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text:
\section{Introduction}
The success of imaging an object with electromagnetic waves is determined as much by the development of new hardware elements
as by the progress in designing efficient algorithms.  These algorithms solve an inverse problem that, most often, is linear. This is the case for example of coherent imaging
when the complex amplitudes of the waves are recorded, or the case of incoherent imaging when only the amplitudes squared, or intensities, are recorded.
In this paper, we present an algorithm for quantitative phase retrieval at terahertz frequencies or above, where it is often difficult to record the phases of the waves directly and, hence, the detectors can only count the number of photons that impinge on them, a measurement that describes intensities. This is a coherent imaging problem, which
is nonlinear in the observed data and, therefore, difficult to solve.

In this context, we consider multiple measurements of a sparse complex-valued object $\vect f$, which in our case is the complex refractive index. The  measurements of this {\em phase object} are of the form
\begin{equation}
\label{eq:measurements_intro}
\displaystyle
|\vect b_i|^2 = \left| {\cal T}(\vect w_i \circ \vect f) \right|^2\, , \,\,i=1,2,\dots,M,
\end{equation}
where $\vect w_i$  is a  mask, and $\circ$ denotes element-wise multiplication.  In (\ref{eq:measurements_intro}), $ {\cal T}$ is a linear transformation which is often the Fourier transform.
Although the available measurements  to recover $\vect f$ are intensities,
the missing phases are still encoded in the recorded intensities, assuming the waves propagate coherently, because there is a fixed phase relationship between waves emerging from different points of $\vect f$. Theoretically, the use of known, or unknown, masks that create enough diversity in the data  has been extensively studied
to address important questions such as the existence and uniqueness of solutions to problem \eqref{eq:measurements_intro} \cite{Balan09,Fannjiang12, Novikov15, Candes15}. In practice, masks are implemented using a spatial light modulator (SLM) or a digital micromirror device (DMD) \cite{Sampsell94,Liu08,Falldorf10, Zheng17}.

Beyond these theoretical questions of the existence and uniqueness of solutions, an efficient method that recovers the phase object $\vect f$
is needed. To invert for the object $\vect f$ from the measurements in (\ref{eq:measurements_intro}), iterative algorithms that serve as approximate inverses are usually employed. Gerchberg and Saxton \cite{GS72}, and later Fienup \cite{Fienup82}, introduced a scheme based on iterative projections which is simple to implement and proved to be very flexible in practice. These methods are fast but, due to the absence of convexity, they do not always converge to the true solution unless good prior information about the sought object is available. In recent years, Chai {\em et al.} \cite{CMP11} and Cand\'es {\em et al.} \cite{Candes13} proposed a different type of algorithm that lifts the  problem (\ref{eq:measurements_intro}) to a higher dimension to make it convex and, thus, guarantees convergence. This type of algorithm solves a low-rank matrix linear system using nuclear norm minimization whose result always converges to the true solution, even without prior information of the object and independently of its complexity. However, when lifting the problem to a higher dimension,  the dimension  increases quadratically in the number of unknowns, and the solution becomes infeasible if the scale of the problem is large.

Instead, we propose to vectorize the matrix linear system to write it as a linear vector equation, and use a {\em Noise Collector} \cite{Moscoso20b} to reduce its dimensionality
dramatically, thus being able to solve large problems as in \cite{Moscoso21}. The reduction of dimensionality is carried out in two steps mimicking the forms in which waves travel. In this way, the algorithm merges coherent and incoherent imaging, as it considers the coherent and incoherent contributions to the data sequentially. The main feature of this algorithm is that it can image transparent objects that are not visible in attenuation- or absorption-based images, as in conventional X-ray images or CT images, because it also produces quantitative phase-contrast images. As a by-product, the proposed algorithm can also be used for phase retrieval with partially coherent observations without any modification and without loss of resolution, which is important for X-ray phase-contrast imaging as fully coherent X-ray sources are hard to produce \cite{Jingshan14}. With the proposed algorithm, only the signal-to-noise ratio (SNR) of the created images is affected when partially coherent observations are used. If the observations are fully incoherent, then the transparent objects cannot be recovered.

In the first step of the proposed algorithm, we  assume that the intensities add incoherently, and treat the coherent contribution to the data as a modeling error which is absorbed by a {\em Noise Collector} \cite{Moscoso20b}. The {\em Noise Collector} is able to absorb this modeling error efficiently, even if it is large, provided the sought object is simple enough to be sparsely represented in a given basis. This step would produce exact reconstructions in, for example, ghost imaging, an absorption-contrast technique
that interrogates the object using a sequence of incoherent illumination patterns \cite{Gibson20}. This is so, because in ghost imaging the phases are completely randomized by a rotating ground glass plate placed right after the object, or by a random medium,
and, hence, the imaging problem is fully incoherent and the resulting linear problem for the recorded intensities is exact.

If, however, the waves are weakly absorbed, absorption-based images lack contrast and, therefore,
a second step  that takes into account the coherent contribution to the data in (\ref{eq:measurements_intro}) is implemented. This step is crucial for phase-contrast imaging that visualizes transparent, or semi-transparent, objects which are otherwise invisible \cite{Zernike55}. Indeed, when a wave is transmitted through an object it is not only absorbed, but also bent inducing phase changes that facilitate the depiction of fine and low absorption-contrast features.
Finally, once the support of both the strong and weakly absorbing objects are known, a straightforward third step that solves the complete problem,
is implemented for precise quantitative phase imaging. This third step uses both the coherent and the incoherent contributions to the measured intensities.

As phase shifts themselves are invisible, many methods have been proposed to make them visible as brightness variations in
the images. Some examples of such methods are  Zernike phase-contrast microscopy \cite{Zernike55}, analyzer-based methods \cite{Davis95a,Davis95b}, and grating-based methods \cite{Pfeiffer06,Pfeiffer07}. All these techniques work well, but they require specialized optical elements that are hard, and some expensive, to manufacture. We, instead, consider another approach, known as
propagation-based  phase-contrast imaging, that does not require lenses, and where phase-contrast appears as the result of the free propagation of the waves, which transform phase variations in the object into detectable intensity variations in the images \cite{Cloetens96,Nugent96,Wilkins96}. This technique appears to be one of the most  relevant and easy to implement for clinical and biomedical research purposes \cite{Lewis05, Gureyev09}. We refer the reader to \cite{Bravin13} for an extensive discussion on
phase-contrast imaging in clinical and biomedical applications, and to \cite{Park18} for recent advances in optical phase imaging for investigating cells and tissues in biomedicine.

Finally, we mention that the proposed algorithm also finds applications in other fields such as exploration seismology in which strong phase errors in the data can make the classical algorithms fail  because they try to match the incorrect phase information, so the measured phases are discarded and the inverse problem is formed for intensities-only  \cite{Gholami14, Aghamiry20}.

The paper is organized as follows. In Section \ref{sec:model}, we introduce the model that produces the measurements for absorption and phase contrast imaging. In Section \ref{sec:algo}, we present the algorithm that produces these images. Section \ref{sec:imple} discusses practical issues for the efficient implementation of the algorithm.
Section \ref{sec:numerics} shows the numerical experiments. Section \ref{sec:conclusions}  summarizes our conclusions.

\section{Model}
\label{sec:model}

When a wave propagates through an object, both its amplitude and phase are altered.
Intuitively, the amplitude of the transmitted wave depends on the absorption of the wave,
while its phase shift depends on the refraction.

Consider a planar object of thickness $l$  illuminated perpendicularly by a monochromatic plane wave of wavenumber $k=2\pi/\lambda$ traveling in the $z$ direction. If the wavelength $\lambda$ is small compared to its
dimension $l$, the interaction of the wave with the object can be described in terms of integrals of the
complex refractive index $n(x,y)$, so its  transmissivity is given by
\begin{equation}
\label{eq:transmissivity0}
t(x',y') =  \displaystyle e^{{i k \int_{l} n(x',y') dz'}}\, .
\end{equation}
Then, the diffracted complex amplitude in the far-field, using the Fraunhofer
approximation, is
\begin{equation}
\label{eq:fourier2d}
b(x,y) = - i \frac{e^{ikL}}{\lambda L}\int t(x',y') e^{i2\pi(xx'+yy')}dx'dy'\, ,
\end{equation}
where $L\gg 1$ is the distance between the object plane and the detector plane. We use prime symbols for the coordinates in the object plane to avoid confusion with the $(x,y)$ coordinates in the detector plane. Mathematically, the complex refractive index in (\ref{eq:transmissivity0}), is expressed as
\begin{equation}
\label{eq:rindex}
n(x',y')=1 - \delta(x',y')+ i\beta(x'y')\,.
\end{equation}
It is the ratio of the wavenumber in the object $\tilde k(x',y')$ and the wave number in the vacuum $k$ and, hence, is a measure of how fast the waves travel through the object.
The real and imaginary components
${\displaystyle  \delta }$ and ${\displaystyle \beta}$ determine the refraction and absorption effects of the interaction wave-matter, respectively.
For a thin planar object, the phase shift and the absorption coefficient are well approximated by
\begin{eqnarray}
\label{eq:phaseshift}
\phi(x',y') &=& k \int_{l}\delta(x',y') dz' \,\quad \mbox{and}\\
\label{eq:absorption}
\mu(x',y') &=& 2 k \int_{l}\beta(x',y') dz' \,,
\end{eqnarray}
respectively. Here, the integration is taken over the extend of the object ${l}$ along the direction of wave propagation $z'$. Both \eqref{eq:phaseshift} and  \eqref{eq:absorption} are proportional
to the density of electrons at each point $(x',y')$ of the object and, hence, both allow for reconstructions of electron densities.
However, these reconstructions are usually based on wave attenuation only, as phase shifts still remain harder to quantify.

However, phase shifts also induce intensity variations on the images, and this effect can reveal important features of the object's structure that are not visible in the attenuation-based images. This is of great important at high energy ranges where wave attenuation is  small, or even negligible. Indeed, the refractive index decrement  $\delta$ and the extinction coefficient $\beta$ have a strong dependence upon  the energy of the incident wave $E$, but they behave very differently as the energy increases. In particular,
for X-rays, $\delta$ decreases approximately as $1/E^2$, while $\beta$ approximately as $1/E^4$, so at these energies $\delta$ is between one and three orders of magnitude larger than $\beta$.  For example, at $E=30 keV$, $\delta\approx 2.56 10^{-7}$ and $\beta\approx 1.36 10^{-10}$ for water.

