 0.
\end{equation}

Let us assume first that $\theta_0 \equiv 0$.
The test procedure can be decomposed as follows:
\begin{itemize}
\item[-] Estimate $\hat \alpha$, as proposed in Section~\ref{sec:esta}.
\item[-] Apply the resampling scheme described in Section~\ref{sec:bs1} above to the HWT coefficients of $X_{\theta_0, \alpha_0, H_0} $ and construct the bootstrap distribution estimate of $\hat \alpha$ from the boostrap estimates $\hat \alpha^{\ast,r}, r= 1,\ldots R$.
\item[-] Set the test significance level $\delta $ for the test.
\item[-] Because when $\theta_0 \equiv 0$, there is no reason to decide a priori that the true $\alpha_0$ will depart from $1$ by being larger or smaller, a bilateral symmetric test is constructed.
Assuming a normal distribution for $\hat \alpha$, the bootstrap-based standard deviation
estimation $\sigma^*$  is used to construct the equi-tailed and symmetric acceptance region $ [-t_{\delta/2}\sigma^*,t_{\delta/2}\sigma^*]$, where $t_{\delta/2}$ denotes the $\delta/2$-th quantile of the zero-mean unit variance Gaussian distribution.
\item[-] Alternatively, the $p$-value of the test can be measured as the minimum between $P(\hat \alpha^{\ast}< \hat \alpha(\hat \theta)) $ and $1 - P(\hat \alpha^{\ast}>\hat \alpha(\hat \theta)) $, divided by $2$.
\end{itemize}

\subsubsection{Test performance}

To assess the validity and performance of the proposed test, it has been compared against Monte Carlo simulations,
based on $100$ independent copies of {\bf OSGRF} $X_{\theta_0=0,\alpha_0,H_0}$ with various parameter settings and for image size  $2^{10}\times 2^{10}$.
Fig.~\ref{fig:test}a) and ~\ref{fig:test}b)  compare the histograms of the estimates of $\alpha_0$ stemming from Monte Carlo simulations against those obtained from bootstrap estimates $\hat \alpha^*$, from a single realization, chosen arbitrarily, for anisotropic (a) and isotropic (b) fields.
For both cases, distributions are found to be in satisfactory agreement.
These figures also show that  $\hat \alpha$ can only take discretized values, because of the finite sample size of the image, as discussed in Section \ref{sec:estillust}.

In Fig.~\ref{fig:test}c) , the significance level of the test has been arbitrarily set to $\delta =0.9$ and the rejection level of the bootstrap test ($R=100$) has been computed as average over  $100$ independant Monte Carlo realizations of {\bf OSGRF} $X_{\theta_0=0,\alpha_0,H_0}$ for various parameter settings.
When  $\alpha_0=1$, {\bf OSGRF} is isotropic and the rejection level $\beta$ is, as expected, found to satisfactorily reproduce the prescribed significance level $1-\delta =0.1$:
 $\hat\beta =0.13$, $0.15$ and $0.12$ respectively for $H_0=0.7, 0.5$ and $0.3$.
When $\alpha_0 \neq 1$, {\bf OSGRF} is anisotropic and the rejection level $\beta$ measures the power of the test.
Interestingly, it is found that the estimated power does not depend on  $H_0$, is symmetric for $\alpha_0$ above and below $1$ and mostly that it increases sharply when $\alpha_0$ departs from $1$.
This is thus indicating a strong potential to detect anisotropy even for small departure of $\alpha_0$ from $1$.
\begin{table}
\begin{center}
\begin{tabular}{||c|c|c|c|c||} \hline\hline
\multirow{2}*{$(\theta_0,\alpha_0,H_0)$} & $\langle \hat{\theta}\rangle-\theta_0$ & $\langle\hat{\alpha}\rangle-\alpha_0$ & $\langle\hat{H}\rangle-H_0$  &  \multirow{2}*{\% rej.} \\
 & (std,MSE) &  (std,MSE) &  (std,MSE) &  \\ \hline \hline
\multirow{2}*{$(\pi/3,0.7,0.6)$} & 0.01  & 0.00 & -0.10 & \multirow{2}*{100} \\
  &(0.03,0.00) & (0.04,0.00) & (0.02,0.01) & \\  \hline
\multirow{2}*{$(\pi/3,0.7,0.2)$} & 0.01  & 0.01 & -0.05 & \multirow{2}*{100} \\
  &(0.03,0.00) & (0.04,0.00) & (0.02,0.00) & \\  \hline
\multirow{2}*{$(\pi/3,0.3,0.2)$} & -0.01  & 0.01 & -0.09 & \multirow{2}*{100}\\
  &(0.02,0.00) & (0.05,0.00) & (0.09,0.02) & \\  \hline
\multirow{2}*{$(0,1,0.6)$} & 0.07  & 0.00 & -0.01 & \cellcolor[gray]{0.8}  \\
  &(0.31,0.01) & (0.08,0.01) & (0.03,0.00) & \cellcolor[gray]{0.8} \multirow{-2}*{8}\\  \hline
\multirow{2}*{$(0,1,0.2)$} & 0.08  & -0.01 & -0.01 & \cellcolor[gray]{0.8}\\
  &(0.25,0.01) & (0.08,0.01) & (0.03,0.00) & \cellcolor[gray]{0.8} \multirow{-2}*{11}\\  \hline \hline
\end{tabular}
\caption{\label{table:test} Biases, standard deviations and mean square errors obtained from $100$ independent copies of {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$ (image size $(2^{10},2^{10})$). The right column reports the corresponding  rejection rate of the anisotropy test described in Section~\ref{sec:testtetha0not0}, with $R=100$ bootstrap surrogates. The significance level is set to $1- \delta = 10\%$.}
\end{center}
\end{table}

\subsubsection{Test procedure for  $\theta_0 \neq 0$}
\label{sec:testtetha0not0}

When $\theta_0$ is unknown and needs to be estimated, the procedure to test isotropy must be slightly amended, as follows:
\begin{itemize}
\item[-] Apply estimation procedure for $\theta_0, \alpha_0, H_0$ as in Section~\ref{sec:estb}.
\item[-] For $\hat \theta$, store the estimate $\hat \alpha(\hat \theta$) and the rotated field $\tilde{X}_{\hat \theta}$.
\item[-] Apply the resampling scheme described in Section~\ref{sec:bs1} above to the HWT coefficients of $\tilde{X}_{\hat \theta}$ and construct the bootstrap distribution estimate of $\hat \alpha$ from the boostrap estimates $\hat \alpha^{\ast,r}, r= 1,\ldots R$.
\item[-] Set the test significance level $\delta $ for the test.
\item[-] Because the estimated $\hat \alpha$ necessarily takes values in  $[0,1]$ a monolateral test must be constructed and the acceptance region is thus defined as:  $ [-t_{\delta}\sigma^*, 1]$.
\item[-] Alternatively, the $p$-value of the test can be computed as $P(\hat \alpha^{\ast}< \hat \alpha(\hat \theta))$.
\end{itemize}
{Table \ref{table:test} (right column) reports the rejection rates of the procedure applied to several anisotropic and isotropic  {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$ fields of size $(2^{10},2^{10})$. For isotropic cases, the rejection rates matches closely the significance level, as expected.
For anisotropic fields, the power of the test is found very high as soon as $\alpha_0$ departs, even slightly, from $1$.}

\section{Other isotropic and anisotopic random fields}
\label{sec:othermodel}

So far, the analysis (estimation and test) procedures proposed here were applied only to the {\bf OSGRF} $X_{\theta_0,\alpha_0,H_0}$, defined in Section~\ref{sec:OSGRF}, and chosen as a convenient reference model, with three parameters accounting jointly for rotation, (an)isotropy and self-similarity.
However, one can naturally wonder whether the isotropy test described above would satisfactorily perform to detect anisotropy for other models, i.e., whether $\hat \alpha =1 $ or not.
In this section, a number of isotropic and non isotropic self-similar models commonly encountered in the image processing and statistics literature are used to test the level of generality of the approach proposed here.

