text:
\section{Introduction}
Over the past couple of decades, $\ell_1$ regularization techniques such as total variation have become increasingly popular methods for image and signal denoising and reconstruction problems.  Along with TV \cite{ROF}, a large variety of approaches for similar $\ell_1$ regularization approaches have been proposed for an array of problems.  {Signal and image recovery methods continue to attract a great deal of interest due to the wide variety of potential applications and ever increasing means of various sensing mechanisms to acquire data.} To name a few, synthetic aperture radar {(SAR)} \cite{wei2010sparse,bhattacharya2007fast}, {magnetic resonance imaging (MRI) \cite{lustig2007sparse,1257394,4587391},} electron tomography\cite{Leary,SGP-ET}, and inpainting \cite{sanders2017subsampling,king2013image} are all image recovery {applications} that have advanced in part due to $\ell_1$ regularization methods, and in each case the approach can be tailored to the challenges that the particular application poses. With many problems such as MRI and electron tomography, the challenge is often to acquire as little data as necessary due to possible damage of the subject being imaged or because of time constraints, driving the need for inverse methods that can achieve the absolute best results from very limited and noisy data.

The mathematical description of the general problem we are interested in is to recover a signal or image $f\in \mathbb R^N$, from noisy measurements $b$ of the form $b = Af+\epsilon$, where $A\in \mathbb R^{m \times N}$ is some sensing matrix that approximates the physical model of the particular problem.  Then the $\ell_1$ regularized solution is given by
\begin{equation}\label{gen-l1}
f_{rec} = \arg \min_f \Big\{ \| Af - b \|_2^2 + \lambda \| T f \|_1 \Big\} ,
\end{equation}
where $T$ is some sparsifying linear transform and $\lambda$ is a parameter that balances the effects of the data and regularization terms.  The appropriateness of this approach is that some prior knowledge of the signal suggests that $Tf$ is sparse, and that the formulation with the $\ell_1$ norm encourages such sparsity \cite{eldar2012compressed,CSincoherence,candes2006robust}.  In many applications, some knowledge of the appropriate transform is available, particularly with images and for other signals, this knowledge is in the form of some ``smoothness.''

In the case of TV, the sparsifying transform is given by $T : \mathbb R^N \rightarrow \mathbb R^{N-1}$, where $(Tf)_i = f_{i+1} - f_i$.  The general idea for this approach is that the signal $f$ is assumed to be piecewise constant with a few discontinuities, in which case $Tf$ is sparse.  If this is not precisely true, this approach still effectively reduces unwanted oscillations at the cost of the well documented stair-casing effect \cite{HOTV,blomgren1997total}.  However, for more {general piecewise smooth functions} higher order TV (HOTV) regularization methods are effective \cite{HOTV,TGV,hu2012higher}, and they do not suffer from the stair-casing effects.  In this case the transform maps $f$ to approximations of discrete derivatives of $f$, e.g. higher order finite differences of $f$.

Another popular choice for $T$ are wavelet transforms \cite{starck2010sparse,mallat2008wavelet,lustig2007sparse}.  For instance, such a transform can be written as $T : \mathbb R^N \rightarrow \mathbb R^{N}$, where $(Tf)_j = \langle f , \psi_j \rangle$ and $\psi_j$  are orthonormal so that $f = \sum_j \langle f , \psi_j \rangle \psi_j$.  The idea here is that for appropriately smooth signals, most of the signal's energy is captured in the few low frequency, larger scaled elements of the basis.  Thus most of the coefficients can be neglected, and thus a sparse approximation of $f$ exists with respect to the basis.

\subsection{Discussion and Contribution}

The crux of general $\ell_1$ regularization methods is that recovering a signal with the most sparse representation, that is recovering the solution with the smallest so called $\ell_0$ norm, is often equivalent to its convexly relaxed variant of recovering the signal with the smallest $\ell_1$ norm, which is a field of study called compressed sensing (CS) \cite{eldar2012compressed,CSincoherence,candes2006robust}.  Although convex $\ell_1$ optimization algorithms are useful in promoting sparsity, some small nonzero coefficients may still persist, an obvious sign that the assumptions needed for the exactness guarantees given by CS theory sometimes do not hold in practice.  This observation is largely the original motivation our present work in developing a multiscale HOTV approach related to multiscale wavelet regularization.

Much work has been devoted to understanding and developing sparsity promoting regularization methods, which are related to our current work.  Numerous variants of higher order TV methods have been proposed \cite{HOTV,Archibald2015,hu2012higher}.  For example, in \cite{Archibald2015} the authors propose an edge detection operator that annihilates polynomials, which leads them to operators close to finite difference matrices. In \cite{HOTV} a combination of a TV regularizer with a quadratic second order regularizer is developed in the continuous domain to eliminate staircasing effects.  Likewise, several authors have shown that using some combination of first and second order methods to be beneficial \cite{VOTV,TGV,setzer2011infimal,chambolle1997image}.  Unfortunately, since there are multiple regularization terms these methods typically introduce additional parameters that need to be tuned.  In terms of theory, it has been well documented that under certain conditions TV and HOTV are equivalent to reconstruction with splines \cite{unser2017splines,steidl2006splines}, i.e. the solution of such methods recovers a piecewise polynomial with a sparse set of jumps.

TV denoising in particular has several very interesting equivalences.  It is well known that TV denoising and other more general first order denoising methods are equivalent to smoothing with a certain nonlinear diffusion models\cite{scherzer2000relations}, a typical result of writing the equivalent Euler-Lagrange equations.  Perhaps discussed less frequently and most related to the observations in our current work, TV denoising is equivalent to soft threshold denoising with the highest frequency basis elements of the Haar wavelets \cite{steidl2002relations,steidl2004equivalence}, in particular with the so called cycle spinning \cite{kamilov2012wavelet}.  In general however, the main difference between these methods is that with TV the smoothness analysis is limited to the finest scales, whereas wavelet regularizations promote function smoothness at multiple scales.  A main contribution of this article is to expand further on the relationship between wavelets in $\ell_1$ regularization and those $\ell_1$ methods related to HOTV.  In regards to extension of wavelets, a number of multidimensional generalizations have been invented including curvelets and shearlets  \cite{guo2007optimally,kutyniok2012shearlets,starck2010sparse}, which are primarily used for sparse function approximation and improve the approximation rates in two and three-dimensions compared with their one-dimensional counterparts.

The method we develop here an alternative for HOTV regularization which we refer to as multiscale HOTV (MHOTV).  In contrast to previous work, our approach considers combining both a multiscale approach and higher order TV methods for the class of image reconstruction problems.  The motivation for such an approach is in observable sub par results due to the relaxation of the sparsity promotion through the $\ell_1$ norm, contrary to the aforementioned results with splines \cite{unser2017splines,steidl2006splines}.  In light of this, we determined this calls for analysis of the function behavior at multiple scales.  As can be deduced, this multiscale {strategy} is similar to the treatment of wavelets, and we argue that our approach is indeed related to the use of Daubechies wavelets, with the main divergence coming in the orthogonality and/or frame conditions prescribed by the wavelets.  {Orthogonality} may be unnecessary for general $\ell_1$ regularization techniques, although fundamental to thresholding denoising techniques, and the relaxation of this condition in our approach allows for better localization of the {transform}.  In the development of MHOTV, we carefully address the computational concerns associated with our approach through the use of both the FFT and operator decompositions.  We are able to show through several numerical examples that MHOTV provides a notable improvement to the current alternatives.

The organization of the remainder of the article is as follows.  In section \ref{sec3} we define the HOTV operators and the corresponding multiscale generalizations.  We also motivate the approach via a numerical example, and make the connection with Daubechies wavelets.  In section \ref{sec4} we precisely define the MHOTV $\ell_1$ regularization model and give precise normalizations to deal with proper parameter selection.  In section \ref{sec5} we address the computational concerns associated with calculating MHOTV coefficients, devising two distinct ways that they can be calculated in an efficient manner.  In section \ref{sec6} we provide numerical results for 1-D and 2-D reconstruction problems, showing that MHOTV is an improvement to the original HOTV and the related Daubechies wavelets.  Some proofs and definitions are provided in the appendix.

\section{HOTV and Multiscale Generalizations}\label{sec3}
As an alternative to TV regularization, {general} order TV methods have been shown to be effective for $\ell_1$ regularization \cite{HOTV,TGV,SGP-ET,Archibald2015}.  The TV transform can be thought of as a finite difference approximation of the first derivative, thus annihilating a function in locations where the function is a constant, i.e. a polynomial of degree zero.  Likewise, higher order TV transforms can be considered higher order finite difference approximations to higher order derivatives, thus annihilating {{the corresponding} degree polynomials.  With this in mind, we have the following definition:

\begin{comment}
\begin{definition}[Finite differences]
Let $\phi_j \in \mathbb R^N$ for $j=1,2,\dots , N$ be given by
\begin{equation}\label{Svector}
(\phi_1)_i = \begin{cases}
(-1)^{k+i-1} {k \choose i-1} & \mbox{if } i\le k+1\\
0 & \mbox{if } i > k+1
\end{cases},
\quad \text{and for } \, j>1, ~
(\phi_j)_i = (\phi_1)_{mod(i-j,N)},
\end{equation}
and let
\begin{equation}\label{Smatrix}
 \Phi_k^T = \left(
 \begin{array}{c}
 \phi_1 ,
 \phi_2 ,
 \dots
 \phi_N
 \end{array}\right) .
\end{equation}
Then the periodic $k^{th}$ order finite difference operator is given by $\Phi_k$.
\end{definition}
\end{comment}

\begin{definition}[Finite Differences]
Let $\phi_k \in \mathbb R^N$ be defined by
\begin{equation}
(\phi_k)_m = \begin{cases}
(-1)^{k} & \mbox{if } \, m=0\\
0 & \mbox{if  } \, 1\le m<N-k \\
(-1)^{k+m+N} {k \choose N-m} & \mbox{if }\, N-k \le m < N
\end{cases}.
\end{equation}
Then for $f\in\mathbb R^N$, the periodic $k^{th}$ order finite difference of $f$ is given by
$$
f*\phi_k,
$$
where $*$ denotes the discrete convolution.
\end{definition}

\begin{remark}
The convolution in this definition (and in general) can be represented by multiplication with a circulant matrix $\Phi_k$, where each row of $\Phi_k$ holds a shifted version of $\phi_k$.  An example of the matrix in the case $k=2$ is given in (\ref{Phi_2}).  Note that this definition uses a periodic extension of $f$ and can be ignored by dropping the last $k$ rows of $\Phi_k$.
\end{remark}
\begin{equation}\label{Phi_2}
\Phi_2 = \left( \begin{array}{c c c c c c}
               1 & -2 & 1 & 0 &  \dots & 0\\
               0 & 1 & -2 & 1 & \dots & 0 \\
               0 & 0 & 1 & -2 & \dots & 0 \\
               \vdots &   & \ddots  &  & \dots & \vdots \\
               1 & 0 & \dots & & 1 & -2\\
               -2 & 1 & \dots & & 0 & 1
              \end{array}\right).
\end{equation}