\subsection{Weak phase-contrast}

If the absorption is negligible so $\beta(x',y')\ll \delta(x',y')$, then the phase shift (\ref{eq:phaseshift})  modulates the detected complex amplitude (\ref{eq:fourier2d}), with transmissivity  given by
\begin{equation}
\label{eq:transmissivity}
t(x',y') = e^{i\phi(x',y')}\, .
\end{equation}
If, in addition,
$\phi(x',y')$ is small so $\beta(x'y')\ll \delta(x',y')\ll 1/l$, (\ref{eq:transmissivity}) can be approximated as
\begin{equation}
\label{eq:transmissivity_approx}
t(x',y') \approx 1 + i \phi(x',y')\,,
\end{equation}
and the phase-contrast is weak. In this case, the diffracted complex amplitude is given by
\begin{equation}
\label{eq:weak}
b_{tot}(x,y) = - i \frac{e^{ikL}}{\lambda L} \int [1 + i \phi(x',y')] e^{i2\pi(xx'+yy')}dx'dy'\, .
\end{equation}
This is called the weak phase object approximation.
Integration of the first term gives a delta function that represents the direct wave that goes through the object
without interaction. We assume that the direct wave is known and, thus, can be removed from the data, so only the diffracted component is considered for imaging.
On the other hand, integration of the second term gives the scattered diffraction pattern
\begin{equation}
\label{eq:phasecontrast}
b(x,y) = \int \phi(x',y') e^{i2\pi(xx'+yy')}dx'dy'\,,
\end{equation}
up to a constant that is set here equal to one.

Assume that, for imaging purposes, the inspected object is discretized using a grid of $K$ pixels, so its phase $\phi(x',y')$ is well approximated by the size $K$ vector
\begin{equation}\label{eq:rho}
\vect \phi=[\phi_{1},\ldots,\phi_{K}]^{\intercal}\in\mathbb{R}^K\, .
\end{equation}
According to (\ref{eq:phasecontrast}), if the object is illuminated by a monochromatic coherent plane wave,
the complex diffraction pattern $\vect b$ measured at an $S$-pixel detector is given by the  size $S$ discrete Fourier transform of the object's phase given by
$$
\vect b = F \vect \phi\, ,
$$
where
${\displaystyle F_{sk} ={\frac {e^{i2\pi(s-1)(k-1)/K}}{\sqrt {K}}}}$,  with $s=1,\dots,S$, and $k=1,\ldots ,K$. An important special case is when $S=K$, so the transformation is the classical  discrete  Fourier transform. If $S<K$, the case considered here, the Fourier transform is said to be {\em undersampled}.

If a known set of $N$ spatially structured patterns or masks
$$
\vect w_i=[w_{i1},\ldots,w_{iK}]^{\intercal}\in\mathbb{C}^K\,, \quad i=1,\dots,N,
$$
illuminate the object, then several complex diffraction patterns
$$
\vect b_i = F W_i \vect \phi\, , \quad i=1,\dots,N,
$$
are available for imaging. Here, $W_i$ is the diagonal matrix $W_i=\mbox{diag}(w_{i1},\ldots,w_{iK})$ corresponding to the $i$-th illumination pattern.
With this notation, the $s$-th component  of the data vector
\begin{eqnarray}
\label{response0}
(\vect b_i)_s = \sum_{k=1}^K  F_{sk}w_{ik} \phi_k \, , \quad s=1,\dots, S,
\end{eqnarray}
represents the complex field at the $s$-th detector when $i$-th illumination impinges on the object.
If the detectors can only measure intensities, then the problem is to recover (\ref{eq:rho}) from measurements of the type
\begin{eqnarray}
\label{response:intensity0}
|(\vect b_i)_s|^2 &= &\left|\sum_{k=1}^K    F_{sk}w_{ik} \phi_k \right|^2 \nonumber \\
\hspace*{-1.5cm} &\hspace*{-1.5cm}& \hspace*{-2cm} = \sum_{k=1}^K  |w_{ik}|^2 |\phi_k|^2 +
\sum_{k=1}^K \sum_{\substack{k'=1\\k'\neq k}}^K  F_{sk} F_{sk'}^*w_{ik} w_{ik'}^* \phi_k\phi_{k'}\, ,
\end{eqnarray}
$i=1,\dots,N,\, s=1,\dots, S$. Note that, in this case, the unknown vector $\vect \phi$ is real. The left side represents the  intensity received
at the $s$-th detector when the
object is illuminated with the $i$-th illumination pattern $\vect w_i=[w_{i1},\ldots,w_{iK}]^{\intercal}$. The first term in the right side of this expression is the incoherent contribution to this intensity, and the second term is the coherent contribution.
The coherent contribution is characterized by a fixed phase relationship between the waves emerging from different points of the object,
while in the incoherent contribution this phase relationship does not exist.
The coherent contribution produces the interferences used in coherent imaging to determine the object's structure, and it is the term
that makes  phase-retrieval  non-linear.

\subsubsection{Incoherent imaging}
According to the previous discussion, if the phases of the waves immediately exiting the object are randomized either by a diffuser or by the medium's inhomogeneities, so  the wavefront is totally scrambled, then the  second term in (\ref{response:intensity0}) is negligible and the inverse problem is linear
in the phase-shifts (squared) $ |\phi_k|^2$. In this case, the intensities received at all the detectors are equal; the data vector does not depend on $s$. Thus, to improve the SNR, the collected intensities  can be averaged and we can form the linear system
\begin{equation}
\label{eq:incohsyst}
  \underbrace{\begin{bmatrix}
   |w_{11}|^2 & |w_{12}|^2 & \hdots & |w_{1K}|^2\\
   |w_{21}|^2 & |w_{22}|^2 & \hdots & |w_{2K}|^2  \\
    \vdots &  \vdots &                   & \vdots \\
    |w_{N1}|^2 & |w_{N2}|^2 & \hdots & |w_{NK}|^2
  \end{bmatrix}}_{ \displaystyle {\cal W }_{incoh} }
  \underbrace{\begin{bmatrix}
    |\phi_1|^2  \\
    |\phi_2|^2  \\
    \vdots \\
    |\phi_K|^2
  \end{bmatrix}}_{\vect \chi} =
  \underbrace{ \begin{bmatrix}
    d_1  \\
    d_2  \\
    \vdots \\
    d_N
  \end{bmatrix}}_{\vect d_{avg}}\, .
\end{equation}
Here, each row of the matrix ${\cal W}_{incoh}$ contains the intensities upon the pixelated object produced by the $N$  masks $\vect w_i$. The linear system
\begin{equation}
{\cal W}_{incoh} \,\, \vect \chi = \vect d_{avg}
\end{equation}
can be easily solved by means of any $\ell_1$- or $\ell_2$-method depending on the number of masks $N$
used for imaging and the sparsity of the vector  $\vect \chi$.

\subsection{Non-weak phase-contrast}
If the absorption is negligible but the phase contrast is not weak but strong, then (\ref{eq:transmissivity}) cannot be approximated by
(\ref{eq:transmissivity_approx}) and we should image the transmissivity vector
\begin{equation}\label{eq:transmissivity_vector}
\vect t=[t_1,\ldots,t_K]^{\intercal}=[e^{i\phi_{1}},\ldots,e^{i\phi_{K}}]^{\intercal}\in\mathbb{C}^K\, ,
\end{equation}
which is now complex. Then, (\ref{response:intensity0}) becomes
\begin{eqnarray}
\label{response:intensity1}
|(\vect b_i)_s|^2 &=& \left|\sum_{k=1}^K    F_{sk}w_{ik} t_k \right|^2 \nonumber \\
&&\hspace*{-1.3cm} = \sum_{k=1}^K  |w_{ik}|^2 +
\sum_{k=1}^K  \sum_{\substack{k'=1\\k'\neq k}}^K F_{sk} F_{sk'}^*w_{ik} w_{ik'}^* t_kt_{k'}^*\, ,
\end{eqnarray}
$i=1,\dots,N,\, s=1,\dots, S$. In \eqref{response:intensity1}, the first term in the right side is the total intensity upon the object, which is known, and that does not provide any information for imaging. As in the case of a weak contrast object, all the information for imaging
is contained in the interferences of  the scattered wave represented by the second term.

At least two approaches are possible to solve the nonlinear problem (\ref{response:intensity1}). One is to solve it iteratively for the $K$ complex unknowns $t_i$ as in \cite{GS72, Fienup82}. This phase retrieval algorithms work well in practice if good prior information about the object of interest is known; otherwise, convergence to the true solution is not guarantee. The second option is to reformulate (\ref{response:intensity1})  as a linear  matrix problem  for the $K^2$ unknowns $t_it_j^*$, $i,j=1,\dots,K$, as in \cite{CMP11, Candes13}, and solve the resulting problem by using nuclear norm minimization. This option guarantees convergence to the unique solution without any prior information about the sought object, but the number of unknowns grows quadratically with the number of unknowns $K$ and, thus, the solution becomes unfeasible for high resolution images with large $K$.

Alternatively, one can define the cross-correlation
vector of $\vect t$
\begin{equation}\label{eq:crossrho}
\vect \chi_{cross}=[t_{1}t_{2}^*,t_{1}t_{3}^*,\ldots,t_{1}t_{K}^*,t_{2}t_{1}^*,t_{2}t_{3}^*,\ldots,t_{2}t_{K}^*,t_{3}t_{1}^*,\ldots,]\, ,
\end{equation}
excluding the real terms $|t_{i}|^2=1$,
and solve the huge linear system
\begin{equation}
\label{eq:hugelinearsystem}
 {\cal W}_{coh}\,\,\vect \chi_{cross} = \vect d\, ,
\end{equation}
given the data
$\vect d = [\vect d_1^{T}, \vect d_2^{T},\dots, \vect d_S^{T}]^T$. Here, $\vect d_s = [|(\vect b_1)_s|^2, |(\vect b_2)_s|^2,\dots,|(\vect b_N)_s|^2]^T$ are the $N$ intensities recorded at the detector $s$.
In \eqref{eq:hugelinearsystem},
\begin{equation}
\label{eq:coh_matrix}
 {\cal W}_{coh} = [({\cal W}_{1,coh})^{T}, ({\cal W}_{2,coh})^{T}, \dots, ({\cal W}_{S,coh})^{T}]^T
\end{equation}
is a huge matrix of size $N S \times K(K-1)$, where ${\cal W}_{s,coh}$ is defined in (\ref{eq:coh_matrix_n})
\begin{table*}[htbp]
\begin{equation}
\label{eq:coh_matrix_n}
{\cal W}_{s,coh}=
  \begin{bmatrix}
  c_{s1,12} & c_{s1,13} & \hdots & c_{s1,1K} & c_{s1,21} & c_{s1,23} & \ldots & c_{s1,2K} & c_{s1,31} &\ldots \\
 c_{s2,12} & c_{s2,13} & \hdots & c_{s2,1K} & c_{s2,21} & c_{s2,23} & \ldots & c_{s2,2K} & c_{s2,31} &\ldots \\
    \vdots &  \vdots &      \ldots &  \vdots  &\vdots &  \vdots              & \ldots &  \vdots              & \vdots &  \ldots              \\
   c_{sN,12} & c_{sN,13} & \hdots & c_{sN,1K} & c_{sN,21} & c_{sN,23} & \ldots & c_{sN,2K} & c_{sN,31} &\ldots \\
  \end{bmatrix}\, ,
\end{equation}
\end{table*}
with $c_{si,lm}=F_{sl} F_{sm}^*w_{il}w_{im}^*$. This matrix models the coherent component of the intensities received
at the $s$-th detector corresponding to the $N$ illumination patterns $\vect w_i$ used to {\em filter} the object. If the resolution is low, so the number of pixels $K$ is small, one could easily solve (\ref{eq:hugelinearsystem}) using an appropriate solver. However, as in the approach suggested in \cite{CMP11, Candes13}, the size of the problem  increases quadratically with $K$, so the search of a solution rapidly becomes prohibitive for large values of $K$, as well.