\subsection{Random fields}

\subsubsection{Another OSGRF}

In \cite{br2011}, another interesting instance of {\bf OSGRF} has been explored. It is defined from Eq. (\ref{eq:modela}) with:
\begin{equation}
f(\underline{\xi})=(|\xi_1|^2+|\xi_2|^{2a})^{-\beta},
\label{eq:modelBE3}
\end{equation}
where $\beta=H_1+(1+1/a)/2$ and $a=H_2/H_1$ for $0<H_1<H_2<1$.
This process resembles {\bf OSGRF} $X_{\theta_0,\alpha_0, H_0}$, in Eq. (\ref{eq:model}), with $\alpha_0=2a/(1+a)$, $H_0=2aH_1/(1+a)$ and $\theta_0=0$.
It is thus anisotropic as soon as  $a \neq 1$.

\subsubsection{Extended Fractional Brownian Fields}
Another class of possibly anisotropic Gaussian field,  referred to as \emph{Extended Fractional Brownian Filed}, was first introduced in \cite{BE03}.
Its definition, $ {X_{f}(\underline{x})} =  \int_{\mathbb{R}^2} (e^{i\langle \underline{x}, \, \underline{\xi} \rangle} -1) f(\underline{\xi})^{1/2} d\widehat{W}(\underline{\xi})\, $,  relies on an admissible function $f$ of the form:
\begin{equation}
f(\underline{\xi})=|\underline{\xi}|^{-2h(arg(\underline{\xi}))-2},
\label{eq:modelBE2}
\end{equation}
where $arg(\underline{\xi})$ is the direction of the frequency $\underline{\xi}$ and $h$ an even measurable periodic function taking values in $(0,1)$.
Fractional Brownian field is a particular and isotropic case of {\bf EFBF}, where $h$ is a constant function, but {\bf EFBF}, is in general anisotropic when $h$ is not constant function. Strictly speaking, {\bf EFBF} is not exactly selfsimilar (except in cases where $h$ is  a constant function).
However,  {\bf EFBF} shows scale invariance properties that are empirically close to those of strictly selfsimilar fields.
Fig.~\ref{fig:modeleEFBF}a) shows a sample field of anisotropic {\bf EFBF}, with
\begin{equation}
h(arg(\underline{\xi}))=H_2\times(\cos(2\times arg(\underline{\xi}))+\epsilon)^2/(1+\epsilon)^2,
\end{equation}
where $\epsilon=1+2\sqrt{H_1/(H_2-H_1)}$.
Function $h$ is $\pi$-periodic and takes values in $[H_1, H_2]$.
Fig. \ref{fig:modeleEFBF}c)  shows one sample-field, obtained with  parameters $H_1=0.2$ and $H_2=0.8$.

\begin{table}
\begin{center}
\begin{tabular}{||c||c|c|c||c|c|c||}  \hline\hline
$H_1$  & \multicolumn{3}{c||}{ $0.7$ } & \multicolumn{3}{c||}{ $0.4$ } \\
\cline{1-7}
$H_2$ & $0.5$ & $0.6$ & $0.7$ & $0.2$ & $0.3$ & $0.4$ \\ \hline \hline
OSGRF & 100 & 84 & \cellcolor[gray]{0.8}{15} & 100  & 99 & \cellcolor[gray]{0.8}{7} \\ \hline
EFBF & 42 & 31 & \cellcolor[gray]{0.8}{15} & 49 & 28 & \cellcolor[gray]{0.8}{10} \\ \hline
FBS & 89 & 78 & 58 & 92  & 87 & 61 \\ \hline \hline
\end{tabular}
\caption{\label{table:testa} {\bf Isotropie test: Rejection rates.} Obtained for three different classes of processes (from $R=100$ bootstraps on each of the $100$ $1024\times 1024$ realizations, significance level of $\delta = 90\%$).}
\end{center}
\end{table}

\subsubsection{Fractional Brownian Sheet }

Fractional Brownian Sheet ({\bf FBS}), introduced in~\cite{Kam96}, provides us with another class of (an)isotropic self-similar Gaussian field.
It can be defined through its harmonizable representation, for any $(H_1,H_2)$ in $(0,1)^2$ (see \cite{ALP02}) :
\begin{equation}\label{eq:defDBF}
B_{H_{1},H_{2}}(x) =\int_{\mathbb{R}^{2}}
\frac{(e^{i<x_{1},\xi_{1}>}-1)(e^{i<x_{2},\xi_{2}>}-1)}{|\xi_{1}|^{H_{1}+\frac{1}{2}}|\xi_{2}|^{H_{2}+\frac{1}{2}}}
d\widehat{W}_{\xi_{1},\xi_{2}},
\end{equation}
where $dW_{x_1,x_2}$ is a Brownian measure on $\mathbb{R}^{2}$ and $d\widehat{W}_{\xi_1,\xi_2}$ its Fourier transform.
{\bf FBS} is a Gaussian field with stationary rectangular increments, satisfying the following scaling property $\forall (a_1,a_2)\in(\mathbb{R}_+^*)^2$
\begin{equation}
\label{}
\{B_{H_1,H_2}(a_1 x_1,a_2 x_2)\}\overset{\mathcal{L}}{=}\{a_1^{H_1}a_2^{H_2}B_{H_1,H_2}(x_1,x_2)\}\;.
\end{equation}

\begin{figure}[t]
\begin{minipage}[t]{\linewidth}
\includegraphics[width=\linewidth]{figEFBFter.pdf}
\end{minipage}
\vspace{-.6cm}
\caption{\label{fig:modeleEFBF} {\bf Other (an)isotropic selfsimilar Gaussian fields.} Left column, sample fields with $(H_1,H_2)=(0.2,0.8)$ for {\bf OSGRF} (top), {\bf EFBF} (middle), {\bf FBS} (bottom).  Right column, $\hat H(\alpha)$ obtained for averages over $100$ realizations, with $H_2=0.8$ and $H_1=0.8$ ($\circ$), $0.6$ ($\star$), $0.4$ ($\vartriangle$) and $0.2$ ($\triangledown$).  $\hat H(\alpha)$ clearly shows a maximum for $\alpha \neq 1$ when fields are anisotropic.}
\end{figure}

\subsection{Testing anisotropy}

The estimation and test procedures described above were applied to these three classes of processes, for various setting of $[H_1, H_2]$.
Estimated function $\hat H(\alpha) $, averaged over $100$ realizations (size $2^{10}\times 2^{10}$), are reported in Fig. \ref{fig:modeleEFBF}, right column.
Isotropy rejection rates,  obtained from $R=100$ bootstrap surrogates for each of the $100$ realizations, are reported in Table \ref{table:testa}.