With this definition, the HOTV model can be said to recover
\begin{equation}\label{HOTV-model}
 f_{rec} = \arg \min_f \Big\{\|Af - b\|_2^2 + \lambda \|\Phi_k f\|_1 \Big\}.
\end{equation}

Unfortunately for many real world imaging problems the equivalence between $\ell_1$ and $\ell_0$ may not hold in practice, yet the $\ell_1$ regularization still tends to encourage favorable solutions.  In terms of the sparsity promoting transform, this means that the transform of the recovered function may not be truly sparse, but most of the values are instead relatively close to zero.  For HOTV, this means that a local Taylor expansion of the recovered function will still contain some small nonzero higher order coefficients, yet essentially unobservable at the very local scale.  In other words, at some point $t$, there exists a polynomial expansion of minimal degree of $f$ given by
\begin{equation}\label{taylor}
 f(x) {\approx} \sum_{m=0}^M \alpha_m(t) \frac{(x-t)^m}{m!} ,
\end{equation}
which holds for all $x$ within some small interval $I$ around the point $t$.  Ideally a solution given by the order $k$ HOTV model recovers a solution so that the coefficients $\alpha_m(t)$ vanish for $m\ge k$.  The $\ell_1$ model allows for these coefficients to remain, although very small, and the function still \emph{appears} to essentially be a polynomial of degree less than $k$.
However, when this behavior persists over many points at a larger scale, the result can be a function which looks more like a trigonometric polynomial rather than an algebraic one.

This phenomenon is demonstrated in Figure \ref{fig1}, where a piecewise polynomial of degree two was reconstructed from random noisy samples with 50\% sampling\footnote{The number of samples is half the number of grid points.} using TV and HOTV regularizations.  The sampling matrix $A\in \mathbb R^{N/2 \times N}$ is constructed so that a random 10\% of its entries are set to be nonzero, where these nonzero values are uniformly distributed over $[0,1]$.  The samples were corrupted with normaly distributed mean zero noise.  Two different grid sizes are demonstrated, 256 and 1024, and it can be observed that these small oscillations become increasingly abundant with more grid points.  However, in the bottom of the figure, the third order finite difference of the HOTV3 solution plotted in logarithmic scale shows that locally this oscillatory behavior {results in almost exact low order polynomials, although \emph{very} small amplitudes persist in the transformed domain and} thus not truly sparse in the $\ell_0$ sense.  Nevertheless, all regularization approaches should still be deemed useful, as evidenced by the least squares solution shown.

\begin{figure}
 \centering
 \includegraphics[width=0.5\textwidth]{fig1-2.pdf}
 \caption{Rows 2-4 and 7-9 reconstruction of a piecewise polynomial function of degree two shown in the top row over 256 (top 5 plots) and 1024 (bottom 5 plots) {points} from random sampling at 50\%.  {The corresponding least squares solution is shown in the fifth plot, and the 3rd order finite difference of the HOTV3 solution over the 1024 grid is shown on the bottom.}}
 \label{fig1}
\end{figure}

Due to this phenomena we propose a multiscale HOTV approach, which considers the regularization transform at multiple scales.  The idea is that a larger stencil would penalize these oscillations even with the $\ell_1$ norm.  As TV generalizes to the Haar wavelet by stretching and scaling of the elements, we propose the same with HOTV.  {To this end we} give the following definition.

\begin{comment}
\begin{definition}[Multi-scale finite differences]
Let $\phi_j^m \in \mathbb R^N$ for $j=1,2,\dots, N$, be given by
\begin{equation}\label{Mvector}
(\phi_1^m)_i = \begin{cases}
(-1)^{k+\lceil \frac{i}{m} \rceil-1} {k \choose \lceil \frac{i}{m} \rceil -1} & \mbox{if } i \le \frac{k+1}{m} \\
0 & \mbox{if } i > \frac{k+1}{m}
\end{cases},
\quad \text{and for } \, j>1, ~
(\phi_j)_i = (\phi_1)_{mod(i-j,N)},
\end{equation}
and let
\begin{equation}\label{Mmatrix}
 \Phi_{k,m}^T = \left(
 \begin{array}{c}
 \phi_1^m,
 \phi_2^m ,
 \dots
 \phi_N^m
 \end{array}\right) .
\end{equation}
Then the periodic $k^{th}$ order finite difference operator of scale $m$ is given by $\Phi$.
\end{definition}
\end{comment}
\begin{definition}[Multiscale Finite Differences]
Let $\phi_{k,j} \in \mathbb R^N$ be defined by
\begin{align}
& (\phi_{k,j})_m =\label{MHOFD-vect} \\
& \begin{cases}
(-1)^{k} & \mbox{if } \, m=0\\
0 & \mbox{if }  \, 1 \le m \le  N-j(k+1)\\
(-1)^{k+\lfloor \frac{N-m}{j} \rfloor} {k \choose \lfloor \frac{N-m}{j} \rfloor}
& \mbox{if } \, N-j(k+1) < m < N
\end{cases}. \nonumber
\end{align}
Then for $f\in \mathbb R^N$, the periodic $k^{th}$ order finite difference of scale $j$ of $f$ is given by
$$
f*\phi_{k,j},
$$
where $*$ denotes the discrete convolution.
\end{definition}

\begin{remark}
Again, this convolution can be represented as multiplication with a circulant matrix $\Phi_{k,j}$.  An example of $\Phi_{k,j}$ in the case $k=2$ and $j=2$ is given in (\ref{Phi_22}).
\end{remark}
\begin{equation}\label{Phi_22}
\Phi_{2,2} = \left( \begin{array}{c c c c c c c c c}
               1 & 1 & -2 & -2 & 1 & 1 & 0 & \dots & 0\\
               0 & 1 & 1 & -2 & -2 & 1 & 1 & \dots & 0 \\
               0 & 0 & 1 & 1 & -2 & -2 & 1 & \dots & 0 \\
               \vdots &   &  & \ddots &  && &\dots &  \vdots \\
               -2 & -2 & 1 & 1 & 0& 0 &  \dots & 1 & 1\\
               1 & -2 &-2 & 1 & 1 & 0 & \dots & 0 & 1
              \end{array}\right).
\end{equation}

\subsection{MHOTV Reconstruction Model}\label{sec4}
We now present the general model for MHOTV reconstruction.  Generally speaking, we still use the model presented in (\ref{gen-l1}), where $A$ maps the unknown function $f$ to some perhaps noisy measurements given by $b$, from which we use to reconstruct $f$.  Our sparsity promoting transforms are now given by the matrices $\Phi_{k,2^j}$, for $j=0,1,\dots, \ell$, where $\ell$ is the maximum scaling of the operator used and $k$ is the chosen order.  Setting our maximum scaling to $\ell=0$ corresponds to the traditional HOTV approach.  Although not completely necessary, we choose a dyadic scaling of the operators, similar to the treatment of wavelets.  As with wavelets, we will show that this is convenient for computational purposes.  Finally then our reconstruction model is given by
\begin{equation}\label{MHOTV-model}
 f_{rec} = \arg \min_f \Big\{ \| A f - b\|_2^2 + \frac{\lambda}{\ell+1} \sum_{j=0}^\ell 2^{-(j+k-1)} \| \Phi_{k,2^j} f \|_1\Big\}.
\end{equation}
The factor of $2^{-j}$ is a normalization that accounts for the increasing norms of each operator, which would otherwise weigh too heavily to the largest scaling operator \footnote{This is akin to the dyadic scaling of the wavelet basis elements after the dyadic stretching.}.  The scaling of the parameter $\lambda$ by $\ell+1$ simplifies the selection of the parameter, which would otherwise need to be manually scaled by such a factor to account for the number of scales being used. By similar reasoning, the additional scaling by $2^{-k+1}$ is used to account for the order $k$ of the method \cite{sanders2018parameter}.