The main purpose of this paper is to propose a dimension reduction technique
that allows  to find the solution of this type of problems  efficiently for large $K$.  This algorithm is presented in Section \ref{sec:algo}.

\subsection{Absorption and phase contrast}
Absorption can be easily taken into account by using  (\ref{eq:phaseshift}) and (\ref{eq:absorption}) in (\ref{eq:transmissivity0}). If absorption is not negligible, then the transmissivity is given by
\begin{equation}
\label{eq:transmissivity1}
t(x',y') = e^{-\mu(x',y')/2} e^{i\phi(x',y')} = |t(x',y')| e^{i\phi(x',y')}\, .
\end{equation}
This is the most general case in which absorption and refraction effects are mixed in the images. In these cases, the transmissivity vector is given by the complex vector
\begin{equation}\label{eq:transmissivity_vector_absorption}
\vect t=[t_1,\ldots,t_K]^{\intercal}=[|t_1|e^{i\phi_{1}},\ldots,|t_k|e^{i\phi_{K}}]^{\intercal}\in\mathbb{C}^K\, ,
\end{equation}
with amplitudes different than $1$. Then, the problem is to find (\ref{eq:transmissivity_vector_absorption}) from measurements of the form
\begin{eqnarray}
\label{response:intensity2}
|(\vect b_i)_s|^2 &=& \left|\sum_{k=1}^K    F_{sk}w_{ik} t_k \right|^2 \nonumber \\
&&\hspace*{-1.8cm}= \sum_{k=1}^K  |w_{ik}|^2 |t_k|^2 +
\sum_{k=1}^K \sum_{\substack{k'=1\\k'\neq k}}^K  F_{sk} F_{sk'}^*w_{ik} w_{ik'}^* t_k t_{k'}^*\,  ,
\end{eqnarray}
$i=1,\dots,N,\, s=1,\dots, S$. This problem has, of course, the same form as before, but now the incoherent contribution to the intensity is modulated by the absorption. This makes attenuation-based imaging possible with incoherent sources.

The set of equations  \eqref{response:intensity2} can be written in matrix form as
\begin{equation}
\label{eq:hugelinearsystem_absorption}
  {\cal W}_{incoh}\,\,\vect \chi + {\cal W}_{coh}\,\,\vect \chi_{cross} = \vect d\, ,
\end{equation}
with ${\cal W}_{incoh}$ and ${\cal W}_{coh}$ defined in \eqref{eq:incohsyst} and \eqref{eq:coh_matrix}, respectively. Solution of (\ref{eq:hugelinearsystem_absorption}) provides {\em complete} images, with  distributions of both
the real and imaginary parts of the complex refractive index \eqref{eq:rindex}.
However, the bottleneck is still the size of the problem, which is enormous if one wants to form high resolution images. An image with only $1000\times 1000$ pixels, amounts to solving a linear system with $10^{12}$ unknowns!

Next, we describe the proposed algorithm that makes possible to find the desired solution in polynomial-time if the  object to be image is sparse in some appropriate basis. For simplicity, we will assume that the sought object is sparse in the real space, meaning that only a few pixels in the object plane absorb or bend the waves. We note, though, that many images are naturally compressible by using appropriate sparsifying transforms such as wavelets, or other dictionaries that are directly adapted to the data \cite{Ravishankar13}.

\section{Algorithm}
\label{sec:algo}

Instead of solving the linear system with $K^2$ variables (\ref{eq:hugelinearsystem_absorption})
for non-weak phase objects, with or without absorption, we propose to reduce its dimensionality constructing a linear problem for only $O(K)$ significant unknowns, and absorb the error corresponding to the contribution of the unmodeled unknowns by using a {\em Noise Collector} \cite{Moscoso20b}. Mathematically, we propose to solve the linear system
\begin{equation}
\label{eq:nc}
{\cal A} \vect \chi + {\cal C}\vect \eta = \vect d\, ,
\end{equation}
where ${\cal A}$ is a matrix with $O(K)$ subsampled columns of the matrix $[ {\cal W}_{incoh} | {\cal W}_{coh}]$, each column of size $N S$. In this formulation, $\vect \chi $ is a sparse vector that represents the object, $\vect \eta$ is an unwanted vector with no physical meaning that absorbs the noise, and $\cal C$ is a {\em Noise Collector} matrix  with $N^\beta$ columns drawn independently and at random on the unit sphere. The number of columns is determined by the  $O(1)$  parameter $\beta$, which is typically close to one (we use $\beta=1.5$ in our simulations).
For the construction and properties of the {\em Noise Collector}, we refer the reader to \cite{Moscoso20b, Moscoso21}.

The algorithm has three steps.
\begin{itemize}
\item[(1)] In the first step, we seek the strong absorbing objects. We set ${\cal A}={\cal W}_{incoh}$ so ${\cal W}_{coh}=0$, and solve \eqref{eq:nc}  for $\vect \chi=[|t_1|^2,|t_2|^2,\dots,|t_k|^2]^T$. The term
${\cal C}\vect \eta$ in \eqref{eq:nc}, where  $\cal C$ is a small matrix with $N^\beta$ columns, absorbs the coherent contributions to the intensities that are treated in this step as noise. Since the model we solve is not exact, only the strong absorbing objects are detected.
\item[(2)] In the second step, we seek the non-absorbing objects. Since these objects are almost transparent and do not have a significant impact on the recorded intensities, we look for their phases, encoded in the vector $\vect \chi_{cross}$ defined in \eqref{eq:crossrho}. To this end, we first subtract the incoherent contribution to the recorded intensities due to the detected objects in the first step. Then,
we set ${\cal W}_{incoh}=0$ and ${\cal A}=({\cal W}_{coh})_{sub}$, where $({\cal W}_{coh})_{sub}$ is a small subsampled matrix of the huge matrix ${\cal W}_{coh}$. It only contains  the $m\,(K-1)$ columns that correspond to the interactions between the $m$ detected objects in the first step and the other pixels in the image. Since we are not modeling the incoherent contributions of the remaining $(K-m)$ pixels, the system we solve is not exact neither. Hence, we also use a Noise Collector matrix $\cal C$ with $N^\beta$ columns to absorb the noise.
\item[(3)] The third step is optional. It is used to obtain more precise quantitative images. Once the strong and weakly absorbing objects are found, we solve the full problem \eqref{eq:hugelinearsystem_absorption} restricted to the recovered support. This is now a small problem that can be solved using an $\ell_2$ minimization method that gives very accurate results.
\end{itemize}

We stress that, for this dimension reduction strategy, it is necessary that the unknown object can be represented as a sparse vector. Otherwise, the (modeling) errors are too big to be absorbed.

\section{Implementation}
\label{sec:imple}

To find an $M$-sparsest solution of (\ref{eq:nc}), we solve the $\ell_1$-norm minimization problem
\begin{eqnarray}\label{rho_t}
\left( \mbox{\boldmath$\chi$}_{\tau}, \vect \eta_{\tau} \right) =
\arg\min_{ \small \vect {\mbox{\boldmath$\chi$}} , \small \vect \eta} \left(    \tau  \| \vect {\mbox{\boldmath$\chi$}}  \|_{\ell_1} +  \| \vect \eta \|_{\ell_1}  \right),\\
 \hbox{ subject to } {\cal A}  \, {\mbox{\boldmath$\chi$}} + {\cal C} \vect \eta =\mbox{\boldmath$d$}  , \nonumber
\end{eqnarray}
with an $O(1)$ no-phatom weight $\tau$.    In (\ref{rho_t}),  ${\cal A}$ is a matrix with $O(K)$ subsampled columns of the matrix $[ {\cal W}_{incoh} | {\cal W}_{coh}]$,
${\cal C}$ an ${\cal N}\times {N}^\beta$ matrix with $\beta$ close to one, $\vect \chi$ a $K\times 1$ vector, and $\vect \eta$ a $N^\beta \times 1$ vector. For the first step of the algorithm ${\cal N}=N$ while for the second ${\cal N}=N S$.

There are two main ingredients here.  First, the
{\em Noise Collector} matrix ${\cal C}$, whose $N^\beta$ columns are chosen independently and at random on the unit sphere
 $\mathbb{S}^{{\cal N}-1}$. Second, the weight
$\tau>1$, whose value should be chosen large enough so it is expensive to approximate the  noise with the columns
of ${\cal A}$, but should not be taken too large or, otherwise, the unknown $\vect \chi$ is lost, also absorbed by the {\it Noise Collector}. The optimal value of $\tau$ is the minimal value for which $\vect \chi = 0$ when the data is pure noise and, therefore, it can be chosen in advance.

It can be shown \cite{Moscoso20b, Moscoso21}, that if the matrix ${\cal A}$ is incoherent enough, so its columns are not almost parallel,
the minimizer in (\ref{rho_t}) has no false positives for any level of noise, with probability that tends to one as the dimension of the data increases to infinity.
To find the solution of (\ref{rho_t}), we  define the function
\begin{eqnarray}
 F(\mbox{\boldmath$\chi$}, \vect \eta, \vect z) &=& \lambda\,(\tau \| \mbox{\boldmath$\chi$} \|_{\ell_1} +  \| \vect \eta \|_{\ell_1}) \\
&+& \frac{1}{2} \| {\cal A}  \mbox{\boldmath$\chi$}  + {\cal C}  \vect \eta - \mbox{\boldmath$d$} \|^2_{\ell_2} + \langle \vect z, \mbox{\boldmath$d$} - {\cal A} \mbox{\boldmath$\chi$} - {\cal C}  \vect \eta \rangle \nonumber
\end{eqnarray}
and determine the solution as
\vspace{-0.2cm}
\begin{equation}\label{min-max}
\max_{\vect z} \min_{\mbox{\boldmath$\chi$},\vect \eta} F(\mbox{\boldmath$\chi$},\vect \eta,\vect z) .
\end{equation}
This strategy finds the minimum in (\ref{rho_t}) exactly for all values of the regularization parameter $\lambda$. Thus,
the  method is fully automated, meaning that it
has no tuning parameters. To determine the exact extremum in (\ref{min-max}), we use the iterative soft thresholding algorithm GeLMA~\cite{Moscoso12}
 that works as follows.