For the $3$ class of processes, when $H_1 \neq H_2$, it is observed that $\hat H(\alpha) $ has a maximum for $\alpha $ that clearly departs from $1$ and simultaneously that the isotropy rejection rates is far larger than the chosen $1 - \delta = 10\%$ significance level of the test.
This is the case even for as small discrepancies between $H_1$ and $H_2$, as $H_2-H_1= 0.2$.
These results clearly show that the proposed procedures clearly \emph{detect} anisotropy.

For {\bf EFBF}, it is reported in \cite{br2011} that the test anisotropy proposed therein failed to detect anisotropy (i.e., test reject in $0 \%$ of cases), when $H_1=0.5$ and $H_2=0.7$.
Trying as careful a comparison as possible, using the same model and parameter setting, it is found that the bootstrap test described in Section~\ref{sec:BSTest}, yields rejection of isotropy, with the $1-\delta = 10\% $ significance level, in $42 \%$ of cases, hence showing a much improved power (cf. Table \ref{table:testa}).

Conversely, for $H_1 = H_2$, it is observed for {\bf EFBF} and {\bf OSGRF} that $\hat H(\alpha) $ has a maximum for $\alpha =1$ and simultaneously that the isotropy rejection rates reproduces the targeted significance level, hence confirming that these processes are isotropic.
For {\bf FBS},  $\hat H(\alpha) $ remains flat for all $\alpha$s, while the rejection rates are higher than the  targeted significance level, this is thus questioning isotropy of {\bf FBS}, even when $H_1 = H_2$, a theoretically opened issue.

\begin{figure}[t]
\begin{minipage}[t]{\linewidth}
\includegraphics[width=\linewidth]{figsinusrot.pdf}
\end{minipage}
\vspace{-.6cm}
\caption{\label{fig:sinus} (a) Sine wave of direction $\theta_0=\pi/8$ with an isotropic field $(H_0=0.5,\alpha_0=1)$; (b) anisotropic self-similar fields with $(H_0=0.5, \alpha_0=0.6,\theta_0=\pi/8)$. $\hat \alpha$ (c) and $\hat H$ (d) versus the angle analysis $\theta$. The symbols ($\circ$) represent the results obtained for the field in (a) and the ($\star$) for the field in (b).}
\end{figure}

\subsection{Anisotropic field with surimposed regular texture}

To finish, let us come back to the original issue disentangling self-similar with a true built-in anisotropy from isotropic selfsimilar processes to which an unrelated anisotropic texture is additively superimposed.
To address this issue, let us compare a truly isotropic {\bf OSGRF} $X_{\theta_0=0, \alpha_0=1, H_0=0.5}$, as defined in Eq. (\ref{eq:model}), to which a sine waveform trend,  with orientation $\theta_0=\pi/8$ is additively superimposed (Fig.~\ref{fig:sinus}a) to a truly anisotropic {\bf OSGRF} $X_{\theta_0=\pi/8, \alpha_0=0.6, H_0=0.5}$.
The estimation and test procedures described above are applied to $100$ realizations of both processes, and $\hat \alpha(\theta)$ and $H(\theta)$ are displayed in Fig.~\ref{fig:sinus}c and d, respectively.
For the truly anisotropic field ($\circ$), $\hat \alpha(\theta)$ displays a clear minimum for $\hat \theta= \theta_0$, with estimated anisotropy ($\hat \alpha=\alpha(\hat \theta)=0.66$) and selfsimilarity ($\hat H(\hat \theta)= 0.36$) parameters in close agreement with the true ones ($\circ$ in \ref{fig:sinus}d).
This is thus clearly validating anisotropy.
For the isotropic field, to which the directional sine wave trend has been added, $\alpha(\theta) $ shows no clear minimum and instead a rather constant behavior in $\theta$ is observed, thus leading to conclude that  anisotropy,  clearly visible on the sample field, is superimposed on rather than built-in self-similarity.

This example leads us to conclude that the  procedure  proposed in the present contribution provides practitioners with a reliable tool to analyze self-similarity in presence of anisotropy and enables them to clearly disentangle built-in anisotropy from independent and superimposed added anistropic trends.

\section{Conclusions and perspectives}
\label{sec:discussion}

The present contribution aimed at studying images or fields where self-similarity is potentially tied to anisotropy.
Replacing the standard 2D-DWT with the HWT, thus permitting to use different dilation factors along horizontal  and vertical directions, enabled us first to estimate the rotation and anisotropy parameters.
In turn, this permitted a correct estimation of the self-similarity parameter along the estimated anisotropy direction.
This direction selection would not be permitted by the use of the sole 2D-DWT coefficients and thus constitutes the major benefits of the use of the HWT, and therefore the key feature of the present contribution.

Additionally, bootstrap based procedures, performed in the HWT coefficient domain, furnish confidence intervals for the estimates and an isotropy test, that can be applied to a single image.

Though studied in depth for a specific Gaussian self-similar model, the proposed analysis is shown to enable the detection of  anisotropy for a large variety of classes of  Gaussian self-similar processes.
Also, true built-in anisotropy is clearly discriminated from isotropy to which an anisotropic trend is added. \\

Extensions of the applicability of the present method or further developments geared towards the analysis of more general classes of processes modeling. Textures with scale invariance, that are not necessarily exactly self-similar and that may, weakly or significantly, depart from Gaussian distributions, are under current investigations.
Notably, this study paves the way toward the far more difficult topic of multifractal analysis and formalim in presence of anisotropy, to which future efforts are devoted.

{\sc Matlab} routines, designed by ourselves, implementing field synthesis and parameter estimation and test will be made publicly available at the time of publication.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


Jawad, Mary Anne and I are in–sorta–and will spend the next couple of days getting a bit more organized prior to actually sleeping there. But, as far as the move from storage to the new place last Friday and Saturday, it could not have been better. Cory, Mike and Darrell were a great team–they clearly enjoyed working together and couldn't have been more helpful to us. They knew what they were doing, worked hard and did us a fine job. They were a pleasure to have in our home and we would have them back in a heartbeat! The move to the new place was letter perfect–thanks to you for that– and a special thanks to Cory, Mike and Darrell. They are a real credit to your organization. Not sure if or when there will be another move in our plans but in the meantime, I will not hesitate to recommend 1st Choice Movers to friends– you all earned it.