\subsection{Relationship to Daubechies Wavelets}
\begin{comment}
\begin{figure}
\begin{myblock}{Characterization of Daubechies Wavelet Transforms}
\footnotesize
\begin{itemize}
\setlength\itemsep{0em}
 \item The elements are generated by shifting, dyadic stretching, and scaling of a single element.
 \item They are specified by a number of vanishing moments (i.e. the highest order polynomial to which they are orthogonal).
 \item Uniqueness of the elements is given by minimizing their support.
 \item They form an orthonormal basis, or in the case of a redundant system, they form a tight frame.
\end{itemize}
\end{myblock}
\begin{myblock}{Characterization of MHOTV Operators}
\footnotesize
\begin{itemize}
\setlength\itemsep{0em}
 \item The elements are generated by shifting and dyadic stretching of a single element.
 \item They are specified by the highest order polynomial to which they are orthogonal (i.e. the number of vanishing moments).
 \item Uniqueness of the elements is given by minimizing their support.
\end{itemize}
\end{myblock}
\caption{Summary of the relationship between the two tranforms.  Notice the first three bullet points characterizing Daubechies wavelets and MHOTV are essentially equivalent.}
\label{newfig}
\end{figure}
\end{comment}

Wavelets can be characterized as an orthonormal basis that is generated through a multiresolutional analysis \cite{daubechies1992ten,mallat2008wavelet}.  The multiresolutional analysis leads to the shifting and dyadic stretching and scaling of a single generating mother wavelet, analogous to our treatment of MHOTV by shifting and stretching of a single row or element of the matrices $\Phi_k$.
From this very general characterization, there are a number of parameters in the design of the wavelets.  {For Daubechies wavelets} the smoothness is characterized by the number of vanishing moments, i.e. the number of polynomial orders to which the wavelet is orthogonal.  A wavelet with $k$ vanishing moments acts as a multiscale differential operator of order $k$.  As a trade off, an increasing number of vanishing moments chosen for the wavelet basis results in an increase in the support of the wavelet functions, and Daubechies wavelets are designed to yield the orthonormal wavelet basis of minimal support given a selected number of vanishing moments \cite{mallat2008wavelet}.

\begin{comment}
{To develop a basic mathematical formulation of wavelet expansion, let us suppose we would like a orthonormal wavelet basis for a subset of functions on $L_2[0,1]$. In the interest of digitization, let's take the space of functions to be \emph{uniform pixelated} functions of the form
\begin{equation}
f(x) = \sum_{j=0}^{N-1} c_j \mathbbm{1}_{ I_j } (x) , \quad \text{where} \quad I_j = \Big[ \frac{j}{N},\frac{j+1}{N} \Big).
\end{equation}
\end{comment}

To develop a basic mathematical description of a wavelet expansion, suppose we want to represent a \emph{uniform pixelated} function with $2^n$ pixels on $[0,1]$ in terms of the wavelet basis.
Then denoting our scaling function and mother wavelet with $k$ vanishing moments by $\varphi_k$ and $\psi_k$ respectively, we have the following orthonormal wavelet representation
\begin{equation}\label{wave-eq}
f = \sum_{t=0}^{2^{\ell} - 1} \langle f , \varphi_{k,\ell,t} \rangle \varphi_{k,\ell,t}
	+ \sum_{j=\ell}^{n-1} \sum_{t=0}^{2^{j}-1} \langle f , \psi_{k,j,t} \rangle \psi_{k,j,t} .
\end{equation}
Here,
$
\psi_{k,j,t}(x) = 2^{j/2} \psi_k \left(2^j x - t \right)$
and similarly for $\varphi_{k,j,t}$,
i.e. shrinking and scaling of the of the generating wavelet functions.  The parameter $\ell$ is a positive integer with $0\le \ell \le n$, and the value
$n-\ell$ is said to be the number of \emph{scales} in the wavelet expansion \footnote{For $\ell=n$ it is understood that the second sum is removed.}.
With the representation in (\ref{wave-eq}), the coefficients for the scaling functions in the first sum capture most of the energy of the signal, and the wavelet coefficients $c_{k,j,t} = \langle f,\psi_{k,j,t} \rangle$ \emph{vanish} for values of $t$ where $f$ is a polynomial of degree $k-1$ over the support of $\psi_{k,j,t}$.  {For $\ell_1$ regularization, we only need to be concerned with regularization of the wavelet coefficients in (\ref{wave-eq}), and thus the coefficients for the scaling functions in the first sum are not included in the regularization.
}

\begin{figure}
\includegraphics[width=.5\textwidth]{db-3levels-3orders.png}
\includegraphics[width=.5\textwidth]{mhotv-3levels-3orders.png}
\caption{The filters in Fourier space of wavelet and MHOTV convolution functions.}
\label{fig: filters}
\end{figure}

Connecting  these ideas to HOTV, we see that these {transforms} are playing similar roles.  Both are prescribed by the number of vanishing moments, or in the language of HOTV, the highest order polynomial that is annihilated by the approach.  Furthermore, both are designed to yield minimal support given the number of vanishing moments.  The crucial difference lies in the orthogonality condition prescribed by wavelets, which further increases the support of the wavelet elements.  We emphasize again, that this condition is fundamental to compression and threshold denoising methods, but not necessarily useful with general image reconstruction problems.

Finally, one additional technique utilized for $\ell_1$ regularization and denoising as well is the use a wavelet frame by taking all possible shifts for each scaling of the wavelets, which is sometimes referred to as translational invariant cycle spinning \cite{spinning,temizel2005wavelet,kamilov2012wavelet}.  This eliminates the lack of translation invariance of a wavelet basis that can otherwise result in unwanted artifacts near discontinuities.  With this in mind, we may define the wavelet frame elements by
$$
\tilde \psi_{k,j,t}(x)  = 2^{j/2} \psi_k \left( 2^j (x - t2^{-n})\right), \quad t = 0 , 1 , \dots, 2^{n-1}.
$$
Then the \emph{averaged} wavelet frame representation of $f$ may be written as
\begin{align*}\label{wave-eq-frame}
f & = \sum_{t=0}^{2^{\ell} - 1} \langle f , \varphi_{k,\ell,t} \rangle \varphi_{k,\ell,t}
	+ \sum_{j=\ell}^{n-1}2^{j-n} \sum_{t=0}^{2^{n}-1} \langle f , \tilde \psi_{k,j,t} \rangle \tilde \psi_{k,j,t} \\
	& = \sum_{t=0}^{2^{\ell} - 1} \langle f , \varphi_{k,\ell,t} \rangle \varphi_{k,\ell,t}
	+ \sum_{j= \ell}^{n-1} 2^{j-n} \Psi_{k,j}^T (f * \psi_{k,j,0}(-x)),
\end{align*}
where $\Psi_{k,j}^T = (\tilde \psi_{k,j,0}, \tilde \psi_{k,j,1}, \dots , \tilde \psi_{k,j,2^n - 1})$.  Hence a wavelet approach promotes sparsity with respect to the vectors $f*\psi_{k,j,0}$, or equivalently with respect to $\Psi_{k,j}f $.  Then a regularization norm in this setting takes the form
\begin{equation}\label{wave-reg-norm}
\sum_{j=\ell}^{n-1} \| \Psi_{k,j} f \|_1,
\end{equation}
which is analogous to our regularization norm in (\ref{MHOTV-model}).  For wavelets, the scalings are inherent to function definitions, and the dyadic stretching of the elements is indicated by $j$ as opposed to $2^j$.

The case when $\ell=n-1$ would be most closely related to the original HOTV, and for smaller values of $\ell$ the wavelets are more comparable to the MHOTV development in this article.}

Since computing both MHOTV operators and wavelets coefficients are convolutional operations, we may visualize their corresponding filters in Fourier space, providing us another basis for comparison, which we have done in Figure \ref{fig: filters}.  Each of these can be interpreted as high pass filters, where the higher levels pass increasingly lower frequencies.  A very close similarity of the wavelet filters and MHOTV filters can be observed in Figure \ref{fig: filters}, providing a strong visual confirmation to our preceding discussion on the close relationship between the two.

\section{Fast Calculation of MHOTV Operators} \label{sec5}
Calculation of traditional HOTV coefficients is a computationally inexpensive task, due to the sparsity of the matrix operator.  However, with increasing dyadic scales the direct calculation increases exponentially.  Due to this, in the proceeding section we develop two distinct approaches that show that these calculations can be carried out with linear increase in the flop count with respect to the number of scales used.

Fast computation of standard HOTV can be done in several ways.  One can construct the sparse matrix $\Phi_k$ and perform matrix computations directly, a calculation with runtime of $kN$ flops.  One could make use of other {procedures}, such as MATLAB's ``diff" command which  {requires the same} flop {count} without storing the matrix.  With MHOTV, these approaches become less appealing.  With matrix construction, if one is using several scales, then several matrices need to be computed and stored, and the matrices become significantly less sparse for larger scales.  The``diff" command cannot be implemented directly for larger scale HOTV operators.

Another alternative is to use the Fourier convolution theorem to perform the convolution {operation} via a product in Fourier space.  For the traditional HOTV operators, this can be fairly slow compared with the matrix and ``diff" approach, since the necessary two discrete Fourier transforms would require $\sim 2 N\log_2 N$ flops compared with the $kN$ flops for the alternative {implementations}.  However, this {method} is relatively comparable for MHOTV, since the Fourier transforms only need to be computed once to determine the coefficients at all scales.

We outline two {procedures} for efficient calculation of MHOTV.  First, we describe the Fourier approach, where we derive precise formulas for the MHOTV Fourier filters.  Second, we describe an alternative efficient approach by decomposition of the MHOTV matrix operators.