Pick a value for the no-phantom weight $\tau$; for optimal results calibrate $\tau$ to be the smallest value for which $\vect \chi=0$ when the algorithm is fed with pure noise.
In our numerical experiments we use $\tau= 2$.
Next, pick a value for the regularization parameter, for example $\lambda=1$, and choose step sizes $\Delta t_1< 2/\|[{\cal A} \, | \, {\cal C}]\|^2$ and
$\Delta t_2< \lambda/\|{\cal A} \|$\footnote{Choosing two step sizes instead of the smaller one $\Delta t_1$ improves the convergence speed.}. Set $\vect \mbox{\boldmath$\chi$}_0= \vect 0$, $\vect \eta_0=\vect 0$, $\vect z_0=\vect 0$, and
iterate for $k\geq 0$:
\begin{eqnarray}
&& \vect r = \mbox{\boldmath$d$} - {\cal A}  \,\mbox{\boldmath$\chi$}_k - {\cal C} \,\vect\eta_k\nonumber \, ,\\
&&\vect \mbox{\boldmath$\chi$}_{k+1}=\mathcal{S}_{ \, \tau \, \lambda \Delta t_1} ( \mbox{\boldmath$\chi$}_k +\Delta t_1 \, {\cal A} ^*(\vect z_k+ \vect r))
\nonumber \, ,\\
&&\vect \eta_{k+1}=\mathcal{S}_{\lambda \Delta t_1} ( \vect\eta_k +\Delta t_1  \, {\cal C}^*(\vect z_k+ \vect r))
\nonumber \, ,\\
&&\vect z_{k+1} = \vect z_k + \Delta t_2 \, \vect r \label{eq:algo}\, ,
\end{eqnarray}
where  $\mathcal{S}_{r}(y_i)=\mbox{sign}(y_i)\max\{0,|y_i|-r\}$.
Terminate the iterations when the distance $\|\vect \mbox{\boldmath$\chi$}_{k+1} - \vect \mbox{\boldmath$\chi$}_{k} \|$ between two consecutive iterates
is below a given tolerance.

\section{Numerical experiments}
\label{sec:numerics}

\begin{figure}[htbp]
\begin{center}
 true $| \rho_k |^2$ \\
 \includegraphics[scale=0.24]{./RHO_true_nscat4_Nilum300} \\
  \begin{tabular}{cc}
 $\ell_1$ no noise & $\ell_1$ SNR$=5$~dB \\
 \includegraphics[scale=0.24]{./RHO_L1_nscat4_Nilum300_optw2_inf} &
 \includegraphics[scale=0.24]{./RHO_L1_nscat4_Nilum300_optw2_5db}  \\
  \includegraphics[scale=0.24]{./xcc_nscat4_Nilum300_optw2_inf} &
 \includegraphics[scale=0.24]{./xcc_nscat4_Nilum300_optw2_5db} \\
   \end{tabular}
\end{center}
 \caption{First step of the algorithm. Imaging $M=4$ absorbing objects using the total power received on the array from $N=300$ illumination patterns.  The top plot shows the true distribution of absorbers; two are strong (red squares) and two are weak (white crosses).  Bottom panel of four figures: reconstructions with no noise (left column) and with additive noise so the SNR is $30$dB (right column).
 }
 \label{fig1}
 \end{figure}

In this work, a thin object is illuminated with quasi-monochromatic coherent, or partially coherent, sources ($\lambda=500$nm, corresponding to $600$THz). These sources are located on a two dimensional array of size $8$mm$\times 8$mm, at a distance of $8$mm from the object. There are $21 \times 21$ sources, evenly distributed. The data is collected $8$mm downstream at $5 \times 5$ receivers located on a plane parallel to the object and to the transmitting array. The  receiving array also has an aperture of $8$mm$\times 8$mm. The  collected data are the diffracted intensities corresponding to $N=300$ illumination patterns.

In order to form an image, the object plane is discretized using $31\times 31$ pixels, with pixel size equal to half a wavelength, {\em i.e.,}  $250$nm in both directions.  Although we consider a typical transmission setup, same results are obtained for reflection or for more complicated sources and receivers layouts.

We consider absorbing and non-absorbing objects that change the phases of the waves that go through them. Since we image only with the scattered intensities, we form images of  $t(x',y') -1$. Therefore, the absorbing objects have amplitudes close to one and phases close to $\pi$, i.e., $|\rho_i| \simeq 1$ and ${\rm arg}(\rho_i)\simeq \pi$, while the non-absorbing objects have amplitudes close to zero and phases close to $\pi/2$. For the non-absorbing objects we set $|\rho_i| \simeq 0.1$ and ${\rm arg}(\rho_i)\simeq \pi/2$. These objects are very hard to image, if only diffracted intensities are recorded, because  the waves go through them changing only slightly.

In the first step of the proposed approach, we solve for $|\rho_i|^2$, using as data the total scattered intensity for each of the $N=300$ illumination patterns.  The contribution to these data coming from all the cross-terms $\rho_i^* \rho_j$ with $i \neq j$ are considered as noise, which is absorbed by the {\em Noise Collector}.  Note that in $|\rho_i|^2$ the ratio between the weak and the strong absorbing objects is very small, of the order of $1/100$, making the detection of the weak absorbers very difficult. We show in Figure \ref{fig1} an example for four objects, two weak (white crosses) and two strong (red squares). The left column of Figure \ref{fig1} are the results for noise free data, and the right column the results for data with  SNR$=30$~dB. In both cases, the locations of the strong absorbing objects are recovered exactly. Moreover, their amplitudes $|\rho_i|^2$ are recovered with quite good accuracy. In the first and second rows of Figure \ref{fig1} we display $|\rho |^2$ as a two dimensional image, while in the third row we plot $| \rho |^2$ as a vector. In this third row, we plot the exact $| \rho |^2$ vector with green circles, and the recovered one with red stars. The black stars are the non-physical unknown $\eta$ introduced in the algorithm to absorb the contribution to the data due to the cross-terms $\rho_i^* \rho_j$, with $i \neq j$. In both cases, with or without noise in the data, the weakly absorbing objects are not recovered because the neglected contribution to the data of the cross-terms $\rho_i^* \rho_j$, with $i \neq j$ , is larger than the contribution of the weakly absorbing objects.

Hence, in the second step of the proposed approach, and after removing from the data the contribution of the found strong absorbers, we seek these cross-terms $\rho_i^* \rho_j$. However, to keep the system small, we only solve for the interactions between the $m$ recovered objects during the first step and all the other $K-1$ pixels in the object plane.
Thus, the number of unknowns is only $m(K-1)$.
For the example shown in Figure \ref{fig1}, we found $m=2$ strong absorbers, so we have $2K-2$ unknowns.
The results of this second step are shown in Figure \ref{fig2}. On the left column we display the unknown recovered by considering the interactions with the first scatterer while on the center column we display the unknown recovered by considering the interactions with the second scatterer. In the third column we plot the unknown $\rho_i^* \rho_j$; the green circles represent the true solution, the red stars the unknown recovered by $\ell_1$ minimization, and the black stars the non-physical part of the unknown corresponding to the {\em Noise Collector.}  This second step finds the weak absorbers with amplitudes and phases recovered with good accuracy; see the results in Figures \ref{fig3} and \ref{fig3b}.

\begin{figure}[htbp]
\begin{center}
 \begin{tabular}{cc}
 $\ell_1$ no noise & $\ell_1$ SNR=$30$dB \\
 \includegraphics[scale=0.24]{./RHO2nda_L1_nscat4_Nilum300_optw2_inf}  &  \includegraphics[scale=0.24]{./RHO2nda_L1_nscat4_Nilum300_optw2_5db}\\
 \includegraphics[scale=0.24]{./RHO2nd2b_L1_nscat4_Nilum300_optw2_inf} &  \includegraphics[scale=0.24]{./RHO2nd2b_L1_nscat4_Nilum300_optw2_5db} \\
\includegraphics[scale=0.24]{./xcc2nd_nscat4_Nilum300_optw2_inf}  & \includegraphics[scale=0.24]{./xcc2nd_nscat4_Nilum300_optw2_5db} \\
    \end{tabular}\\

\end{center}
 \caption{Second step. Imaging $M=4$ scatterers using intensity measurements over the receiving array.  Left column:  noiseless data. Right column:  SNR $=30$dB.
 }
 \label{fig2}
 \end{figure}

\begin{figure}[htbp]
\begin{center}
 \begin{tabular}{cc}
$\ell_1$ no noise & $\ell_1$ SNR=$30$dB\\
 \includegraphics[scale=0.24]{./xcc2ndabs_nscat4_IW_Nilum300_optw2_inf} &  \includegraphics[scale=0.24]{./xcc2ndabs_nscat4_IW_Nilum300_optw2_5db}\\
\includegraphics[scale=0.24]{./xcc2ndarg_nscat4_IW_Nilum300_optw2_inf}  & \includegraphics[scale=0.24]{./xcc2ndarg_nscat4_IW_Nilum300_optw2_5db}
    \end{tabular}\\
   \end{center}
 \caption{Second step of the algorithm. Left column:  noiseless data. Right column:  SNR $=30$dB. The top row shows the recovered amplitudes and the bottom row the recovered phases.
 }
 \label{fig3}
 \end{figure}

 \begin{figure}[htbp]
\begin{center}
true \\[5pt]
 \includegraphics[scale=0.24]{./phase_true_nscat4_Nilum300_optw2_inf}\\
 \begin{tabular}{cc}
  inf SNR & SNR=30dB \\
\includegraphics[scale=0.24]{./phase_final_nscat4_Nilum300_optw2_inf} &
 \includegraphics[scale=0.24]{./phase_final_nscat4_Nilum300_optw2_5db} \\
    \end{tabular}
   \end{center}
 \caption{Phase maps corresponding to Figure \ref{fig1}. The top plot is the true phase distribution, and the bottom left and right plots the recovered phase distributions without noise and with noise, respectively.}
 \label{fig3b}
 \end{figure}

The next numerical experiment considers  two strong absorbing objects and four weak ones. Measurement noise is added to the data so the SNR is $30$dB. The results in Figures \ref{fig4a}, \ref{fig4b} and \ref{fig4c} resemble those shown in the previous experiment. Imaging with the total power received from $N=300$ illuminations allows to image the two strong absorbers only; see the right column of Figure \ref{fig4a} that misses the four weak absorbers. The second step of the algorithm that uses the phase information contained in the cross-terms $\rho_i^* \rho_j$ is, however, capable of imaging these weak absorbers; see Figures \ref{fig4b} and \ref{fig4c}. Again, amplitudes and phases of all the absorbing objects are recovered accurately.