Upwork is the world's largest freelancing website, making it easier to get work done anytime, anywhere. With talented freelancers offering services in more than 3,500 areas of expertise on the platform, Upwork makes it easy for businesses and independent workers to connect—and team up on projects that range from web development to accounting.
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Luther Strode is an average teenage nerd until he sends away for The Hercules Method, a bodybuilding course that promises to give him the body of his dreams But the powers he unleashes make him a target for a murder cult as old as mankind, cost him almost everything, and take him across the world in the fight to take back his life.The complete series collected for the firLuther Strode is an average teenage nerd until he sends away for The Hercules Method, a bodybuilding course that promises to give him the body of his dreams But the powers he unleashes make him a target for a murder cult as old as mankind, cost him almost everything, and take him across the world in the fight to take back his life.The complete series collected for the first time in one edition.Collects The Strange Talent of Luther Strode issues 1 through 6, The Legend of the Luther Strode issues 1 through 6, and The Legacy of Luther Strode issues 1 through 6.
Guilty pleasure hyper-violence meets B dialog and cheesy romance. It hints of that Captain America and Flex Mentallo theme of "weakling becomes a hero." Some of the cleanest, most brilliant action sequences ever illustrated.
Flat writing, some dynamic hyper stylized art though.
Fun, five star, way-way-way over the top hyper kinetic art, a thin wisp of a two star story.
A bit of the ultraviolence here. There is a story but its cliche and boring and really doesn't matter. Read this for the gore and guts and blood. Not terrible but definitely not my thing.
Read this on Hoopla MUCH BLOOD.