\subsection{Computation via Fourier Transforms}
By the Fourier convolution theorem, the MHOTV operators can be computed {as multiplications} in Fourier space, i.e.
\begin{equation}
 f*\phi_{k,j} = F^{-1} \left( F(f)\cdot F(\phi_{k,j})\right) ,
\end{equation}
where $F$ denotes the discrete Fourier transform.  Although this can be numerically computed, it is a convenient to have an exact formula for the discrete Fourier transform of $\phi_{k,j}$.  Moreover, analytic determination of $F(\phi_{k,j})$ allows us to generalize the MHOTV to fractional orders.
\begin{proposition}
The DFT of the vector $\phi_{k,j}$ defined in (\ref{MHOFD-vect}) has an explicit expression given by
\begin{equation}\label{eq: DFTeq}
 F( \phi_{k,j} )_\xi = \frac{\left( e^{ \frac{i2\pi\xi j}{N}} - 1\right)^{k+1}}{e^{\frac{i2\pi\xi}{N}}-1},
\end{equation}
for $\xi = 0, 1, \dots, N-1$.
\end{proposition}
\begin{proof}
The expression for the $\xi^{th}$ Fourier coefficient in the DFT of $\phi_{k,j}$ is given by
\begin{equation}\label{DFT}
F( \phi_{k,j} )_\xi = \sum_{m=0}^{N-1} (\phi_{k,j})_m e^{\frac{-i2\pi\xi}{N}m}.
\end{equation}
Notice that the terms $1\le m \le N - j(k+1)$ vanish by definition of $\phi_{k,j}$.  For the latter terms, we make the substitution $n=N-m$ and flip the sum to give the expression
\begin{equation}
 F(\phi_{k,j})_\xi = \sum_{n=0}^{j(k+1)-1} (-1)^{k + \lfloor{\frac{n}{j}}\rfloor} {k \choose \lfloor \frac{n}{j} \rfloor } e^{\frac{-i2\pi\xi}{N}(N-n)},
\end{equation}
where the term $n=0$ corresponds to $j=0$ and the following indices $n=1,2,\dots, m(k+1)-1$, correpsond to $j=N-1,N-2, \dots, N-m(k+1)+1$, respectively.
Notice that we may drop the $N$ in the numerator of the exponential and that the values of $\phi_{k,j}$ repeat over strings of length $j$.  Therefore each of these corresponding strings of exponential terms in (\ref{DFT}) get the same weights, leading to the following sum:
\begin{equation}
 F( \phi_{k,j} )_\xi
   = \sum_{m=0}^k \left( (-1)^{m+k}{k\choose m} \left[ \sum_{\ell=0}^{j-1} e^{\frac{i2\pi\xi}{N}(jm+\ell)} \right] \right) .
\end{equation}
Here the inner sum represents the $j$ consecutive terms in (\ref{DFT}) that receive the same weights from $\phi_{k,j}$, namely $(-1)^{m+k}{k\choose m}$.  Switching the order of summation, we recognize the sum over $m$ as a binomial expansion leading to
\begin{align*}
 F( \phi_{k,j} )_\xi  &= \sum_{\ell=0}^{j-1} \sum_{m=0}^k (-1)^{m+k}{k\choose m} e^{\frac{i2\pi\xi}{N}(jm+\ell)}\\
 & =  \sum_{\ell=0}^{j-1} \left( e^{\frac{i2\pi\xi}{N}j} - 1 \right)^k e^{\frac{i2\pi\xi}{N}\ell}.
\end{align*}
The remainder of the proof follows by elementary calculations.
\begin{comment}
 \begin{align*}
  (\hat \varphi_{k,m})_\xi
  & = \sum_{j=0}^k \left( (-1)^{j+k}{k\choose j} \left[ \sum_{\ell=0}^{m-1} e^{\frac{-i2\pi\xi}{N}(jm+\ell)} \right] \right) \\
  & = \sum_{\ell=0}^{m-1} \sum_{j=0}^k (-1)^{j+k}{k\choose j} e^{\frac{-i2\pi\xi}{N}(jm+\ell)} \\
  & = (-1)^k \sum_{\ell=0}^{m-1} \left( e^{\frac{-i2\pi\xi}{N}m} - 1 \right)^k e^{\frac{-i2\pi\xi}{N}\ell} \\
  & = (-1)^k \left( e^{\frac{-i2\pi\xi}{N}m} - 1 \right)^k \frac{e^{\frac{-i2\pi\xi}{N}m} - 1}{e^{\frac{-i2\pi\xi}{N}}-1}\\
  & =  \frac{\left( e^{ \frac{-i2\pi\xi m}{N}} - 1\right)^{k+1}}{e^\frac{-i2\pi\xi}{N}-1}
 \end{align*}
\end{comment}
\end{proof}

\subsection{Fast Computation via Operator Decomposition}
In this section, we give a decomposition for the matrix operator $\Phi_{k,2^j}$ and describe how this decomposition can be used for rapid calculation of MHOTV operators.  The decomposition of $\Phi_{k,2^j}$ is given in the following theorem.

\begin{comment}
\begin{proposition}
 Let the matrix $P$ be defined by $P = |\Phi_1|$, that is all ones on the diagonal, all ones on the upper off diagonal, and one in the $(N,1)$ entry.  Then we have the following relations
 \begin{equation}\label{decomp1}
 \Phi_{k,2} = P^{k+1} \Phi_k = \Phi_k P^{k+1} = P^{k+1} \Phi_1^k.
 \end{equation}
 Moreover we have the following relation between $P$ and $\Phi_k$,
 \begin{equation}
 P^{k} = |\Phi_k|.
 \end{equation}
\end{proposition}
\end{comment}

\begin{theorem}\label{thm1}
 Let the matrix $P_j$ with entries $\{p_{m,n}\}_{m,n=1}^N $ be defined by
 \begin{equation}\label{pvalues}
 p_{m,n} = \begin{cases}
            1 & \mbox{if } m=n\\
            1 & \mbox{if } n= (m+j-1)\bmod{(N)} + 1 \\
            0 & \mbox{if } otherwise
           \end{cases}.
 \end{equation}
Then the following holds:
\begin{enumerate}
\item The entries of $P_j^{k+1}$, which we denote by $\{p_{m,n}(j,k)\}_{m,n=1}^N$, are given by
$$
p_{m,n}(j,k) = \begin{cases}
              {k+1 \choose \ell} &\mbox{if } n = (m+j\ell-1)\bmod({N)}+1\\
              0 &\mbox{if } otherwise
             \end{cases},
$$
where it is implied $\ell$ is an integer satisfying $0\le \ell \le k+1$.
\item  $\Phi_{k,2^j}$ has the decomposition
\begin{equation}\label{decomp2}
 \Phi_{k,2^j} = P_j^{k+1} P_{j-1}^{k+1} \cdots P_1^{k+1} \Phi_k
\end{equation}
and therefore
\begin{equation}\label{decomp3}
 \Phi_{k,2^j} = P_j^{k+1} \Phi_{k,2^{j-1}}.
\end{equation}

\item The equality in (\ref{decomp2}) holds for any rearrangement of the product of matrices.
\end{enumerate}
\end{theorem}
{The proof of this theorem is given in the appendix. The matrices $P_2$ and $P_2^2$ are shown below to illustrate the sparse structure of these operators:}
\begin{align*}\label{pmatrices}
 P_2 &= \left( \begin{array}{cccccc}
        1 & 0 & 1 & 0 & \dots & 0\\
        0 & 1 & 0 & 1 & \dots & 0\\
        \vdots & & \ddots & & \dots &  \vdots\\
        0 & 1 & 0 & \dots & & 1
       \end{array}\right) , \\
P_2^2 &= \left( \begin{array}{ccccccc}
                1 & 0 & 2 & 0 & 1 & \dots & 0\\
                0 & 1 & 0 & 2 & 0 & \dots & 0\\
                \vdots & & \ddots & & & \dots &  \vdots\\
                0 & 2 & 0 & 1 & 0 & \dots & 1
               \end{array}\right).
\end{align*}

\begin{proposition}\label{prop2}
 Direct calculation of $\Phi_{k,2^j}$ requires $2^j Nk$ flops.  The same calculation using the decomposition in (\ref{decomp2}) requires $j N(k+1) + Nk$ flops.  The same calculation using the Fourier method requires $ 2Nlog_2 N + N$.
\end{proposition}
Proposition 2 is a direct result of Theorem \ref{thm1}, the Fourier convolution theorem combined with the FFT, and the flops required for the direct calculation.  We assume that the FFT and inverse FFT can be computed in $N\log_2 N$ flops, although the exact count is somewhat vague, depending on the precise algorithm and if $N$ is a power of 2.  To compute the full set of operators, we can get away with less flops then adding the flops for each level.  If we use the decomposition approach to calculate the operators as determined by (\ref{decomp2}), the associated computations are limited to that at the highest scale, since the intermediate scales are determined in this calculation as pointed out in (\ref{decomp3}).  If we use the Fourier approach for calculating the coefficients in (\ref{MHOTV-model}), only one foward FFT is required for the function $f$.  Then the product of $F(f)$ and $F(\phi_{k,2^j})$ must be computed for each $j$, as well as the inverse FFT for each of these products. The observations lead to the following corollary.

\begin{corollary}
 Let $T$ be the matrix containing the complete set of $\ell +1$ operators involved in the MHOTV $\ell_1$ regularization norm, so that $T^T= [\Phi_{k,1}^T , \Phi_{k,2}^T , \dots , \Phi_{k,2^\ell}^T].$  Then calculating $T$ using the operator decomposition given in Theorem 1 requires $\ell N(k+1) + Nk$ flops.  Calculating $T$ using the Fourier approach requires a total flop count of $(\ell +2) N\log_2 N+(\ell+1) N$.
\end{corollary}

\begin{comment}
\begin{remark}
When concerned with solving (\ref{MHOTV-model}), if we use the decomposition approach to calculate the operators as determined by (\ref{decomp2}), the associated computations are limited to that at the highest scale, since the intermediate scales are determined in this calculation as pointed out in (\ref{decomp3}).  Thus with $\ell +1$ scales, then by Proposition \ref{prop2} the total flop count for computing all $\ell +1$ scales is $\ell N(k+1) + Nk$.
\end{remark}

\begin{remark}
 If we use the Fourier approach for calculating the coefficients in (\ref{MHOTV-model}), only one foward FFT is required for the function $f$.  Then the product of $F(f)$ and $F(\phi_{k,2^j})$ must be computed for each $j$, as well as the inverse FFT for each of these products.  With $\ell +1$ scales, this gives a flop total to $(\ell +2) N\log_2 N+(\ell+1) N$.
\end{remark}
\end{comment}
A few concluding remarks are in order.