 \begin{figure}[htbp]
\begin{center}
  \begin{tabular}{cc}
  true $| \rho_k |^2$ & $\ell_1$ 1st step \\
 \includegraphics[scale=0.24]{./RHO_true_nscat6_Nilum300} &
 \includegraphics[scale=0.24]{./RHO_L1_nscat6_Nilum300_optw2_5db} \\
 $\chi_\tau$ and $\eta_\tau$ &  $\chi_\tau$ \\
  \includegraphics[scale=0.24]{./xcc_nscat6_Nilum300_optw2_5db} &
    \includegraphics[scale=0.24]{./xcc_nscat6_IW_Nilum300_optw2_5db} \\
        \end{tabular}
   \end{center}
 \caption{First step of the algorithm for $M=6$ absorbing objects; two are strong (red squares) and four are weak (white crosses).
 Top left and top right: true and recovered $|\rho |^2$ distributions, respectively. Bottom left: $|\rho |^2$ reconstruction as a vector (physical unknown with red stars, and  non-physical unknown $\eta$ with black stars). The exact vector is shown with
 green circles.  Bottom right: Only the exact (green circles) and reconstructed physical part (red stars) are shown. SNR=30dB.
 }
 \label{fig4a}
 \end{figure}

 \begin{figure}[htbp]
\begin{center}
  \begin{tabular}{ccc}
  true $| \rho_k |^2$ & $| \chi_\tau| $ and $|\eta_\tau|$  \\
 \includegraphics[scale=0.24]{./RHO_true_nscat6_Nilum300} & \includegraphics[scale=0.24]{./xcc2nd_nscat6_Nilum300_optw2_5db}\\
 $\ell_1$ 2nd step  &  $|\chi_\tau|$\\
 \includegraphics[scale=0.24]{./RHO2nda_L1_nscat6_Nilum300_optw2_5db}  &  \includegraphics[scale=0.24]{./xcc2ndabs_nscat6_IW_Nilum300_optw2_5db} \\
 $\ell_1$ 2nd step & $\mbox{arg}( \chi_\tau)$ \\
 \includegraphics[scale=0.24]{./RHO2nd2b_L1_nscat6_Nilum300_optw2_5db}
&
 \includegraphics[scale=0.24]{./xcc2ndarg_nscat6_IW_Nilum300_optw2_5db} \\
        \end{tabular}
   \end{center}
 \caption{Second step for $M=6$ absorbers.   SNR$=30$dB.}
 \label{fig4b}
 \end{figure}

\begin{figure}[htbp]
\begin{center}
 \begin{tabular}{cc}
 true & SNR=30dB \\
 \includegraphics[scale=0.24]{./phase_true_nscat6_Nilum300_optw2_5db} &
\includegraphics[scale=0.24]{./phase_final_nscat6_Nilum300_optw2_5db} \\
    \end{tabular}
   \end{center}
 \caption{Recovered phase maps for  $M=6$ scatterers.  SNR$=30$dB.
 }
 \label{fig4c}
 \end{figure}

The last example shows the results for ten absorbing objects, with only a strong one, and $\mbox{SNR}=30$dB; see Figure \ref{fig5}.  As in the previous examples,  only the strong absorber is found  with the first step of the algorithm in the absorption-based image. In the second step, as expected, we also find all the weak ones in the phase contrast image.
The last row of Figure  \ref{fig5} shows the reconstructed amplitudes (left) and phases (right) of the weakly absorbing objects plotted as vectors. The exact values are represented with green circles and the reconstructed values with red stars.

We observe from the results in Figure \ref{fig5} that the phases of the weakly absorbing objects are recovered with much more accuracy than their amplitudes. This is because the error induced by the neglected terms in the second step increases with the number of objects. Indeed, in the second step we  only account for the interactions between the strong and the weak absorbers ($M-1=9$ terms here), but we neglect all the interactions between the latter ($(M-1)^2=81$ terms here).

Better results can be easily obtained by considering, in a third step, the full problem  \eqref{eq:hugelinearsystem_absorption} restricted to the recovered support.
This third step allows us to recover the unknown $X_{supp}=\rho_{supp} \rho_{supp}^*$ accurately using an $\ell_2$ minimization method because the locations of all the absorbers are recovered exactly and because the contributions to data from all terms in $X$  are taken into account. Figure \ref{fig6} shows that this provides a great accuracy in the recovered values of both the amplitudes and the phases. Figure \ref{fig6b}, that shows the true and recovered phase distributions, illustrates the potential of the proposed imaging method with intensity-only measurements.

\begin{figure}[htbp]
\begin{center}
true $| \rho_k |^2$  \\
  \includegraphics[scale=0.24]{./RHO_true_nscat10_Nilum300} \\
\begin{tabular}{cc}
 $\ell_1$ 1st step & $\ell_1$ 2nd step \\
  \includegraphics[scale=0.24]{./RHO_L1_nscat10_Nilum300_optw2_5db} &
  \includegraphics[scale=0.24]{./RHO2nda_L1_nscat10_Nilum300_optw2_5db} \\
  \end{tabular}
  \begin{tabular}{cc}
  \includegraphics[scale=0.24]{./xcc2ndabs_nscat10_IW_Nilum300_optw2_5db} &
  \includegraphics[scale=0.24]{./xcc2ndarg_nscat10_IW_Nilum300_optw2_5db} \\
  \end{tabular}
\end{center}
 \caption{First step and second steps for $M=10$ scatterers.  The bottom left and  bottom right plots show the recovered amplitudes and the recovered phases, respectively. SNR$=30$dB.}
 \label{fig5}
 \end{figure}

\begin{figure}[htbp]
\begin{center}
 \begin{tabular}{cc}
   \includegraphics[scale=0.24]{./xl2abs_nscat10_Nilum300_5db} &
  \includegraphics[scale=0.24]{./xl2arg_nscat10_Nilum300_5db} \\
   \end{tabular}
   \end{center}
 \caption{Third step for the full unknown $X=\rho \rho^*$ restricted to the recovered support. SNR$=30$dB.
 }
 \label{fig6}
 \end{figure}

\begin{figure}[t]
\begin{center}
 \begin{tabular}{cc}
 true & SNR=30dB \\
 \includegraphics[scale=0.24]{./phase_true_nscat10_Nilum300_optw2_5db} &
\includegraphics[scale=0.24]{./phase_final_nscat10_Nilum300_optw2_5db} \\
    \end{tabular}
   \end{center}
 \caption{True and recovered phase maps for the $M=10$ absorbers.  SNR$=30$dB. }
 \label{fig6b}
 \end{figure}

Finally, we present results for the case in which the illuminations are partially incoherent. We use the following model to generate the data
\begin{eqnarray}
\label{response:intensity_pcoh}
|(\vect b_i)_s|^2 &=& \sum_{k=1}^K  |w_{ik}|^2 |t_k|^2 \nonumber\\
&+ &\displaystyle \alpha_{coh}
\sum_{k=1}^K \sum_{\substack{k'=1\\k'\neq k}}^K  F_{sk} F_{sk'}^*w_{ik} w_{ik'}^* t_k t_{k'}^*\, ,
\end{eqnarray}
with $0 \le \alpha_{coh} \le 1$. If $\alpha_{coh} = 1$, the sources are fully coherent, and if $\alpha_{coh} = 0$ they are fully incoherent. This parameter is unknown for the inversion of the data.

Figure \ref{fig:incoh} shows the results when the illumination used for imaging is partially coherent; $\alpha_{coh} = 0.5$ in this numerical experiment. The first and second rows show the outputs of the first and second steps of the algorithm, respectively. Because the illumination is partially coherent and, thus, the modeling error in the first step is smaller, we observe that the first step recovers the amplitudes of the strong absorbers with great accuracy. The second step is still able to recover the locations of the weak absorbers exactly. However, as expected, we observe that there is a SNR issue, and that if $\alpha_{coh}$ decreases below a certain threshold, we would not be able to image them. This threshold depends on the transparency of these objects, their number, and the noise in the data. For the numerical experiment shown here, the phases of all the absorbers are recover with precision as in the previous experiment (results not shown).

\begin{figure}[htbp]
\begin{center}
  \begin{tabular}{cc}
  \includegraphics[scale=0.24]{./RHO_L1_nscat4_Nilum300_optw2_ci_0p5} &\includegraphics[scale=0.24]{./RHO2nda_L1_nscat4_Nilum300_optw2_ci_0p5}\\
\includegraphics[scale=0.24]{./xcc_nscat4_Nilum300_optw2_ci_0p5}  &  \includegraphics[scale=0.24]{./RHO2nd2b_L1_nscat4_Nilum300_optw2_ci_0p5} \\
 &\includegraphics[scale=0.24]{./xcc2nd_nscat4_Nilum300_optw2_ci_0p5} \\
  \end{tabular}
\end{center}
 \caption{Imaging $M=4$  with partially coherent illumination; $\alpha_{coh}=0.5$. Left and right columns: first and second steps of the algorithm, respectively. There is no noise in the data.
 }
 \label{fig:incoh}
 \end{figure}

\section{Conclusions}
\label{sec:conclusions}

In this paper we have presented a two steps algorithm for phase retrieval. This algorithm is very efficient because its cost is linear in the number of pixels of the image and, thus, it can be employed for high resolution imaging. It guarantees exact recovery if the image is sparse with respect to a given basis, and it can be used, without any modification, when the illumination is partially coherent. This is very important for, for example,  phase-contrast X-ray imaging because fully coherent sources of X-rays are very hard to be obtained.

\section*{Acknowledgments}
The work of M. Moscoso was partially supported by the grant PID2020-115088RB-I00. The work of A.Novikov was partially supported by NSF DMS-1813943 and AFOSR FA9550-20-1-0026.
The work of C. Tsogka was partially supported by AFOSR FA9550-21-1-0196.

redpajama_set_name: RedPajamaArXiv


text:
Nreal's $599 mixed reality glasses are launching in the US on Verizon
Compatible with Samsung and OnePlus phones
By Adi Robertson@thedextriarchy Nov 18, 2021, 10:28am EST
Share All sharing options for: Nreal's $599 mixed reality glasses are launching in the US on Verizon
Nreal
Nreal's augmented reality glasses are coming to the US through a Verizon partnership. The $599 Nreal Light glasses will appear in 20 Verizon stores nationwide on November 30th and be available for purchase online December 2nd. The glasses are compatible with several recent 5G-enabled Samsung phones and one OnePlus 5G device.
The Nreal Light is one of the only consumer-focused AR headsets on the market. The Light glasses launched first in Korea last year, and they've since expanded to Japan and Germany. They support projecting 3D images into real space using reflected micro OLED displays, while outward-facing cameras let the glasses scan real space and pin virtual objects — like screens or game boards — into particular locations. The visual experience is broadly similar to the Microsoft HoloLens or Magic Leap One, although the Light uses different optics technology. Nreal particularly touts the glasses' wider-than-average 53 degree field of view, as well as a light 106-gram design.
Nreal's glasses are powered by an Android smartphone — in the US, they'll work with the Samsung Galaxy S21 Ultra 5G, Samsung Galaxy S21 Plus 5G, Samsung Galaxy S21 5G, Samsung Galaxy Z Fold3 5G, Samsung Galaxy S20 FE 5G UW, Samsung Galaxy S20 5G UW, Samsung Galaxy Note 20 Ultra 5G, and OnePlus 8 5G UW. Once you've plugged it into the glasses, you can either directly mirror your phone screen, including streaming video apps, or enter a full mixed reality experience that includes some AR games and productivity tools. The phone itself acts as a controller, although Nreal also lets developers build apps with hand tracking.
Earlier this year, Nreal announced that it was launching a second, cheaper pair of camera-free glasses called the Nreal Air. Those glasses are set to debut in China, Japan, and Korea, with no launch date set for the US.

redpajama_set_name: RedPajamaCommonCrawl


The Women's Center is one of the three campus community centers. It hosts programs and events for faculty, staff and students around issues of gender. It is currently located in the Original Students Center next to Mandeville.
The Women's Center provides support and education on gender issues for students, staff, faculty, and the community. Everyone is welcome to use the facility. Events include speakers, workshops, discussion groups, legal clinics, and other programs. The center's library includes a collection of books, videos, and magazines, including sections focusing on women in academia, women in science, parenting, health, lgbt and multicultural issues. There is a small children's area that parents can use (children must be supervised at all times). The center has a quarterly calendar and runs two listservs (a general listserv and one specifically for women in science). For additional information contact the Women's Center at 965 University Center, (858) 822-0074 or ude.dscu|nemow#ude.dscu|nemow.