text:
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Revolution #60, September 10, 2006
A Jagged, Unjust, and Obsolete World:
A Critique of Thomas Friedman's The World is Flat
By Raymond Lotta
The World Is Flat (Updated and Expanded Edition)
Farrar, Straus and Giroux, 2006.
IMPERIAL TUNNEL VISION AND THE REAL WORLD
Thomas Friedman's The World Is Flat has been on The New York Times bestseller list for over a year now. The paperback version will be out soon. The World Is Flat is an undisguised celebration of globalization—chock full of conversations with CEOs like Michael Dell and Bill Gates and politicians like the former president of Mexico, and brimming over with dumbstruck accounts of info-tech gadgetry and tributes to the cutting-edge (read: cut-throat) business models of companies like Wal-Mart.
This is imperialist globalization as seen at the top. This is imperialist globalization from the perspective of those controlling and benefiting at the top.
The World Is Flat has gotten enormous play in the U.S. media and has become the stuff of National Public Radio specials and business school curriculums. But the book is more than clever cheerleading for corporate capitalism in the 21st century. The World Is Flat has an ideological subtext. Friedman's take-home message is that this "flat world" is the best and only of all possible worlds…and the world that must be sustained and defended.
The Contours of the World
The incredible hype surrounding this book is understandable given the scale and temerity of Thomas Friedman's ambition.
We live in a world of enormous inequalities: the rich countries have 20 percent of the world's people but 80 percent of its gross domestic product (or income), while the 20 percent living in the poorest countries have 3 percent of world income. This is a world that is stalked by disease, malnutrition, and life-destroying poverty: some 2.6 billion people, about 40 percent of the world's population, live on less than $2 a day, and 850 million suffer from hunger and malnutrition.1 This is a world in which thousands of lives can be mangled by a single computer keystroke taken in electronic trading rooms, as globe-straddling investors shift capital from one profit-opportunity to another.
If we had the capacity to look at planet Earth from above with the aid of an economic and social telescope, we would find a landscape marked by staggeringly high peaks of controlling wealth and power, by vast valleys of exploited and impoverished humanity, by a deep fault-line between rich oppressor and poor oppressed nations, and by cruel geographies of uneven and unequal development. We would see rivers of blood and bone-littered canyons carved out by colonial conquests and annexations, imperialist interventions, two inter-imperialist world wars, and imperialist-sponsored proxy wars in places like Africa.
But seated in the business class of his analytical jetliner, Thomas Friedman does not see these inequalities and horrors. He gazes out at this same world and declares, like a conjuror, that it is flat.
What Is the "Flat World"?
Now when Friedman says the world is flat, what he means is that the world economy is fast becoming a "level playing field" in which opportunities are converging. Friedman's book is basically a theorization of how this "level playing field" came about, what its implications are, and how it might further develop.
Friedman points to several related factors that have produced his "flat world." He discusses "networked" production and trade. Companies do less and less "in-house" and on their home shores. They carry on "offshore production" in affiliates and subsidiaries and outsource to subcontractors in many different countries around the world. Friedman alleges that this global dispersion and scattering, this creation and management of "global supply chains," is bringing down the "vertical" walls (like protectionist policies) of national-states and indeed undermining the dominance of national states.
Another factor he cites: the collapse of the Soviet bloc and the capitalist restructuring of China (along with the further "opening" of India's economy and other parts of the Third World) have brought huge populations into a single global marketplace.
Most importantly in Friedman's account: new "technological platforms" associated with computer, Internet, search engine, and wireless technology now allow entrepreneurs and would-be entrepreneurs to enter the global market from anywhere in the world.
According to Friedman, this new globalized economics is "flattening" the world, connecting people to one another in entirely new ways. Central to Friedman's whole account is that this process of global flattening, as it further develops, will be increasingly driven by individuals. This is a world, as Friedman presents it, of interconnected individuals—collaborating and competing individuals. Those who think big and jump at opportunity will, he says, become winners.
What Is Wrong With This Picture?
There is so much that is wrong, so much that is missing, from this picture.
1) In Friedman's "flat-world," the powerful, monopolistic transnational corporations and banks that dominate world trade and investment have somehow receded into the shadows and unbundled themselves. In fact, the 300 largest capitalist firms in the world control about 25 percent of the world's productive assets. Of the 500 largest corporations in the world, 95 percent are based in the U.S., Japan, and Europe.2
Monopoly Capital and Class Domination
What Friedman obscures is that in almost every sector of the world economy, huge agglomerations of capital, in alliance and principally in competition with each other, are shaping and dominating new structures of production and management. They are driving the "outsourcing" and "networking," the ways in which "work flows" are organized, and the sophisticated logistics systems that Friedman sees as "empowering" of individuals. It is these firms that are the "system integrators," playing suppliers off against each other and squeezing labor costs to the minimum all the way down their "supply chains" as they seek to expand and penetrate markets.
With his template of the (entrepreneurial) individual, Friedman blots out the hierarchies of capitalist class domination and class exploitation. It's truly amazing. When Friedman talks about the Dell "supply chain," he takes the reader on a breathless ride through several countries, most in the Third World, as an order is filled and a computer put together—but, strangely, there are no workers, there is no labor, and there is no exploitation.
2) In the "flat world," there are no imperialist and neocolonial states. Products, services, finance, and ideas just seem to flow effortlessly in Friedman's increasingly "frictionless" world. Power relations seem to vanish.
The Imperialist State
In fact, the imperialist nation-state functions as the main support for imperialist capital— which operates on an international level but which remains rooted in distinct national markets. This state support takes the form of subsidies, rescue operations, fiscal and monetary policies, etc., and the guarantee of "law and order." The imperialist state protects and enforces the global framework within which the process of capitalist accumulation takes place. The state does this through political and diplomatic means (negotiating trade agreements, for instance), by deploying a large arsenal of economic weaponry involving state agencies like the Treasury Department and the Agency for International Development, and through military basing, military pressures, and military interventions.
In the imperialist era, accumulation-which is the production of profit and more profit on the basis of the exploitation of wage labor—proceeds through competition and rivalry: among corporations, banks, financial groups, etc., and, most sharply, among national imperialist states. And in response to strategic necessity and opportunity, imperial powers have at certain historical junctures set out to forcibly recast that global framework.
The United States is certainly the major economic power in the world. But it is not only that. It is a military colossus. It spends some $400 billion a year on military expenditures. It has 700 military bases girdling the globe. It has carried out countless interventions and wars of suppression (such as in Vietnam).
Here it must be said that when Friedman talks about the transformative power of new information-communications technologies, how the global market harnesses all of this, and what he calls the "secret sauce" of America's open capital markets and flexible labor markets, he is being disingenuous with his readers. Back in 1999, he declared: "The hidden hand of the market will never work without a hidden fist. McDonald's cannot flourish without McDonnell Douglas, the designer of the F-15. And the hidden fist that keeps the world safe for Silicon Valley's technologies is called the United States Army, Air Force, Navy, and Marine Corps."3 This is one of the only true statements Thomas Friedman has ever made. Now he chooses to hide that hidden fist from conceptual view.
Imperialism and State Structures in the Third World
Friedman visits Bangalore, the high-tech center in India, where IBM is heavily invested. How does U.S. capital arrive there (or in China for that matter)? And how is it able to operate in an environment favorable to its development? Through the mere workings of an "open global market"? No, U.S. capital passes through the Indian state, which is a dependent client state serving imperialism (although the Indian ruling class has its own regional ambitions). This state guarantees certain wage and labor standards and terms of ownership for foreign capital, as well as the ability of corporations like IBM to repatriate profits back to the home country.
In China, which Friedman regards as a success story of a "flattening world," the Chinese state has created "export-processing zones" that enable foreign capital to super-exploit Chinese labor—where teenage girls are living in barracks-like conditions and which have gained notoriety as environmental disaster zones in the making.
Throughout the Third World, imperialism structures and props up the neocolonial state: through economic aid, military treaties and assistance, training of technical and administrative personnel, and many other mechanisms. Friedman complains of the state of development and the masses' hatred towards the West in the Arab world. But he seems to forget that the U.S. government has propped up brutal and oppressive regimes in the Middle East (from the Shah of Iran to the Saudi royal family) to serve its geopolitical interests.
3) In Friedman's "flat world," imperialist-led institutions like the International Monetary Fund and World Bank play a minor role. Friedman wants us to believe that "deregulation" and opening of domestic markets to foreign capital and goods, "privatization" of economic activities, and the adoption of "intellectual property rights" standards are all part of an inexorable and natural process of globalizing economics.
The World Bank and IMF as Instruments of Market Restructuring
In fact, the world that Friedman is hailing would not exist were the World Bank and International Monetary Fund (IMF) not engaged in the brutal restructuring of Third World economies. By the late 1990s, some 90 Third World countries were subjected to IMF "structural adjustment" programs. Many of these countries had to cut subsidies to the poor for basic necessities like tortillas, cooking oil, and health care as conditions for stretching out repayments of loans and getting new loans.
This very real flattening of social protections has been carried out in the name of "market-efficiency" and is designed precisely to make these economies more hospitable to imperialist investment capital.
India: An Example of What?
Friedman's journey to India is perhaps his masterstroke of imperial tunnel vision. He sings praises to information-technology companies in Bangalore and the MP3-playing programmers who can speak American slang. He would have us believe that this is India's wave of the future. But what is the larger reality?
Only a miniscule fraction—1.3 million out of India's working population of 400 million—is employed in the information and business processing industries. This sector is oriented towards the world market and to the U.S. market in particular. It is a kind of enclave that has limited technological linkages to and employment effects on the rest of India's economy. And at the same time that this information-knowledge sector has prospered (and, yes, some people are doing very well), facilities for primary education have collapsed in large parts of India—as the Indian state has pressed forward with its program of "market reform."4
Two-thirds of India's population lives in the countryside. But Friedman has little to say about the plight of tens of millions of India's farmers and peasants facing the pressures of the imperialist world market and the penetration of Western agro-business. This translates as low crop prices, heavy debts, and the withdrawal of government supports for small farmers. The situation has grown dire. Between 1993 and 2003, 100,000 farmers committed suicide.5 And during India's "globalizing decade" of the 1990s, food-grain available per Indian fell almost every year.
A Dangerous Rationalization
The World Is Flat is being widely looked to as a comprehensive, highly readable, up-to-the-minute account of seismic economic trends. For this reason alone it would be important to summarize and answer some of its key theses and mystifications. But there is more at stake.
Friedman is writing mainly for the educated middle classes in the U.S. Many of these readers are anxious and concerned about what the U.S. is doing in the world, and to the world. And this is what makes the work especially poisonous.
Friedman is offering people a comforting narrative and rationalization: a deregulated global capitalism is good for the world and self-correcting.
Towards the end of this sprawling 593-page book, Friedman tersely acknowledges that there is still great poverty in the world. But to people concerned about deprivation, mistreatment of workers, and environmental degradation, Friedman shouts a loud "worry not."
He advertises efforts by Hewlett-Packard to "lead the way" in establishing labor standards in its supply chains. He heaps praise on McDonalds for its efforts to establish environmentally friendly standards for its food suppliers.6 He assures people that open and competitive markets and the initiatives of Bill Gates will solve the problems of world poverty.
Friedman's "Dell Theory of Conflict Prevention"
But there is more. Friedman offers the balm that the world's economic interconnectedness will create such wealth and common stakes that violent conflicts will disappear. He calls this the "Dell Theory of Conflict Prevention": no two countries will go to war if they are both part of the same globalized supply chain. These supply chains, according to Friedman, will make all nations' economies so locked into and dependent on each other, that they will have no reason to go to war.
Here too, there is so much that is wrong. A supply chain is not a collegial coming together of people in order to share information, technology, and wealth for bettering the world. These chains are built to extract wealth. They are forged and re-forged only insofar as they serve the interests of fiercely competing capitals. Further, these global supply chains, though vast and interconnected, have not merged into a single unit of capital. These chains are part of large, competing agglomerations of capital. They are wielded as weapons in the battle for market share and market control.
And there is more than networked production in the world. Supply chains exist in a geopolitical context. There are contending geoeconomic interests. There are struggles for spheres of geopolitical influence. There is struggle among imperialist states for control over strategic resources. This is part of what is being played out in the Middle East, as the U.S. seeks to establish uncontested hegemony over the region and its oil resources—on which the European and Japanese economies are highly dependent. There is intense jockeying taking place among the major imperialists and regional powers over the Caspian Sea region, home to huge energy reserves.
Cross-country investments and alliances do not obliterate the distinct and contending interests of different national imperialist formations. Just prior to the outbreak of World War 1, there were bourgeois and revisionist Marxist thinkers who were arguing that trade between Germany and England had become too important for the two countries to go to war. (Some years back, Friedman had peddled a similar McDonald's theory of conflict prevention; as Jeff Faux in a review has pointed out, the war in Yugoslavia, with its many McDonald's outlets, landed that theory in the dustbin of history.)
What Friedman wants readers to believe is that, left to its own workings and logic, the market imperatives of globalization will lead to progress, wealth, and stability.
An Unacceptable Choice
But with the optimism also comes grim news. Friedman points to forces out there who oppose what he considers to be the life-enhancing, democratizing, and empowering tide of globalization. He lashes out at the progressive anti-globalization movement, which in the late 1990s and early 2000s galvanized tens of thousands and inspired tens of millions around the world as it exposed and opposed the injustices of imperialist globalization.
Friedman discusses other obstacles to unbridled globalization. But right now, he says, the most dangerous enemy to all the promise held out by his mythic globalization is aggressive Islamic fundamentalism.
He analyzes that these Islamic fundamentalist forces (he calls them "Islamo-Leninists" in order to equate this utterly reactionary movement with communism) are resentful of the dynamism and modernity of globalization, able to marshal the passive support of humiliated populations, and have shown themselves adept and unflinching at using new technologies for evil.
What to do? Friedman declares that the U.S. has to lead the way with its "can-do optimism" and with its strength. There is no world government; so, and this is Friedman's trump card, the U.S. has to play a special role in preserving the "rules of commerce" and the "norms of behavior." In short, to U.S. imperialism falls a special superpower responsibility.
Up until May of this year, Thomas Friedman was an ardent advocate of another kind of flattening that is taking place in the world: the Bush regime's bombing, invasion, and occupation of Iraq. Lately, he seems to have had some reservations about the conduct (not aims) of the U.S. war on Iraq. But in its own perverse way, and against the backdrop of the Bush regime's "war on the world," The World Is Flat concentrates a momentous question of our times.
Friedman would have us believe that humanity's only choice is between imperialist globalization with its advanced technology controlled by and for the benefit of the few, with its sweatshops; with its juggernaut to impose "intellectual property rights" on everything from critical medicines to living organisms; and with its wars…it is either imperialist globalization, or religious fundamentalism, with all that is retrograde and oppressive about that.
Is this the only choice? No, a world without exploitation and oppression, and without ignorance and superstition, a world in which humanity is truly emancipated, is necessary and possible. What stands in the way is the system of world capitalism.
1. Data from United Nations, Human Development Report, 2005. back
2. Data from Medard Gabel and Henry Bruner, Global Inc.: An Atlas of the Multinational Corporation (New York: New Press, 2003). back
3. Thomas Friedman, The New York Times Magazine, March 28, 1999. back
4. Pankjaj Mishra, "The Myth of the New India," The New York Times, July 6, 2006. back
5. Saritha Rai, "India to Import Wheat After a 6-Year Hiatus," The New York Times, June 30, 2006. back
6. Friedman lauds McDonald's partnership with Conservation International (p. 381). For an exposure of this dubious and business-oriented "environmental group," see Aziz Choudry, "Conservation International: Privatizing Nature, Plundering Biodiversity," [www.grain.org/seedling]. back
Next—Part 2: What Globalization Is and Isn't, and Why World Development Is Not Converging
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mild lemon aroma, are used to make medicine. Lemon balm is used alone or as part of various multi-herb combination products. Lemon balm is a perennial herb from the mint family. The leaves, which have a mild lemon aroma, are used to make medical drugs.
OTHER NAME(S): Balm, Bálsamo de Limón, Cure-All, Dropsy Plant, Honey Plant, Melisa, Melissa, Melissa officinalis, Melissae Folium, Mélisse, Mélisse Citronnelle, Mélisse Officinale, Melissenblatt, Monarde, Sweet Balm, Sweet Mary, Toronjil.