\begin{remark}
 All of the results presented are for 1-D signals.  For higher dimensions say 2-D, the operators can be applied along each row and column, and the flop count is only doubled, disregarding the likely increased number of indices.
\end{remark}

\begin{remark}
 To solve (\ref{MHOTV-model}), we use the well establised alternating direction method of multipliers (ADMM) \cite{Li2013,bregman,Zhang}.  This approach introduces splitting variables that allows one to split the objective functional into equivalent subproblems that can be solved relatively fast.  Our algorithm can be downloaded at \cite{toby-web}, and some of the simulations in the proceeding section can also be found there.
\end{remark}

\section{Numerical Experiments}\label{sec6}
\subsection{Repeat of 1-D Simulations}\label{sec: repeat1d}
\begin{figure}
 \centering
 \includegraphics[width=.5\textwidth]{fig2-3.pdf}
 \caption{Reconstruction of a piecewise polynomial function of degree two over 1024 stencil from random sampling at 50\%.  Three scales are used for both the Daubechies wavelets and multiscale HOTV.}
 \label{fig2}
\end{figure}
{To compare MHOTV and wavelet regularized reconstructions we repeat the numerical examples presented in Figure \ref{fig1} with the same noisy data used for the HOTV reconstruction. The corresponding reconstruction with MHOTV and wavelets are presented in Figure~\ref{fig2}.} Recall that the measurements were collected at a 50\% sampling rate and corrupted with normally distributed mean zero noise.  For the multiscale HOTV and wavelets, three scaling levels were used.  The selection of the regularization parameter $\lambda$ was set to the same value for each order for HOTV and the wavelets, where we used a similar normalization approach for the wavelets coefficients as presented in (\ref{MHOTV-model}).

{The results in Figure \ref{fig2} were generated with orders 1, 2, and 3.} The order is indicated with the numbers next to the approach in the legends, e.g. we denote the order $k=3$ MHOTV approach with MHOTV3. For a baseline comparison, the least squares solution is shown as well.  Compared with the corresponding $1024$ reconstructions from HOTV in Figure \ref{fig1}, these solutions show clear improvements, particularly with the higher orders.  As we expect, although the MHOTV1 and Haar wavelet coefficients are computed in a different manner, the resulting reconstruction are identical since the models are theoretically equivalent.  They both exhibit the staircasing and noise effects in precisely the same locations.  The higher order approaches also show many similar effects of the noisy features, exhibiting certain oscillatory features with the same general behavior in precisely the same locations.  However, with the higher orders, these approaches are not equivalent and MHOTV provides regulatory information at finer scales due to the minimal support of the transform elements.  The result appears to be a modest improvement in the resulting reconstructions.

Finally, in the bottom of the figure the third order finite difference of the MHOTV3 solution is plotted in logarithmic scale.  Comparing this with the original HOTV3 finite difference in Figure \ref{fig1}, we observe that the solution exhibits much better sparsity with respect to this transform domain, as desired.

\subsection{2-D Tomographic Simulations}
In this section we investigate the regularization methods on the common 2-D tomographic image reconstruction problem \cite{Natterer2}.  The phantom test image is shown in Figure \ref{fig3} (a).  The data generated for tomography are 1-D line integrals of the image, well-known as Radon transform data.  Formally, the Radon transform of a 2-D function or image $f$ is defined as
\begin{equation}\label{Radon}
 R f (t,\theta) = \int_\Omega f(x,y) \delta(t-(x,y)\cdot(\cos \theta ,\sin \theta)) \, dx \, dy,
\end{equation}
where $\Omega$ is the image domain and $\delta$ is the Dirac delta function.  As in many applications, the data collected for reconstruction are of the form known as parallel beam geometry.  In this setting, the full knowledge of noisy $Rf(t,\theta)$ is known for some finite set of angles, $\theta.$\footnote{There is also a discretization over $t$, but it is small enough to ignore.}  In this numerical experiment, we use a total of 29 angles that are equally distributed across the full 180$\degree$ angular range, which are visualized as a sinogram in Figure \ref{fig3} (b).  Such a limited set of data is sometimes referred to as \emph{limited data} tomography.  Mean zero normally distributed noise was again added to the data values.  Classically tomographic reconstruction from parallel beam geometry can be done by first transforming the data into Fourier space by the Fourier slice theorem, and then applying a chosen ramp filter to this data and the inverse Fourier transform.  This direct approach, called filtered backprojection, is sensitive to noise and is shown in Figure \ref{fig3} (c).

The problem can however be discretized and approximated by a set of linear equations $Af = b$ (see for instance \cite{sanders2015image} on pages 8-9 within section 1.5.), where $A$ is sparse matrix that is a discretized approximation of the Radon transform, $f$ is the vectorized image, and $b$ is a vector holding the data values.  With this set up we can apply regularization models such as (\ref{gen-l1}) and (\ref{MHOTV-model}).  We use a $512\times 512$ pixelated mesh for the image domain in this experiment.  The results for applying these models with HOTV, MHOTV, and Daubechies wavelets all at orders one and three are shown in Figure \ref{fig4}.  Each of the models are also supplemented with a nonnegativity constraint, $f\ge 0$, which is carried out with a projected gradient method.  A baseline comparison obtained by a conjugate gradient least squares solver is also shown in the figure.  To ensure accurate comparison between the methods by appropriate parameter selection and algorithmic convergence, the relative data errors defined by $\| Af - b \|_2 / \| b\|_2$ are shown in the figure, and it confirms that each approach approximately fit the data equally well, with all of the errors contained within an interval of size 0.0129.

As has been observed previously \cite{SGP-ET}, due to a number of reasons including undersampling, noise, fine details between the image features, and nature of the regularization, the order 1 solutions (TV) can leave the fine features under resolved, even though the underlying image is truly a piecewise constant that classical TV was originally designed to recover.  Each of these order 1 images appear relatively similar, with the MHOTV and Daubechies approaches showing modest improvements in resolving some features.  As in the 1-D case, the HOTV3 solution exhibits some small local oscillations that appear as noise in the image.  However, this image, as well as the other order 3 approaches resolve the features notably more clear than the order 1 approaches.  Both of the order 3 multiscale approaches appear less noisy than the HOTV order 3, while still maintaining a good approximation of the image features.

\begin{figure}
 \centering
 \includegraphics[width=0.5\textwidth]{fig3-2.pdf}
 \caption{(a) Phantom image.  (b) Noisy tomographic data in sinogram format, 29 projections in total.  (c) Classical filtered backprojection reconstruction from data.}
 \label{fig3}
\end{figure}

\begin{figure*}
 \centering
 \includegraphics[scale=.8]{fig4.pdf}
 \caption{Reconstructions of phantom image from 29 tomographic projections.  Orders 1 and 3 are shown for the regularization approaches.  Top left: relative data fitting error from each approach shows approximately equivalent data fitting from each approach.  Top right: least squares solution for baseline comparison.}
 \label{fig4}
\end{figure*}

\begin{table*}
\begin{center}
\begin{tabular}{r | c | c | c | c | c | c |}
            SNR   & & Order 1 & Order 2 & Order 3 & Order 1.5 & Order 2.5 \\ \hline
      &          &mhotv \vline\,Daub&mhotv \vline \,Daub&mhotv \vline \,Daub&mhotv& mhotv\\
             \hline
      & 1 level  &  .1624, .1624  & .2039, .1961   & .2464, .2306   &  .1819   & .2328 \\
 2    & 2 levels &  .1612, .1617 & .1742, .1852   & .2135, .2223   &  .1782    & .2183 \\
      & 3 levels &  .1699, .1615  & \textbf{.1513}, .1778   & .1745, .2149   &  .1776    & .1975 \\
      & 4 levels &  .2001, .1647 & .1584, .1745   & .1764, .2104   &  .2031    & .2102 \\
      \hline
      & 1 level  &.0864, .0864& .0971, .0914 & .1293, .1090 & .1025&.1287 \\
 5    & 2 levels &.0858, .0857& .0761, .0838 & .0946, .1004 & .0987&.1172 \\
      & 3 levels &.0926, .0864& \textbf{.0668}, .0805 & .0766, .0982 & .1016&.1133 \\
      & 4 levels &.1100, .0894& .0742, .0801 & .0828, .0981 & .1186&.1276 \\
 \hline
       & 1 level  &.0543, .0542& .0509, .0480 & .0694, .0572 & .0690& .0841 \\
 10    & 2 levels &.0542, .0539& .0400, .0442 & .0489, .0528 & .0657& .0763 \\
       & 3 levels &.0589, .0547& \textbf{.0359}, .0430 & .0399, .0522 & .0694& .0776 \\
       & 4 levels &.0696, .0570& .0413, .0436 & .0442, .0535 & .0802& .0880 \\

       \hline
\end{tabular}
\end{center}
\caption{Average relative reconstruction error over 100 simulations, as a function of the order of the method and number of levels in the multiscale approaches.  The minimums for each SNR are emphasized in bold.}
\label{table1}
\end{table*}

\subsection{Quantitative Results}

We performed two sets of simulations to compare the methods in a more quantitative manner.  The first set of results presented here involved setting up 100 different test problems and then running all of our methods over each time for multiple noise levels, and the mean reconstruction error over all simulations is presented in Table \ref{table1}, with the MHOTV resulting in the left of each column and Daubechies wavelets in the right of each column.  It is important to note here, that the parameter $\lambda$ in \ref{MHOTV-model} was optimized in every reconstruction to yield the solution that minimized the true error between the test signal and the reconstruction, making for objective comparisons.  In order to set up each test problem, a 1D piecewise quadratic polynomial (presumably ideal for order 3) was randomly generated over a 1024 stencil, and the entries in sampling matrix $A \in \mathbb R^{1024\times 1024}$ and added noise to $b$ were randomly generated from a mean zero Gaussian distribution.  Overall, these results show that MHOTV moderately outperforms Daubechies wavelets in each case, and remaining comparisons between the order and number of levels are generally consistent between MHOTV and the wavelets.