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text:
Meet EUROAPI: Sanofi christens to-be-spun-off drug ingredients business, taps CEO
by Fraiser Kansteiner |
EUROAPI, Sanofi's standalone drug ingredients business, is expected to pull in $1.2 billion in sales by 2022. (Sanofi)
Sanofi has tied a name—and a new chief executive—to the European drug ingredients business it plans to spin off next year.
Now dubbed EUROAPI, the active pharmaceutical ingredient (API) business is Sanofi's "made in Europe" solution to the region's heavy reliance on materials sourced from countries like China and India—an issue thrown into relief by the COVID-19 pandemic.
Pegged at €1 billion ($1.2 billion) in sales by 2022, the company "will rank number 1 in small molecules API, and number 2 on the global API market," Sanofi estimates.
Along with the new moniker, Sanofi tapped Karl Rotthier, former CEO of Dutch drugmaker Centrient Pharmaceuticals, to steer the EUROAPI ship. Rotthier joins as CEO of the business, which comprises six Sanofi sites in France, Italy, Germany, Hungary and the U.K., on Jan. 18.
RELATED: Sanofi, despite COVID-19, sees early gains from CEO Hudson's strategy revamp
The company, with 3,200 staffers, will be based out of France, Sanofi said. The French Big Pharma plans to own about a 30% stake in the new company; plus, it says it will be one of EUROAPI's first customers. Sanofi intends to decide by 2022 whether to launch an IPO for the company on Euronext Paris.
Revealed in February 2020, the EUROAPI plan took shape as then-new CEO Paul Hudson embarked on a quest to trim €2 billion from Sanofi's annual expenses by 2022 through a mix of job cuts, tightened manufacturing budgets, diminished travel and more.
As the COVID-19 pandemic unfolded, Sanofi stayed the course and entered talks with potential investors about the company, including France's state investment bank, sources told Reuters in July.
RELATED: Sanofi, intent on slimming down, mulls $200M-plus sale of anti-inflammation meds: report
Still, the spinoff won't be easy, the sources said. Defining the status and contracts of staffers could prove complicated, because Sanofi will only retain a minority stake in the company. Meanwhile, Sanofi's move in July to slash some 1,680 jobs in Europe over the next three years could hamper discussions with employee representatives about the spinoff, the sources said.
The COVID-19 pandemic and political sensitivities about drug supplies threatened to toss another spanner in the works, Reuters added.
For now, the "extremely complex" API split—as one source described it to Reuters—looks a bit more concrete. Once the company takes off, it will boast a portfolio of 200 APIs and tap into a commercial network of more than 80 countries, Sanofi said.
API drug manufacturing spinoff Sanofi France
Fraiser Kansteiner


text:
Next Study: Minnesota drivers suck least
Jail is very bad
Monday, June 3, 2019 by Mike Mullen in News
Abby Rudolph's death sounds like it was awful... and avoidable. Karen Stroh
Abby Rudolph, 19, was arrested for shoplifting in 2016.
Shoplifting.
Who could've possibly imagined such a minor happening was a death sentence for Rudolph, a resident of Moorhead and described in an online obituary as "artistic and creative with blue eyes that smiled and lit up a room."
Abby had developed a drug addiction after suffering a hip injury, per a story by Forum News Service, which details the grim aftermath of Rudolph's arrest the day before Halloween. Picked up for theft, Rudolph went into withdrawal while in the Clay County Jail.
Rudolph's condition soon took a turn for the worse, as noticed by other inmates, who, according to a lawsuit filed by Rudolph's family, asked jail staff to do something for the teenager. Rudolph had been in for just one day when people wrote a note to her jailers saying she wasn't doing well.
And yet it was several days later, as Rudolph refused food, threw up, and grew weaker, that an employee and a nurse "noted that Rudolph appeared sluggish and was cool to the touch," the Herald reports. That was the beginning of the end for Abby Rudolph, a user of pain pills and, on at least one occasion, a shoplifter. (A search of Minnesota court records doesn't indicate Rudolph had other arrests on her record.)
Rudolph was taken to the jail's shower, but had by that time lost control of her body, which jerked sporadically, and was "catatonic," her family's lawsuit states.
Medics arrived but it was too late. Rudolph was declared dead on the afternoon of November 3, fully four days after fellow arrestees told the jail this young woman they'd just met needed help. She died of bronchopneumonia, according to her death certificate, which the Herald characterizes this way:
"The certificate lists the manner of Rudolph's death as natural and indicates neither injury nor trauma contributed to her death."
Dear God. Now who seems like they're "cool to the touch"?
In a shocking turn of events, a subsequent Minnesota Department of Corrections investigation determined no one did anything wrong. Rudolph's treatment was not just compassionate but "very compassionate," the report states, and staff acted not just professionally but "very professionally."
Sure, a small amount of retraining is in order, that report determined, but there was evidently not much more that could've been done in the 100 or so hours between the time wardens got a note about Rudolph and the moment she died in the parking lot outside. Nothing at all.
If this feels like a bit of a gut punch, here's a couple paragraphs to make sure your Monday morning includes some bitter rage:
Rudolph's mother, Jill Rudolph, who filed the wrongful death lawsuit in early 2018, died in December 2018 at the age of 52. In an interview, Craig Casler said it's believed his sister, Abby's mother, likely died of a medical condition after lying down to take a nap, but he suspects there was more to it than that.
"She died of grief. She could not go on," said Casler, who feels his sister simply could not endure another Christmas without Abby.
Maybe we're just biased because we care about people more than property, but if you ask us, there are worse sins than shoplifting. Like letting someone die.
CityPages News
Minnesota badass took a knee during Trump-adjacent national anthem January 16 by Mike Mullen
'Is it bullshit?': Cold and flu edition January 16 by Hannah Jones


My main goal is to provide High Quality design in concept and execution.
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"We wanted a specific concept that could be done with a minimalist design, flexible coloring, and a higher end feel. We felt this design met all the criteria and was the best fit for what we were looking for with our initial concept. "
"Great Job! quick response, got right to what we wanted."
"Bravo - perfect design, exactly what I was expecting. You took the liberty of reusing our logo and it was a good idea !!"
"dsfraser listened thoroughly to my brief, and presented their work well. My brief was for honey, so they presented the logo on a honey jar. they were quick to act on requests, and were always happy to try new ideas"
"Good work, Quick turnaround and easy to work with."
"Very good, responsive designer. Appreciate the designer's patience with us. Excellent work, of course."
"Thank you so much, we love it!"
"We like the design with the font in the middle of the series of font versions. Thank you."
"Great design. Produced a great design on the first try. It was simple yet effective and was so amazing as a black and white design. Made psd changes quickly. Great designer. Really understands how to create an effective, logo design. "
"dsfaser did a phenomenal job for us! Can't thank him enough for being so flexible, easy to work with and we loved the clarity of his design that is now our logo! Super excited! Great experience with 99designs.com. "
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Hidden Lakes offers several options during our camping season of May 1 through October 31. You can rent a site by the day, week, month or all season - the choice is yours. All campers/RVs must have a holding tank. For tent camping there are 5 comfort stations located throughout the campground. There is a dump station available for all our campers.
Monday to Thursday, includes water and electric. Site must be vacated by 12 noon.
Includes water and electric. Site must be vacated by 12 noon.
Includes water and electric. Weekly rates are for 7 days and 6 nights. On the 7th day, site must be vacated by 12 noon.
Includes water and electric. Monthly rates are for 30 days and nights. Site must be vacated by 12 noon on the last day of your stay.
A credit card deposit is required to reserve rental lots. This can be paid with a credit card with a 2.5% processing fee.
Hidden Lakes offer seasonal camping to the community. You may lease a Hidden Lakes lot for the season, or for an additional charge leave your camper on the site for the off-season. However, although you may visit your camper in the winter, winter living is prohibited at Hidden Lakes and will mean the forfit of your lot and fees.
Call the office at 419-946-7050 to be added to our waiting list for 2019.
You may enjoy all the amenities of the campground. You will be issued a gate card for entrance into Hidden Lakes. Consolidated Electric supplys the electricity to the campground. You will need to arrange with them to open an account and you will be billed directly. Consolidated Electric is located at 5255 State Route 95, Mt. Gilead, OH. Or you may call them at 419-946-6600 to arrange your service.
A one-time security deposit is collected at the beginning of your leasing period. When you leave, if the lot is in the same or better condition than when leased, your deposit will be returned. This will be decided after a thorough inspection.
You may decide you like your area and want to keep it for the next year. You may leave your camper on your site during the off season. This also entitles you to occasionally camp, in your contained camper, during the winter.
OFF-SEASON LIVING IS NOT PERMITTED AT HIDDEN LAKES AND IS A VIOLATION OF YOUR LEASE. vIOLATORS WILL BE ASKED TO LEAVE THE CAMPGROUND AND FEES MAY BE FORFEITED. INFORMATION ON THIS PAGE, INCLUding FEES, IS SUBJECT TO CHANGE WITHOUT NOTICE.