On April 28, the V3 Legion will get to witness what upper level wrestling is all about when Mark Martin enters the cage for his professional debut.
Although only having two amateur fights to his name (2-0), combat sports are nothing new to Martin.
His resume includes being a three time state wrestling finalist in Ohio, a two time Ohio state champion, three time NCAA division one national qualifier, FILA Junior National Champion and he was also a starter on Ohio State University's wrestling team that won the national championship in 2015. At Ohio State he trained with several All-American wrestlers including Kenny Courts, Nathan Tomasello and Olympic Gold Medalist, Kyle Snyder.
"Kyle is a special dude and you can see that in his matches, but you really notice it when you see how he lives on a daily basis" states Martin.
Growing up, Martin watched a lot of fighting and also got into several fights, so it was only right to be doing this now. In high school, he attended the prestigious national powerhouse in wrestling, St. Eds, which is also the school that UFC fighter Gray Maynard attended. Maynard would come back and attend some of their tournaments and practices so Martin was exposed to MMA at one of the highest levels.
A week after graduating from Ohio State, he started training at Strongstyle MMA in Cleveland, OH alongside several UFC fighters including heavyweight champion, Stipe Miocic.
"That was sweet because the goal is to be the best and who better to follow than the heavyweight champ" states Martin.
After a few months of thinking back and forth, Martin decided to move to Phoenix, AZ to train full time at the MMA Lab. He felt that that was the best decision for him and his career and now trains with the likes of Benson Henderson and several other high level UFC and Bellator MMA fighters.
"I've been lucky enough to meet and train with tons of world class guys in many different sports, so I know what it takes to be the best" states Martin.
Mark Martin has the potential to do great things in this sport, so get your tickets now and witness the night where it all started!
Tickets can be purchased at www.v3fights.com or at www.cagetix.com.


Faith cometh by hearing and hearing by the word of God. Sorry you were not able to come and listen in person, so here is the next best thing. Listen to the service held at Life Mennonite Fellowship on May 3, 2015 featuring the message "Dedicated Unto The Lord" by Doug Fuhrman.


The Fond du Lac School District Acuity Fund supports new and innovative programs and projects proposed by staff members for their students.
FOND DU LAC - The Fond du Lac School District Board of Education reviewed and authorized the annual Fond du Lac School District Acuity Fund grant recipients.
The Fond du Lac School District Acuity Fund supports new and innovative programs and projects proposed by staff members for their students. The District Acuity Fund became possible via money earned on funds paid to the District by Acuity for naming rights of the Fond du Lac High School Fieldhouse.
For the current school year, a total of $27,460.62 was available; $25,983 via the fund, which is held for the District by the Fond du Lac Area Foundation, and $1,477.62 that was donated by Board member Linda Uselmann.
For more information, call Shawn Bauer, secretary to the superintendent of schools, at 920-906-6502.
Information submitted by the Fond du Lac School District.


text:
Q: Blazor Webassembly with Google Oauth Email confirmation fails with InvalidOperationException: No page named '/Index' I created a new project from .NET 7 templates: Blazor Webassembly app with Individual Accounts - ASP.NET Core hosted and added Microsoft.AspnetCore.Authentication.Google and a Google authentication connection with this:
builder.Services.AddAuthentication().AddGoogle(googleOptions =>
{
    googleOptions.ClientId = builder.Configuration["Authentication:Google:ClientId"] ?? throw new InvalidOperationException("Connection string 'Authentication:Google:ClientId' not found.");
    googleOptions.ClientSecret = builder.Configuration["Authentication:Google:ClientSecret"] ?? throw new InvalidOperationException("Connection string 'Authentication:Google:ClientSecret' not found.");
});

Authentication via Google works properly without Email confirmation; just the final self-confirmation click.
After adding Email confirmation via a new EmailSender service :
builder.Services.AddTransient<IEmailSender, EmailSender>();

The Email send works and creates a link starting with https://localhost:7040/Identity/Account/ConfirmEmail?userId=21c0f.....  When the link is clicked, it produces the error
InvalidOperationException: No page named '/Index' matches the supplied values.