For the single level case (original TV and HOTV), the error generally increases for higher orders, contrary to the results in previous work \cite{Archibald2015}.  Multiple scales show notable improvement for the higher orders, whereas they show a mild reduction in accuracy for order 1.  The most benefit for both orders 2 and 3 is seen when using 3 levels, and order 2 actually outperforms order 3.  Finally, using the fact that (\ref{eq: DFTeq}) gives us a way to compute fractional orders of the method, we present also the results from orders 1.5 and 2.5.  These are notably worse than the integer orders, a testament to the fact that these fractional order derivates result in highly nonlocal differences \footnote{To observe these nonlocal stencils, one can compute the inverse Fourier transform of (\ref{eq: DFTeq}) for fractional orders $k$.}.

In the second set of results presented here we ran a series of numerical simulations and measured the rate of {successful} recovery for each method as a function of the sampling rate. For each simulation we randomly generated a piecewise polynomial of specified maximal degree over a 1024 stencil.  This function was randomly sampled at the specified sampling rate precisely as in the previous 1-D simulations in section \ref{sec: repeat1d}, where the sampling rate is defined by the number of samples divided by the number of grid points.  Each regularization {procedure} was then used for reconstruction, and the $\ell_2$ error between the true function and reconstructed functions is determined.  If the error was less than $10^{-2}$, then the reconstruction was said to yield a \emph{successful recovery.}  This simulation was carried out for each sampling rate in 20 trials, and the fraction of those 20 trials that yielded {successful} recovery is set as our probability of success.  {In each case, the generated test functions had five discontinuities, and the location of the jumps were drawn randomly from a uniform distribution on the approximation interval.}

No noise was added for these simulations, as this can make the likelihood of an exact recovery unlikely.  Therefore, for this case our general $\ell_1$ model as a modification of (\ref{gen-l1}) is given by
\begin{equation}\label{exact-model}
f_{rec}  = \arg \min_f \| T f \|_1 \quad \text{s.t.} \quad Af = b,
\end{equation}
and similarly for our specific MHOTV model in (\ref{MHOTV-model}).
This constrained data fitting problem is solved by reformulating as an unconstrained problem with an augmented Lagrangian function \cite{hestenes1969multiplier, Li2013}.

The results for these simulations are shown in Figure \ref{fig5}.  The results are separated in two ways, by the degree of the piecewise polynomial function that is sampled (varying along the rows) and the order of the regularization method (varying along the columns).  In the first row are results for piecewise constant functions, in the second row are piecewise linear functions, and in the third row are piecewise quadratic functions.  In all cases, the MHOTV yields the highest probability of success, regardless of the degree of the polynomial or order of the regularization, and the Daubechies wavelets success appears to generally lie somewhere between MHOTV and HOTV.  The order 1 regularizations perform well only in the case of piecewise constant functions.  On the other hand, the order 2 and 3 regularizations perform well for all function types, with order 2 again outperforming order 3 both with piecewise linear and quadratic signals.

\begin{figure*}
 \centering
 \includegraphics[width =\textwidth]{fig5-2.pdf}
 \caption{Probability of success for HOTV, MHOTV, and Daubechies wavelets at orders 1 (left column), 2 (middle column) and 3 (right column).  A successful recovery is deemed whenever the relative $\ell_2$ error between the reconstruction and the true signal is less than $10^{-2}$. Top row: piecewise constant functions.  Middle row: piecewise linear functions.  Bottom row: piecewise quadratic functions.}
 \label{fig5}
\end{figure*}

\section{Summary

HOTV circumvents the staircasing often observed in TV solutions and has been shown to be more effective for problems with fine features, where resolution can be improved by increasing the order of derivatives in the regularization term \cite{SGP-ET}. In some applications, however, high order derivatives promote solutions with spurious local oscillations, as shown in Figure~\ref{fig1}. The MHOTV regularization we introduce in this work is shown to mitigate unwanted oscillations while maintaining the resolution power of high order regularization.

{Although the theory for MHOTV reconstructions remains underdeveloped when compared to wavelets regularization \cite{eck1995multiresolution,tenoudji2016wavelets,guo2007optimally,kutyniok2012shearlets,starck2010sparse,gao1998wavelet,taswell2000and}, our experiments indicate that MHOTV can outperform wavelets regularization in practical applications.  Figure~\ref{fig2}, for instance, shows fewer spurious oscillations in the MHOTV reconstruction than for  Daubechies wavelets penalization. A feature that can also be observed for the 2-D tomographic data. Moreover, our results  show that MHOTV regularization requires fewer samples for successful reconstructions than for HOTV and wavelets. Computational efficiency is achieved by performing the transformation in Fourier space or by matrix decomposition, as derived in section \ref{sec5}. The associated matlab algorithms can be downloaded at \cite{toby-web}, and some of the simulations in the proceeding section can also be found there.}

redpajama_set_name: RedPajamaArXiv


Full-text search engines evolved much later than traditional database engines, as corporations and governments found themselves with more and more unstructured textual data in electronic format. These new text documents didn't fit well into the old table-style databases, so the need for unstructured full-text searching was apparent.
Since it was developed later, search engine technology borrowed heavily from the database world, and many search engines still employ some type of traditional table structures in their underlying architecture. Some text retrieval companies were even staffed with employees who came from traditional database company backgrounds. Many of the traditional RDBMS (Relational Database Management System) paradigms have also migrated into search engine technology, though often renamed or recast.
While relational database systems and full-text search engines are optimized to process fundamentally different types of data, there are a number of similarities between the two.
While there are similarities between full-text and relational technologies, there are a number of differences as well because of the fundamental differences between the types of data being indexed and the flexibility of the retrieval options. While the differences can present some challenges, they also present the opportunity to take advantage of the key features of full-text search to provide an innovative solution to the problem at hand.


text:
A foamin' good time despite rain
Kids play in foam which is part of the "That Kid Place" display at the Chenoa Shop-N-Go Spring Fling Saturday.
Saturday's rain wasn't ideal for the Chenoa Shop-N-Go Spring Fling event but that didn't stop folks from venturing out to see the activities.
"The weather is not on our side right now," Shop-N-Go organizer Melissa Cooper told Chenoa News around midday. "We had about 30 vendors back out."
Several vendors and crafters offered something for everyone, plus an auction was held under a big tent along Division Street and the car show filled the area of Owsley and Second streets.
Ryan Abbott brought his "That Kid Place" mobile sensory museum to the park during the Spring Fling. Abbott has put almost 20,000 people through the 192 square foot museum in just two seasons.
"I built this about two years ago. I've never closed once for anything in the last two years," Abbott said.
Abbott raises funds to do events for kids with special needs, traveling all over.
"It's a blast doing it."
The retired State Farm employee said this is his way of giving back to the community and the idea snowballed from the beginning. A foam pit was located outside the trailer and the kids passing by seemed to love it.
"Having exhibits out here doubles my size," added Abbott.
Cooper admits it can be hard to set a rain date as no one knows exactly what the weather will do so they just "go with it." Chenoa Shop-N-Go is a charity event for kids to keep them active in sports and other activities while keeping them away from drugs and alcohol and off the streets.
"We do organized events for kids in the park here and the light-up parade we help with," noted Cooper.
Shop-N-Go also holds the annual Halloween costume contest and puts up Christmas decorations. They are considering holding a few more events during the summer months.
Saturday's Spring Fling at the City Park included numerous food trucks such as Engine One Concessions, Rollin' Smoke Barbeque and Monster Grilled Cheese and More.
In addition to Cooper, Shop-N-Go members include Dylan Cerda and Sally Decker. Carrie Holzhauer helped with the car show.

redpajama_set_name: RedPajamaCommonCrawl


When I took 9 other people to Peru, I wanted to make sure Arequipa was on the itinerary. I wanted to make sure we had enough time to relax and enjoy the second largest city in Peru…. and thank god that we had that time because almost everyone got sick at this point in our trip. I will spare you the details but I will leave you some nuggets of wisdom: eat probiotics before you travel, don't drink the water, and pack some pepto. Besides a handful of people being sick, everyone was a trooper and we still had a great time in this beautiful city.
We arrived early in the morning on a Cruz del Sur bus, traveling 12 hours from Huacachina (read about our stay in Huacachina here) and we checked in to our hostel, Home Sweet Home. When you travel to Arequipa, I recommend making sure your lodging is close to the city center. We were able to walk everywhere, and it made things very convenient.
Plaza de Armas is just a real fun way to say "city square". Every city we visited in Peru had a Plaza de Armas, and they all were beautiful. So when you visit Arequipa, make sure you visit this city square. Make friends with the pigeons, and visit during the evening. The buildings are absolutely beautiful when they are lit up at night, and the atmosphere is perfect (also get ice cream… where ever and whenever you find it).
Located not far from the Plaza de Armas, this monastery is a photographer's dream. This breathtaking place was build in 1579, and has amazing colors and architecture all throughout. If you are lucky (like we were) there won't be too many people here, so you should be able to enjoy things leisurely. Make sure you climb the large building with the dome on top of it to enjoy a roof top view over Arequipa.
More views from the magical dome building roof top.
Ooooo that blue. Work it, Arequipa.
…but watch your pockets (our friend got pick pocketed outside of the market). Even if you aren't a big shopper, this is an awesome market to visit. The colors of the fruit, fabrics and flowers is worth the trip. Its an 8 minute walk from the Plaza de Armas: exit the main square via Calle San Francisco, turn left on Consuelo and walk 2.5 blocks and you'll see entrances to the market.
This was, by far, one of the highlights of our entire stay in Peru. Colca Canyon is said to be one of the deepest canyons in the world, twice as deep at the Grand Canyon. The landscape was as awesome as you would expect it to be. We booked our tour through our hostel and it cost $18 per person for the entire day. We were picked up from our hostel at 2:30 am, followed by a 3 hour bus ride to Colca Canyon.
The entire day was amazing, but the most memorable part for me was when we visited the Mirador Cruz del Condor. The Andean condor is a large species of bird that was almost gone forever due to over hunting. In the last few years, conservation efforts have brought the population back and there are now about 80 Andean condors living at Colca Canyon. We were lucky enough to see 10 of these condors (I even got pooped on by one as it swooped down over my head). We watched in silence as these huge birds soared around the most beautiful backdrop you could ever imagine.
After being mesmorized for an hour, we continued our tour. We got to see alpaca, llamas, and vicuñas off in the distance. If you visit Arequipa without planning on seeing Colca Canyon, you're doing it wrong.
There is, of course, so many other things to do while staying in Arequipa. However, if you find yourself with little time, I encourage you to make sure you do these four things in the small amount of time you do have.
If you are wondering how to fit Arequipa into a Peru trip, make sure to read my blog on my full Peru budget and itinerary for 14 days in this wonderful country.