Meeting Type: Regular Board Mtg.
The ES serves 3 core functions: (1) assures confidentiality, (2) creates a mechanism for Board independence and oversight, and (3)enhances relationships among Board members and with the School Director. While in ES, only Board members and individuals invited by the Chair may be present. At the option of the Chair, the School Director or other Board members may be excused.
Executive Session will include members' assessment of the meeting and suggestions for process improvement.


text:
Is there a term for a sentence that suggests a question but isn't one?
I've noticed some sentences or phrases that suggest a question that people tend to accidentally end with a question mark. Examples: "I wonder why" or "The question is what we should do now".
Is there a term for such a sentence or for the error of punctuating them with a question mark?
Mu MindMu Mind
I believe the term you're looking for is indirect question.
a question that is reported to other people in speech or writing, rather than the exact words of the original question, for example 'He asked me what was wrong.' (Cambridge Dictionary)
Various ESL sites specifically cite the "I wonder..." case as an example of an indirect question.
Direct question: Where have they been?
Indirect question: I wonder where they have been?
Notably, that example does use the question mark at the end. It is not always considered incorrect; it really depends on whether you want it to be perceived as a question or a statement. For more info about that, see this related question and its answers.
Shouldn't the second one end with a period?
@Mehrdad Yes. A question mark converts it to a declarative question. _'Can you read Quentin's last sentence? His writing is worse than mine.' _'I wonder where they have been'?
Is that a "rhetorical question"? I think it is.
CodeswitcherCodeswitcher
No, a rhetorical question is somewhat the opposite — a question asked to make a point, not a statement made to raise a question.
– hobbs
Question mark when ending sentence with "I wonder"?
Is there a term that means "oft-used misquotation"?
Is there a different grammatical term for "If I was" than for "If I were"?
What's the term for introducing an error when you edit a sentence?
Term for greetings that are also valedictions
Is there a term for sentences which structurally reflect their meaning?
Term for a misspelled word, where the "misspelling" is a correctly spelled but out of context word?


This is the largest family of macro moth caterpillars in the British Isles with over 400 species. Included are some of the most commonly sighted caterpillars such as the Angle Shades.
Due to the recent reclassification of some moth species several appearing here will be moved into the Erebidae family.
Most species have three true legs, four prolegs and a rear clasper. However, The Snouts have one less proleg and moths of the Plusiinae family such as Silver Y have 2 fewer prolegs.
Many of these caterpillars such as the Large Yellow Underwing have relatively smooth skins with few obvious hairs and live hidden in the soil or low down at the base of plants.
While others are extraordinarily hairy such as the Miller and Sycamore Moth which feed on the leaves of deciduous trees.
Others such as The Knot Grass, the brightly coloured Mullein Moth caterpillars and Dark and Grey species of Dagger are amongst the top twenty most enquired about caterpillars.
One of the most extraordinarily caterpillars is that of the Chamomile Shark which despite its large size and streaks of colour blends in making it surprisingly difficult to see amongst its food plant, species of Mayweed.
Nut–tree Tussock caterpillar is the only member of the sub family Pantheinae.
Also included are the caterpillars of the Scarce Bordered Straw and Bordered Straw regular migrants and two or the commonest caterpillars to be imported into the British Isles on cultivated flowers.
In the Cucullinae family are three moth species with strikingly marked caterpillars.
The Mullein Moth is one of the most commonly sighted caterpillars in gardens throughout much of the southern half of the UK.
The more local Toadflax Brocade caterpillar is restricted to the south east but spreading.
The fabulously marked and variable caterpillar of the Chamomile Shark is more locally distributed and seen less often in gardens.
In the Acronictinae family are some moths with extraordinarily hairy and colourful caterpillars.
The Knot Grass caterpillar if one of the most commonly sighted caterpillars throughout much of the British Isles.
The extraordinarily hairy Sycamore Moth and Miller Moth caterpillars are often sighted when they leave the foodplant to wander in search of somewhere to pupate.
The spectacular Alder moth and The Poplar Grey are widespread but less commonly seen.
The caterpillars of the Grey Dagger and Dark Dagger both have distinctive dorsal 'humps' and are often recorded in woodland and gardens.
Unusually for noctuid caterpillars there are a few species in the Plusiinae sub family that have fewer prolegs than the normal four and rear clasper.
This group in the British Isles includes both resident and migrant species.
The caterpillars of the Silver Y, the commonest of the larger migrant moths to the UK, are often referred to as the Cabbage looper in other parts of the world where they are considered a pest. In common with other closely related caterpillars they have two prolegs missing.
The caterpillars of other rarer migrant species such as Slender Burnished Brass and Double Gold Spot may also occasionally be imported on agricultural produce, although these fail to survive the cold winters.


text:
require 'rails'

module Jsonapi
  module Rspec
    module Matchers
      class Railtie < Rails::Railtie
      end
    end
  end
end

redpajama_set_name: RedPajamaGithub


Financing the purchase of second hand cars is a complex process for any buyer. You must take great care to finance your car properly, and you should work with one shift in SG that allow you to finance your car the way you want. This article holds a few tips for financing your car that will help make the process a bit less painful.
You may finance your car in advance of a purchase with a pre-approval process at a bank. The lender that finances your car will supply you with a pre-approval letter that states your eligibility for financing. Any dealer you visit will want to know you are paying for the car, and you may show this letter as proof of your ability to pay for the vehicle.
These letters may be general form letters from the lending agent, or you may ask the agent to show how much money you can borrow for the car. A specific amount helps the dealer understand that you are shopping responsibly.
You can get through the loan process much more quickly with the help of the loan officer for getting your new car.
You may finance your car with the mart where you purchase your car. The financing department at the one shift will ask you to submit your application before they show you cars, but they will lead you to the cars in your price range once they have approved your financing.
Financing at the dealer happens quickly because they have a loan officer on the property. You can get through the process much more quickly, and you will get off the lot with your new car in a hurry.
Choose the price range of the vehicle that under your budget.
Shopping for your car requires a good bit of wisdom. You must look for cars that do not push the limits of your budget. Getting pre-approved allows you to limit your shopping to cars that fall in the middle of your price range. You cannot spend so much that you cannot make the payments, and you should look in the lower end of your price range for a gem that you can drive for years.
Setting up payment arrangements for more payment options and flexibility for you to choose.
Financing your vehicle involves payments you make every month to the company lending you the money. Financing your car at the one shift is a good idea if you want to work directly with the source. You can make payments at their office, or you can mail in your checks. Financing your car with a bank gives you more payment options and flexibility. A large bank or lender can afford to let you pay online, and they can help you make different payment arrangements if you have difficulty making your payments in the future.
Purchasing a used car in Singapore involves financing that you must acquire for each purchase. The tips in this article explain how you can finance your car in a responsible way. Set up your financing with a bit of wisdom to make your car purchase a delightful experience.


Yes, you can openly and publicly advertise your Membership for sale if you no longer want or need it. This has always been our policy. We don't believe it is fair to members if they invest their money in a membership, but are prohibited from advertising and selling it. While members often sell memberships to people who were going to buy one from the club, we aren't going to penalize our members because we are greedy and want to block out fair competition. If a member wants to put forth an effort to recover their investment, we support them. Members may not advertise selling their membership on any official HSP websites or social platforms. For example, you are welcome to put your membership up for sale on your own personal Facebook profile, but you may not post it for sale on the HSP Facebook page.


Just a big wheel Nomad or something completely different?
We live in a world seemingly obsessed with hacking. And I'm not talking about influencing elections or accessing compromising cloud-based imagery. I'm referring to the trend for tarting up items of flat-pack furniture or devising the most inventive use for a zip-tie. Blame YouTube and Pinterest if you will, but there's no escaping the irresistible urge to muck about trying to improve stuff. It's human nature and has been going on since caveman A figured out a sharp stick was better for hunting food than caveman B's rock.
Pivots run on large diameter axles and bearings, which Santa Cruz offer free lifetime replacement for. Grease ports on lower links make it serviceable. Photo by Gary Perkin.
But what on earth has hacking got to do with a new Santa Cruz? Well, the brand was ahead of the curve with a half decent 29er – the Tallboy. It then hacked that Tallboy to make the longer travel Tallboy LT. Then it decided the segment deserved its own unique model, so the Hightower was born. And, you guessed it, a few owners immediately hacked it, extending the travel by 10mm. So they released the official Santa Cruz Hightower LT version, which added a touch more travel and fixed the compromises created by the hackers. And life was good for a while, until, inevitably, proper long travel 29ers started to become a thing, and the 150mm travel Hightower LT started to look outgunned and undersized by bikes like the Yeti SB150, Scott Ransom and Specialized Stumpjumper Evo.
Which brings us to the present day, and the new Megatower. A bike that, save for five letters and two 29in wheels, shares more with the current Santa Cruz Nomad and Santa Cruz Bronson than the old, and now discontinued, Hightower LT. This, then, is no hack. It's a fully legit model in its own right. And that's worth celebrating, for a few reasons.
Firstly the Megatower uses the lower link to drive the shock. It's the VPP layout first seen on the Santa Cruz V10, then adapted for the Nomad, and it gives less rate change through the travel. In a nutshell, where the Hightower LT tended to fall into the mid-stroke, now there's a nice consistent progression throughout the travel, giving more support and better bottom-out resistance. It's also easier to set-up and there's now sufficient ramp up to allow riders to choose between air and coil for suspension duties without compromising on end-stroke performance.
The Megatower further improves on the old Hightower LT with the fitment of a bearing at the rear eyelet of the RockShox Super Deluxe shock. This reduces breakaway friction for a more supple and active bump response. And it's also worth noting that all the Megatower builds, as well as the frameset, come with RockShox shocks sporting the same tune, which ensures all models should ride exactly how the product managers intended.
As introduced on the Nomad, two shock positions on the lower link tune the suspension and geometry. Defined Hi and Lo, this chip actually only changes the static ride height by a minimal 3.5mm and tweaks the head angle by a smidge. However, it allows you to maintain BB position if you decide to bump up suspension fork travel to 170mm – something we can see plenty of riders considering. And should you wish to run a coil shock, the Lo position also gives you a more progressive rate.
That's not the only flip-chip on the Megatower however. Cast your eyes to the rear dropout and you'll see that Santa Cruz has added an adjustable chainstay length feature. By switching between the two different inserts and brake adaptors (all included with the bike) you can run either a 435 or 445mm rear centre measurement. While 10mm might not sound like much, when you consider that the Megatower size range runs from small to XXL, the front centre (BB to front axle) varies by 120mm, which has a huge impact on your weight balance. And while most brands, with a few exceptions (Mondraker, Norco, YT, Geometron) accept this compromise, being able to tune the weight distribution is a major bonus, particularly for taller riders.
Take a look at the geometry chart and you'll see that Santa Cruz has taken a big step in the right direction with its sizing. The size large gets a 470mm reach, which seems to be the default number now and well suits riders of average height. But with two frame sizes above there's plenty of headroom if you're over six foot. Indeed the XXL frame has a 515mm reach – that's the same as a large Geometron – and although the seat tube length is longer than Nicolai's offering, there's 20mm more standover on the Megatower.
You do get a larger head angle change (1° slacker) if you go from Hi to Lo and up-fork by 10mm at the same time. Switching between Lo and Hi affects shock rate, with the Lo position giving slightly more progressive rate and more bottom out resistance. Photo by Gary Perkin.
Elsewhere the numbers sit squarely within a window that's proven to work on a number of recent releases. The head angle runs around 65° depending on fork length and chip position while the bottom bracket can stoop just below 340mm with the stock tyres. Only the seat angle errs on the slack side. Yes, Santa Cruz lists it as 76°, but the actual seat angle on our large test bike was closer to 69°, so long legged riders running lofty saddle heights may find themselves a little further over the rear axle than optimal.
It's no deal breaker though, as the VPP pedalling DNA runs strong through the veins of the Megatower. Seated pedalling is superbly stable for such a long travel rig, and the big wheels means it covers ground between the fun bits with real efficiency, but you knew that would be the case anyway, right? Techy, steep climbs with ledges and steps are not the new bike's forte though. Factor in 30 per cent sag and that low BB (even in its tallest position) means you'll need precision timing to avoid pedal strikes.
In short position, the largest tyre recommended is 29×2.5″. In the long chainstay setting, a 29×2.6″ will fit as well. Photo by Gary Perkin.
Once on the fun stuff, the Megatower impressed, delivering that same leap in performance enjoyed by the Nomad and Bronson post-VPP makeover. The first thing you notice is the improved support, allowing you to pump for speed out of turns and compressions – this is especially useful in situations where the low BB prevents pedalling. We felt like we were using all the travel when needed, but never getting jarred at bottom-out. Considering the bike ships with no spacers in the shock as standard, there's plenty of room to add progression if you feel the need. Indeed this would be something we'd like to try as the Fox 36 fork with Grip 2 damper rides high in its travel with great support, so it's unlikely adding a spacer out back would upset the balance, and may even improve it. That's something we'll have to try at a later date though.
On small bumps there's a decent, if not class-leading, level of sensitivity and while square-edge performance is significantly improved with the new lower link design, it isn't perfect and we still hung up hard enough on one particular rock to almost drag us over the bars.
So is the Megatower a big wheel Nomad or something completely different? Well, it definitely can't match the playful agility of the Nomad, but it's still easy to chuck around and a good deal more efficient at covering ground. It's much more of a head down racer's tool than the goof ball Nomad, but it still knows how to party outside office hours.