Trying to navigate directly to the address https://localhost:7040/Identity/Account/ConfirmEmail produces the same error message.  A trace shows that the page should have been initialized:
Microsoft.AspNetCore.Mvc.Razor.Compilation.DefaultViewCompiler: Debug: Initializing Razor view compiler with compiled view: '/Areas/Identity/Pages/Account/ConfirmEmail.cshtml'.

What else needs to be configured or how can this be debugged?

A: This can diagnosed by scaffolding the identity functions into the Server project.  In this case, the confirmation Email contained an invalid escape sequence in the query string arguments:
&code=

I performed this hack in In   {project}\Areas\Identity\Pages\Account\ConfirmEmail.cshtml.cs :
public async Task<IActionResult> OnGetAsync(string userId, string code)
        {
//>>>> BEGIN ADD
            var queryParameters = HttpContext.Request.Query;

            if (code == null)
            {
                if (queryParameters.ContainsKey("amp;code")) code = queryParameters["amp;code"]; // Handle double-encoding
            }
//<<< END ADD
            if (userId == null || code == null)
            {
                return RedirectToPage("/Index");
            }

redpajama_set_name: RedPajamaStackExchange


text:
Q: Is there a way to detect which css actually I'm using on my pages and throw everything else? I have an huge .css file and I was wondering if there was a way to detect which css I'm actually using on my pages so as to eliminate everything else that I do not use. Is there a website? App?

A: use one of the following
Chromes Audit tab in Developer tools
https://addons.mozilla.org/en-US/firefox/addon/css-usage/
https://unused-css.com/


text:
<?php
/**
 * @package    Zend_Cache
 * @subpackage UnitTests
 */

namespace ZendTest\Cache;
use Zend\Cache;

/**
 * @package    Zend_Cache
 * @subpackage UnitTests
 */
class StaticBackendTest extends TestCommonBackend
{

    protected $_instance;
    protected $_instance2;
    protected $_cache_dir;
    protected $_requestUriOld;
    protected $_innerCache;

    public function __construct($name = null, array $data = array(), $dataName = '')
    {
        parent::__construct('\Zend\Cache\Backend\StaticBackend', $data, $dataName);
    }

    public function setUp($notag = false)
    {
        $this->mkdir();
        $this->_cache_dir = $this->mkdir();
        @mkdir($this->_cache_dir.'/tags');

        $this->_innerCache = Cache\Cache::factory('Core','File',
            array('automatic_serialization'=>true), array('cache_dir'=>$this->_cache_dir.'/tags')
        );
        $this->_instance = new Cache\Backend\StaticBackend(array(
            'public_dir' => $this->_cache_dir,
            'tag_cache' => $this->_innerCache
        ));

        $logger = new \Zend\Log\Logger(new \Zend\Log\Writer\Null());
        $this->_instance->setDirectives(array('logger' => $logger));

        $this->_requestUriOld =
            isset($_SERVER['REQUEST_URI']) ? $_SERVER['REQUEST_URI'] : null;
        $_SERVER['REQUEST_URI'] = '/foo';

        $this->_instance->setDirectives(array('logging' => true));

        $this->_instance->save('bar : data to cache', bin2hex('/bar'), array('tag3', 'tag4'));
        $this->_instance->save('bar2 : data to cache', bin2hex('/bar2'), array('tag3', 'tag1'));
        $this->_instance->save('bar3 : data to cache', bin2hex('/bar3'), array('tag2', 'tag3'));
    }

    public function tearDown()
    {
        parent::tearDown();
        unset($this->_instance);
        $_SERVER['REQUEST_URI'] = $this->_requestUriOld;
        $this->rmdir();
    }

    public function testConstructorCorrectCall()
    {
        $test = new Cache\Backend\StaticBackend(array());
    }

    public function testRemoveCorrectCall()
    {
        $this->assertTrue($this->_instance->remove('/bar'));
        $this->assertFalse($this->_instance->test(bin2hex('/bar')));
        $this->assertFalse($this->_instance->remove('/barbar'));
        $this->assertFalse($this->_instance->test(bin2hex('/barbar')));
    }

    public function testOptionsSetTagCache()
    {
        $test = new Cache\Backend\StaticBackend(array('tag_cache'=>$this->_innerCache));
        $this->assertTrue($test->getInnerCache() instanceof Cache\Frontend);
    }

    public function testSaveCorrectCall()
    {
        $res = $this->_instance->save('data to cache', bin2hex('/foo'), array('tag1', 'tag2'));
        $this->assertTrue($res);
    }

    public function testSaveWithNullLifeTime()
    {
        $this->_instance->setDirectives(array('lifetime' => null));
        $res = $this->_instance->save('data to cache', bin2hex('/foo'), array('tag1', 'tag2'));
        $this->assertTrue($res);
    }

    public function testSaveWithSpecificLifeTime()
    {
        $this->_instance->setDirectives(array('lifetime' => 3600));
        $res = $this->_instance->save('data to cache', bin2hex('/foo'), array('tag1', 'tag2'), 10);
        $this->assertTrue($res);
    }

    public function testSaveWithSpecificExtension()
    {
        $res = $this->_instance->save(serialize(array('data to cache', 'xml')), bin2hex('/foo2'));
        $this->assertTrue($this->_instance->test(bin2hex('/foo2')));
        unlink($this->_instance->getOption('public_dir') . '/foo2.xml');
    }

    public function testSaveWithSpecificExtensionWithTag()
    {
        $res = $this->_instance->save(serialize(array('data to cache', 'xml')), bin2hex('/foo'), array('tag1'));
        $this->assertTrue($this->_instance->test(bin2hex('/foo')));
        unlink($this->_instance->getOption('public_dir') . '/foo.xml');
    }

    public function testRemovalWithSpecificExtension()
    {
        $res = $this->_instance->save(serialize(array('data to cache', 'xml')), bin2hex('/foo3'), array('tag1'));
        $this->assertTrue($this->_instance->test(bin2hex('/foo3')));
        $this->assertTrue($this->_instance->remove('/foo3'));
        $this->assertFalse($this->_instance->test(bin2hex('/foo3')));
    }