Revolution officials report "Drastic jump in growth"
From a Revolution press release: Words can't describe the events that took place this weekend. The corps has taken a drastic jump in growth both in actual size and maturity of achievement. The weekend began with the 2003 awards banquet at the Adams Mark hotel in downtown San Antonio, where the 2003 season video debuted. Plans for the 2004 season were also unveiled. The next morning was off to a surprising start. With 64 horns, 85 percussionists, 29 guard and eight drum major auditionees, the corps handled the most successful camp in its four-year history. Not only were attendance records broken, but the corps had some other achievements such as performing the entire 2004 opener. "The size and achievement of the horn line was nearly overwhelming and exceeded all our expectations," said Jeremy "Bird" Grange, brass caption head. "I am glad to see that the interest in corps is growing in Texas and heading our way," added Ajay Riojas, color guard caption head. Camp pictures as well as midi clips will be posted in the coming week. "This is by far the best start to our season we have ever had!" executive director John Rodriguez said. "The turnouts for auditions each year just keep getting better and better and the progress made towards the 2004 season have been extraordinary."


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Ed Ling named CPSA clay shooter of the year
Oympian Ed Ling has been voted clay shooter of the year at the Clay Pigeon Shooting Association (CPSA) Awards
From left: Steve Scott, Ed Ling and Olympian Sharron Davies MBE who hosted the CPSA Awards	Credit: CPSA
TAGS: CPSA
The Taunton born shooter received the award having been voted for by CPSA members from a shortlist of five contenders, which also featured fellow Olympic bronze medallist Steve Scott.
When presented with his award Ed said: "It's a great honour to be up here and I'd like to thank the CPSA for organising the event. I must also thank my family for all the support they have given me over the years. This is a fantastic way to round off what has been a great year for me."
Ed Ling, Clay Shooter of the Year
Voted for by CPSA members
A total of nine awards were presented on the night, each focusing on a different aspect of the sport. All the nominees and winners of the awards were voted for by CPSA members.
Among the highlights of the night, Julie Sperling and Cheryl Hall were both clearly delighted to receive awards highlighting their voluntary and development work for the sport.
Following the success of the World English Sporting in 2016, E.J. Churchill Shooting Ground in High Wycombe was named ground of the year.
Rob Fenwick of EJ Churchill commented: "I am really delighted with the award. The team worked so hard last year with all the World Championships along with everything else we do. All of which were a great success. It's a great award to win as it's the members voting and so it's good to know they are also enjoying what we are doing!"
Nick Fellows, Chief Executive of the CPSA said: "We're delighted to see so many people recognised and honoured here tonight at what has been a fantastic celebration of all that is good in our sport. All of the nominees were put forward by their peers and the winners voted for by CPSA members, so it's a great event to raise our profile too. The Awards dinner has grown year on year and has fantastic support from the shooting industry as a whole and we thank our sponsors in particular for their support in making tonight a change in how we recognise the many achievements of our members."
EJ Churchill was voted CPSA shooting ground of the year
Full list of winners
Clay Shooter of the Year: Ed Ling
Special Contribution Award: Martin Barker
County Committee of the Year: Cheshire
Disabled Shooter of the Year: Andy Womble
Ground of the Year: E.J. Churchill
Young Shot of the Year: Amy Easeman
Volunteer of the Year: Julie Sperling
Development Initiative of the Year: Cheryl Hall
Coach of the Year: Linda Grayson
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The obligation of the borrower and the cosigner to repay a SELF loan shall be forgiven upon documentation of the death of the borrower. Payments made after the borrower's death prior to the submission of a death certificate will not be reimbursed.
Total and permanent disability of borrower.
The obligation of the borrower and the cosigner to repay the SELF loan shall be forgiven upon written request to the office if the office determines that a borrower has a total and permanent disability and if the total and permanent disability of the borrower occurred after all disbursements of the loan were made. The forgiveness is effective the date the total and permanent disability documentation is approved and accepted by the office.


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You are at:Home » Football » European Football » Eintracht Frankfurt v Mainz Betting Preview: Silva to continue to shine
Eintracht Frankfurt v Mainz Betting Preview: Silva to continue to shine
By Matthew Kirby on	 May 1, 2020 European Football, Football, German Football Tips, Tips
MATCHDAY 30 of Bundesliga continues on Saturday afternoon as Eintracht Frankfurt host Mainz. Matthew Kirby (@M_Kirby95) shares his research.
Eintracht Frankfurt v Mainz | Saturday 6th June 2020, 14:30 | BT Sport
Consecutive away wins for Eintracht Frankfurt have seen them pull away from the drop zone. And this looks a brilliant chance to make it three on the bounce against a Mainz side who haven't won in five.
Adi Hütter made just one change for Wednesday's trip to Bremen with Almamy Toure coming in at right wing-back. But it was substitute Stefan Ilsanker was the man of the moment. He scored 14 seconds after coming off the bench and added his second eight minutes later to take his Bundesliga goals tally to two.
Mainz are treading water with their heads just above the water. They sit a point above the relegation play-off spot. They are winless in five with just two points collected since the return but have been picked up in away games.
Achim Beierlorzer's side are in a desperate need of a win, but they are struggling to create that many clear cut chances. Ridle Baku is one man that has caught the eye down their right side, but he's banned for this trip.
Die Nullfunfer have scored 22 goals in 14 away games, but they're outperforming their Expected Goals (xG) creating just 18.15xG on their travels. Having racked up a few games under 1xG, then it's very likely to come back to haunt them.
Kostic to unlock Mainz defence
One of Frankfurt's key men is Filip Kostic. He's chipped in with 10 assists so far this season making him just one of eight players to have reached double figures. He added to his tally in Wednesday's win with a pinpoint cross finding Andre Silva. And that was just one of 16 crosses he attempted against Werder Bremen.
The Serbian is a key cog in this Frankfurt side and he does rack the crosses up. However, looking at Ladbrokes' Bet Builder feature, the bar is already set at 14+ crosses (17/20), which isn't something I'd consider although he will be on corner duty.
Given his role in the team, it could be worth looking at the Mainz right-back for tackles and a card. But with Baku suspended, that's still a bit of an unknown. The Bundesliga site names Daniel Brosinski as the most likely to come into the side, but there's a chance Ronael Pierre-Gabriel will get the nod.
From a betting perspective, the latter would be a better option too. He's made five appearances in 2020 and completed 15 tackles across those 337 minutes, so if he does start then he's one for the tackles market.
Silva to continue to shine
The AC Milan loanee is flying with four goals in five games taking him to eight in the Bundesliga. He can be quiet in games but always seems to pop up in space and elude the opposition defence.
He's averaging 2.1 shots per game, and given this Mainz defence have only kept one away clean sheet, then he's likely to keep this hot streak going. The in-form forward is worth a look in the goalscorer markets. You can get 6/4 on him to score anytime and 6/1 (William Hill) for him to open the scoring, which he's done in the last two games.
With Frankfurt's upturn in form and Mainz barren spell combining the two will appeal to some, so Silva to score in a home win at 5/1 with Bet365.
The betting angles
I'm keen to side with an Andre Silva goal and two other angles appeal. I'll start with the 5/6 shot and it's for a goal to be scored from the 76th minute to full-time on Bet365. Frankfurt games this season average 3.52 goals per game with 66% of goals coming in the second half, and for home games, it's 62% of goals scored come after the break.
Just looking at the hosts date then you'll see the longer the game goes, the more chance of a goal. Goal times by every 15 minutes is consistent at around 13% until you hit the hour mark. From 61 to 75 minutes it rises to 19% then it increases to 30% for the final 15 minutes.
We saw that highlighted by their strong finish against Bremen. And at home, Frankfurt have scored 13 goals in that final-15 minute spell. For Mainz, 17 goals in their 14 away league games have come late in the game – again it's 30%.
With this being the third game in seven days for Frankfurt then there may be a few tired legs in Hutter's side, so late fatigue could be a factor to add into this line of thinking.
The final bet I've lined up is a juicy 7/1 bet and comes from something I've noticed when watching Frankfurt of late.
It's Daichi Kamada to be carded, which I feel many will overlook given his disciplinary record is pretty exemplary. He's only received 11 cautions in 164 first-team appearances for his clubs, but Wednesday's game highlighted a certain cynical side to his game, which I'm surprised Patrick Ittrich didn't punish with a yellow card.
It's six yellows across the Bundesliga and Europa League, so six in 2262 minutes of football. So he averages a card every 377 minutes, but with an increased foul count since the return of the league, then I'm sure a few German officials will be making a mental note of this more aggressive play.
He's committed eight fouls across the five games, although only playing 19 minutes against Bayern. He was booked in the away win at Wolfsburg last weekend and is one to look at given his recent foul count. And at 7/1 with Sky Bet, it's a good price for me to get involved in.
Eintracht Frankfurt v Mainz – Goal after 75:59 (5/6 Bet365)
Eintracht Frankfurt v Mainz – Andre Silva to score (6/4 Sky Bet)
Eintracht Frankfurt v Mainz – Daichi Kamada to be carded (7/1 Sky Bet)
Matthew Kirby
Matt is a graduate in sports journalism and is currently plying his trade as a content editor for a major betting company. He was bitten by the betting bug when backing horses and continues to do so. His main sporting interests are football, horse racing and darts. Matt is a Stoke season ticket holder and enjoys going to gigs.