This image copyright White Wolf © 2006.
The cover for the new Werewolf: The forsaken supplement.
It could be used as poster for a werewolf film.
spectacular would be an understatement to me but I cant find the best word to coment your masterpiece( . i guess you are possesed, but Ilove your work.
I also am a "Forsaken" player and greatly admire all of the art that you contributed.
Of course I like it! As a player and as an artist (but making pictures for Polish RPG handbooks is not very nice job ).
That is a wonderful picture, I have seen a lot of werewolf drawings, but I have to admit that this one is one of the best! Keep up the good work!
Thanks..Glad you like the artwork.
"Forsaken" players and readers (for example me) are very content to see your works in the handbooks. Good job! You're the best! And this one is absolutely fantastic! Keep up the good work!
Thanks Matthew, You kind words are very much appreciated...BTW Your not so bad yourself Superb illustrations buddy.
outstanding, what a great painting! Congratulations on a well-deserved editors pick.
nice work on this. The way that you handled the color is very striking. The woman becomes the center of the piece not only by being centered but also by the being brighter in color.
Scary as hell. Excellent piece of work. Congrats on Editor's Pick. It's been a long time since your last one.
Great work, I particularly like the wolves. Good job on the Ed's Pick too.
Great piece abi!... Hoooooowww ...haha!


text:
Mirabeau steht für

 Mirabeau (Métro Paris), eine Station der Pariser Métro
 Draâ Ben Khedda, algerische Gemeinde, wilaya von Tizi-Ouzou, während der Kolonialzeit Mirabeau genannt.

Mirabeau heißen folgende Gemeinden in Frankreich:
 Mirabeau (Alpes-de-Haute-Provence) im Département Alpes-de-Haute-Provence
 Mirabeau (Vaucluse) im Département Vaucluse
 Le Bignon-Mirabeau im Département Loiret
 Les Pennes-Mirabeau im Département Bouches-du-Rhône

Mirabeau als Personenname:
 ein männlicher Vorname, siehe die weibliche Form Mirabella (Vorname)
Familienname:
 Jean-Baptiste de Mirabaud (1675–1760), französischer Philosoph
 Victor Riqueti de Mirabeau (1715–1789), französischer Ökonom und Philosoph; Vater von Gabriel und André; älterer Bruder von Jean-Antoine:
 Honoré Gabriel de Riqueti, comte de Mirabeau (1749–1791), französischer Revolutionspolitiker
 André Boniface Louis Riquetti de Mirabeau (1754–1792), genannt Mirabeau-Tonneau, französischer Politiker
 Jean-Antoine Riqueti de Mirabeau (1717–1794), Bailli des Malteserordens
 Sibylle Gabrielle Riquetti de Mirabeau (1849–1932), französische Schriftstellerin

Siehe auch:
 Pont Mirabeau, Mirabeau-Turm, Cours Mirabeau

redpajama_set_name: RedPajamaWikipedia


text:
import os, glob
import shutil, errno
import sys

# --------------------------------------------------------------
# Change this to the location of a blank/empty Lightroom catalog
# --------------------------------------------------------------
_LRTEMPLATE = "/Users/mcgraths/Pictures/Workflow/_LRTEMPLATE"
# --------------------------------------------------------------

def copyfolder(src, dst):
    try:
        shutil.copytree(src, dst)
    except OSError as exc: # python >2.5
        if exc.errno == errno.ENOTDIR:
            shutil.copy(src, dst)
        else: raise

def create_directory(directory):
	if not os.path.exists(os.getcwd() + "/" + directory):
		os.makedirs(os.getcwd() + "/" + directory)
		print "Folder created: " + directory

def rename(dir, pattern, find, replace):
	#print os.path.join(dir, pattern)
	for pathAndFilename in glob.iglob(os.path.join(dir, pattern)):
		title, ext = os.path.splitext(os.path.basename(pathAndFilename))
		os.rename(pathAndFilename, os.path.join(dir, title.replace(find, replace) + ext))
		#print pathAndFilename
		#print title.replace(find, replace) + ext

def init():
	create_directory('Throwaways')
	create_directory('USB')

	folder = os.path.basename(os.getcwd())
	dest = os.getcwd() + "/" + folder + " Catalog"

	#copy Lightroom catalog template
	if not os.path.exists(dest):
		copyfolder(_LRTEMPLATE, dest)
		rename(dest, "_LRTEMPLATE*", "_LRTEMPLATE", folder)
		print "Lightroom catalog created"

	print "[Session Initialized]"

def init_wedding():
	create_directory('Throwaways')
	create_directory('USB')
	create_directory('Book')
	create_directory('Prints')

	folder = os.path.basename(os.getcwd())
	dest = os.getcwd() + "/" + folder + " Catalog"

	#copy Lightroom catalog template
	if not os.path.exists(dest):
		copyfolder("/Users/mcgraths/Pictures/Workflow/_LRTEMPLATE", dest)
		rename(dest, "_LRTEMPLATE*", "_LRTEMPLATE", folder)
		print "Lightroom catalog created"

	print "[Wedding Initialized]"

if len(sys.argv) < 2:
	print "ERROR: Please supply at least two arguments"
	print "Available commands:"
	print "- init"
	print "- init wedding"

elif len(sys.argv) == 2:
	init()
else:
	init_wedding()


"Moisture Protection & Firestop Specialists"
Midwest Sealant Supply will ship via the most economical Carrier. Shipping charges are the responsibility of the purchaser. Do not refuse any delivery. Refusal of any delivery will likely result in additional charges to the purchaser.
Returns accepted up to 30 days after purchase for stocked items. Non-stocked items, if returnable, require a re-stocking charge. No returns on custom orders.
For materials shipped in error, Midwest Sealant Supply will cover the return shipping charges.
For all other returns, return shipping charges are the responsibility of the purchaser.
Please call for a material return authorization and shipping instructions.


Since the mid-1700s, collectors have prized these tiny, exquisitely detailed boxes. Handmade of pedigreed porcelain, each handpainted box is hinged with brass fittings and a decorative fastener. The dog boxes range from 2" to 3"l.


Last week Shayla and I embarked on our first journey to the pacific islands. We spent loads of time on google researching what we could do while on our holiday. We had this picture in our minds that reflected paradise.
After a month planning we flew three hours to reach our destination.
There is something about looking out a small oval window above the trees and clouds that is so exciting. It is as if you have never seen anything like it before. Well I guess we hadn't.
Our first 24hours in Tonga – let me get creative here. We drove through living nightmares to reach the so called paradise.
Don't get me wrong the sun was out, the heat was definitely in our face but this was no lively city. Imagine driving from the airport in the dark passing street markets, dogs and passing poverty. We drove on a back road in a small rental car with at 10kms over endless potholes and no street lights – good one google maps. We reached our accommodation one hour later – it should have taken you 30mins.
Our accommodation was right on the beach shore – the evident crashing of waves made the situation a little better. We could not wait to wake up in the morning to watch the sunset whilst lying on the beach.
We woke at 2pm, 4pm and 6am- the heat was not our cup of tea. Our baked beans and cold fried eggs awaits us on our official wake at 9am. What a fantastic start to our paradise.
The next few days were much the same. We were hot. We were hungry. We were tired.
We drove for a hour a day passing houses with no windows and sheets for doors. We had only just missed some street dogs that own the roads- there were some that would not move and the way they engage eye contact with you truly speaks a desire for death. Many people stand at markets looking to get food at a reasonable price.
Now I could journal all the exciting things we did while in Tonga and believe me we saw some beautiful beaches, travelled to some beautiful islands and swam in caves but I will stop there. Tonga, like many of the cook islands and other countries around the world open your eyes to reality beyond our own. This year I have bought a new phone, laptop and Ipad so I can be up with new technology and in the know. What was going to happen if I did not get any of those new gadgets, probably nothing. Why did I? Because I know no different. The media tell us too. Our friends tell us to. Most importantly we tell ourselves we have to. My highest moments this year revolve around new toys such as the above. My lowest was realising that the people I saw in Tonga were never going to know how wealthy we live in countries like New Zealand. They live each day with what they have and in the conditions they are.
No one is equal. No one is ever always happy. We are who we can be.
Next postFuck it, we're eloping.
Hey guys! This is such an interesting post. I've never heard of this place, but I saw some of this in places as mainstream as the Dominican Republic. The tourist areas are beautiful, but the rest of the island is completely impoverished. Very sad, but makes you think a lot about how good we have life in countries like New Zealand or the USA.
On another note- I really like your blog! You're doing a cool thing by doing it together. That's an edge that a lot of other twenty-something blogs don't have. Best of luck, I'm following!
Hey Niicolee00 – thank you for visiting and for your kind words! Tonga was an eye opener, that's for sure. We are yet to venture that far away from NZ but it's interesting to hear that our experience isn't exclusive to Tonga.
I'm going to pop over to your blog now for a nosey!


These folks know wheel chairs! I needed 2 springs for my daughter 1989 Quickee RX and Dave knew exactly what I needed. I also wanted HD inner tubes which they have in stock. Good folks to deal with.
Excellent service and delivery for very fine products I used to repair a wheelchair, which I will donate to Catholic Chairities.
A great buy and a speedy delivery.
Thank you for your genuine concern in providing exceptional customer service and products.