    public function testTestWithAnExistingCacheId()
    {
        $res = $this->_instance->test(bin2hex('/bar'));
        if (!$res) {
            $this->fail('test() return false');
        }
        return;
    }

    public function testTestWithANonExistingCacheId()
    {
        $this->assertFalse($this->_instance->test(bin2hex('/barbar')));
    }

    public function testTestWithAnExistingCacheIdAndANullLifeTime()
    {
        $this->_instance->setDirectives(array('lifetime' => null));
        $res = $this->_instance->test(bin2hex('/bar'));
        if (!$res) {
            $this->fail('test() return false');
        }
        return;
    }

    public function testGetWithANonExistingCacheId()
    {
        $this->assertFalse($this->_instance->load(bin2hex('/barbar')));
    }

    public function testGetWithAnExistingCacheId()
    {
        $this->assertEquals('bar : data to cache', $this->_instance->load(bin2hex('/bar')));
    }

    public function testGetWithAnExistingCacheIdAndUTFCharacters()
    {
        $data = '"""""' . "'" . '\n' . 'ééééé';
        $this->_instance->save($data, bin2hex('/foo'));
        $this->assertEquals($data, $this->_instance->load(bin2hex('/foo')));
    }

    public function testCleanModeMatchingTags()
    {
        $this->assertTrue($this->_instance->clean('matchingTag', array('tag3')));
        $this->assertFalse($this->_instance->test(bin2hex('/bar')));
        $this->assertFalse($this->_instance->test(bin2hex('/bar2')));
    }

    public function testCleanModeMatchingTags2()
    {
        $this->assertTrue($this->_instance->clean('matchingTag', array('tag3', 'tag4')));
        $this->assertFalse($this->_instance->test(bin2hex('/bar')));
    }

    public function testCleanModeNotMatchingTags()
    {
        $this->assertTrue($this->_instance->clean('notMatchingTag', array('tag3')));
        $this->assertTrue($this->_instance->test(bin2hex('/bar')));
        $this->assertTrue($this->_instance->test(bin2hex('/bar2')));
    }

    public function testCleanModeNotMatchingTags2()
    {
        $this->assertTrue($this->_instance->clean('notMatchingTag', array('tag4')));
        $this->assertTrue($this->_instance->test(bin2hex('/bar')));
        $this->assertFalse($this->_instance->test(bin2hex('/bar2')));
    }

    public function testCleanModeNotMatchingTags3()
    {
        $this->assertTrue($this->_instance->clean('notMatchingTag', array('tag4', 'tag1')));
        $this->assertTrue($this->_instance->test(bin2hex('/bar')));
        $this->assertTrue($this->_instance->test(bin2hex('/bar2')));
        $this->assertFalse($this->_instance->test(bin2hex('/bar3')));
    }

    public function testCleanModeAll()
    {
        $this->assertTrue($this->_instance->clean('all'));
        $this->assertFalse($this->_instance->test(bin2hex('bar')));
        $this->assertFalse($this->_instance->test(bin2hex('bar2')));
    }

    // Irrelevant Tests (from common tests)

    public function testGetWithAnExpiredCacheId()
    {
        $this->markTestSkipped('Irrelevant Test');
    }

    public function testCleanModeOld()
    {
        $this->markTestSkipped('Irrelevant Test');
    }

    // Helper Methods

    public function mkdir()
    {
        $tmp = $this->getTmpDir();
        @mkdir($tmp);
        return $tmp;
    }

    public function rmdir()
    {
        $tmpDir = $this->getTmpDir(false);
        foreach (glob("$tmpDir*") as $dirname) {
            @rmdir($dirname);
        }
    }

    public function getTmpDir($date = true)
    {
        $suffix = '';
        $tmp = sys_get_temp_dir();
        if ($date) {
            $suffix = date('mdyHis');
        }
        if (is_writeable($tmp)) {
            return $tmp . DIRECTORY_SEPARATOR . 'zend_cache_tmp_dir_' . $suffix;
        } else {
            throw new Exception("no writable tmpdir found");
        }
    }

}

redpajama_set_name: RedPajamaGithub


text:
Home News Election PR Battle, Supreme Court MIA, Economic Update
Greg Hunter's Weekly News Wrap-Up for December 18th, 2020
by Greg Hunter
USA Watchdog
The election battle is as much about public relations (PR) as it is about actual evidence. Massive amounts of evidence have come out about things like voting machines flipping votes, huge drops of ballots in the middle of the night and reports of outright fraud in multiple states. Still, not even the Supreme Court will give these claims a fair hearing. So, the PR and education from the Trump side continues to gain steam ahead of the all-important vote in Congress to make official the Presidential Election. Congress has the last word in who actually won the election. Did Donald J. Trump win in November with a record amount of votes for a sitting President? Or, is Congress going to certify Biden after campaigning out of his basement?
Continue Reading at USAWatchdog.com…


text:
Q: Evaluate $\int _0^1\:\frac{2-x^2}{(1+x)\sqrt{1-x^2}} \, dx$
Evaluate $$\int _0^1\:\frac{2-x^2}{(1+x)\sqrt{1-x^2}}dx$$

$$
\int _0^1\:\frac{2-x^2}{(1+x)\sqrt{1-x^2}} \, dx = \int_0^1 \bigg[\sqrt{\frac{1+x}{1-x}}+\frac{1}{(1-x)^{3/2}\sqrt{1+x}}\bigg] \, dx=\int_0^1[f(x)+f'(x)]\,dx
$$
Does making it into the above final form helps somehow solve the given definite integral ?
Note: The solution given is $\pi/2$

A: Let $x=\sin t$. Then,
$$\int _0^1 \frac{2-x^2}{\left(1+x\right)\sqrt{1-x^2}}dx
=\int _0^1\left(\frac{1}{\left(1+x\right)\sqrt{1-x^2}}
+\frac{\sqrt{1-x^2}}{1+x}\right)dx$$
$$=\int_0^{\pi/2} \left( \frac{dt}{1+\sin t}+1-\sin t\right)dt$$
$$=\left(-\frac{\cos t}{1+\sin t} + t + \cos t\right)_0^{\pi/2}=\frac\pi2$$

A: $$x=:\sin u\implies dx=\cos u\,du\implies\int_0^1\frac{2-x^2}{(1+x)\sqrt{1-x^2}}dx=\int_0^{\pi/2}\frac{2-\sin^2u}{(1+\sin u)}\,du=$$
$$=\int_0^{\pi/2}\left(\frac1{1+\sin u}+1-\sin u\right)du=\int_0^{\pi/2}\frac{1-\sin u}{\cos^2u}+\left.\frac\pi2+\cos u\right|_0^{\pi/2}=$$
$$=\left.\left(\tan u-\frac1{\cos u}\right)\right|_0^{\pi/2}+\frac\pi2-1=\frac\pi2$$
since
$$\lim_{u\to\frac\pi2^-}\left(\tan u-\frac1{\cos u}\right)=\lim_{u\to\frac\pi2^-}\frac{\sin u-1}{\cos u}\stackrel{\text{L'Hospital}}=0$$