Scoop up the ideal outfit, gift, or specialty item at the East Bay's finest retail establishments.
Whether you're shopping for a custom engagement ring, a high-end timepiece, or a dazzling bauble, Heller Jewelers offers designs that suit every style. The shop also repairs jewelry, often restoring treasured heirlooms to their original state.
It's no surprise, then, that customers have remained loyal to the store over the years. In fact, the owners note, clients who purchased their wedding rings at Heller Jewelers decades ago are now coming in to pick out 20th anniversary gifts. "It's an honor to see everyone go through their journeys in life," says Catherine, "and to be able to celebrate that with them." San Ramon, hellerjewelers.com. —V.S.
"Rakestraw Books has an incredibly diverse set of high-quality books. The staff is friendly but not intrusive if you just want to browse. They are also all incredibly knowledgeable, and their recommendations are always spot-on. You can order a book online in a jiffy, but you'll never get the insight and personalized recommendations [that you'd get] at our wonderful local bookstore." Danville, rakestrawbooks.com.
Fans of Venture Quality Goods know they can score 12 brands of clothing and shoes that are made or designed in California—such as SeaVees, Patagonia, and Johnnie-O—at the Lafayette shop. Continue the Golden State love and pick up a pair of Warriors socks by Stance. From images of the Splash Brothers shooting three-pointers, to artistic portraits, to classic logos, there are 16 styles to choose from—allowing you to telegraph your team spirit year-round. Lafayette, venturegoods.com. —C.M.
This family-owned Walnut Creek boutique feels like an artfully arranged home, featuring several unique rooms stocked with carefully curated gifts for every type of recipient—party hosts; babies, kids, and tweens; wine lovers; jewelry junkies; and of course, fun-loving men and women. With many products sourced from local artisans, a friendly staff that goes out of their way to help each customer, and a giving-back-to-the-community ethos (Wish hosts fundraisers and partners with local nonprofits), the shop deserves its best-in-class status. Walnut Creek, wishwalnutcreek.com. —V.S.
You don't have to be a master chef to appreciate the liquid gold on display at this olive oil purveyor's two East Bay stores. Vats of premium extra virgin and infused olive oils share shelf space with aged balsamic vinegars in numerous tempting flavors such as black fig and pomegranate. The family-owned business works in direct partnership with farmers in both hemispheres to deliver the freshest goods year-round, in addition to producing many of its own olive oils and infused vinegars. With some 50 products on tap—not to mention a highly trained staff ready to offer expert advice, recipes, or help creating the perfect gift basket—Amphora Nueva has a delicacy to tantalize every taste. Berkeley and Lafayette, amphoranueva.com. —R.O.
Achieving decor perfection is simple with a visit to Elsie Green. Need a few chic items to enhance your summer soiree? Nab some wine and cocktail sets made from recycled Moroccan glass in a gorgeous shade of light blue, and a bamboo peg rack with French country flair for guests to hang their jackets on. Lighten the mood with a breezy tablecloth, and top it off with some sunny yellow candles in the shape of songbirds. Concord, elsiegreen.com. —C.M.
Tootsies has provided foot candy for locals since its much-loved Rockridge location opened in 2005. After its second shop debuted in Alameda four years ago, it became easier than ever to pick up stylish footwear. Try a ballet-inspired flat from Gentle Souls, a Kenneth Cole brand that boasts flexibility, breathability, and extra cushioning. Or work a Mediterranean vibe with a classic espadrille by Eileen Fisher. For a versatile shoe that goes with everything, step into a cream-hued, chunky-heel sandal by Vince Camuto. Alameda and Oakland, www.tootsiesboutique.com. —C.M.
For more than 70 years, Lafayette has been home to the gardening oasis Orchard Nursery. The five-acre haven offers a diverse selection of high-quality plants and landscaping materials—and it's all too easy to spend hours wandering through the expertly curated outdoor collection, the houseplant atrium, and the Lazy K gift shop. Detailed signage helps visitors keep track of their favorite foliage, although with so much impressive greenery, the shopping list never seems to end. Lafayette, orchardnursery.com. —M.M.
The East Bay's go-to stop for all things bike, 42-year-old Livermore Cyclery offers everything from repairs to rebuilds, cruisers to racers, and accessories to rentals. Knowledgeable staff members—many of whom are fit certified—make sure your bike meets your personal comfort standards and can recommend the best two-wheeler for your sport, whether you are a casual rider, an off-road adventurer, or a professional triathlete. And to maximize your cycling experience, all three locations host clinics on minor maintenance and nutrition, as well as yoga classes. Alamo, Dublin, and Livermore; livermorecyclery.com. —K.H.
Having celebrated a decade in business last fall, this trendsetting women's boutique is seriously hitting its stride, thanks to its owner, local style icon Sherri McMullen. A former buyer for Neiman Marcus who was recently profiled in Vogue, McMullen travels to New York and Paris Fashion Weeks to fill her shop with chic looks and on-the-cusp designer lines, bringing a sophisticated style sense to the East Bay. (She's the exclusive U.S. retail purveyor of the cult-favorite Nigerian label Maki Oh.) Next up: McMullen will launch a new shop in Uptown Oakland this fall. Piedmont, shopmcmullen.com. — L.B.
The bad news: This Walnut Creek antique boutique closed in March. The good news: The Spotted Cow will relaunch in Martinez in September. Owner Mackenzie Lesher says the new location—an old metal barn in Martinez's revitalized downtown—is perfect for her store's early-1900s farmhouse aesthetic. The venue will open only four times a year for weeklong or 10-day extravaganzas featuring music, cocktails, food, and workshops, in addition to a wide array of vintage finds. Here are more details about The Spotted Cow's reincarnation. Martinez, spottedcowvintage.com. —V.S.
6–8: Antiques vendors—in addition to the shop's owners and special guests—who sell their wares at the store.
30+: Years the Martinez barn stood vacant before The Spotted Cow moved in.
50: Approximate number of vintage treasures the owners picked up on a recent sourcing trip to Paris.
1,700: Size, in square feet, of the new space (excluding the outdoor area, which will also host vendors and activities).
Warm summer months can mean a plethora of outdoor events. No matter what your calendar holds, Vici offers racks stuffed to capacity with ever-changing, fashion-forward clothes, along with shoes and accessories. Diablo asked co-owner Aimee Dudum Colorado what to wear for various occasions, and she supplied some tips for completing the perfect outfit. Walnut Creek, vicicollection.com. —C.M.
Offering high-end designer brands at a fraction of their original retail price, Labels Luxury Consignment has been dressing the area's most fashionable women for more than 20 years. (The store even carries coveted Hermès and Chanel handbags.) Diablo asked owner Lynn Nice about the smart styles buyers can score in her boutiques. Berkeley and Walnut Creek, labelsluxury.com. —C.M.
You know best: Diablo readers recommend these local specialists.


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Tag: mansion
Leith Street: Cheong Fatt Tze Blue Mansion
Cheong Fatt Tze Mansion is a historical and cultural rich building located along Leith Street in George Town, Penang. A winner of multiple historical and heritage awards including the "Most Excellent Project" Award at the UNESCO Heritage Awards 2000, the blue mansion is one of the most visited places and a definite tourist destination in Penang. The mansion is said to be one of the two of its kind outside China which also features architecture design of both Chinese and Western influences. The mansion was built by a famous Chinese merchant in 1880s named Cheong Fatt Tze and it has 38 rooms, 5 granite-paved courtyards, 7 staircases with 220 vernacular timber louvre windows.
During his time, Cheong Fatt Tze was also known as the "the Rockefeller of the East" and had helped a lot in building business and diplomatic relationship between the Chinese and Western governments. His legacy and contributions in the were so influential that during his death, top Chinese officials came all the way from China to attend his funeral while both the British and the Dutch authorities ordered flags to be flown at half mast in honour of him.
Featured in a number of movies (such as Crazy Rich Asians), the mansion now also houses a boutique hotel and a restaurant. It is also opens for public guided tour for a few slots of time of the day.
Penang Perspective: A Blue Mansion
Cheong Fatt Tze Mansion, Leith Street, George Town.
George Town Street View: 23 Love Lane
A renovated and refurbished pre-war heritage mansion located in Love Lane that had been converted to a cosy little boutique hotel, which is also popular with tourists these days. The mansion itself is apparently unique for its 5 architectural periods, that are reflected by the Anglo Indian Bungalow, Indian Chinese Terrace, Straits Eclectic Building, 1920's Jack Roof Annex Block and the Indian Shophouses.
The hotel's website can be located here: www.23lovelane.com.
George Town Street View: Hostel 100 Cintra Street
100 Cintra Street Mansion
Hostel 100 Cintra Street is an old 19th century pre-war mansion that used to have served as an inn in the old days and has since hosted various functions, including a departmental store, a museum and now a hostel. The blue-painted heritage building can be found at Cintra Street, George Town.
A View from the Hill Mansion
View of George Town from the peak of Penang Hill.
A Glimpse of Suffolk House
Suffolk House, Penang.
Penang Chinese Recreation Club
The Penang Chinese Recreation Club (or also known as CRC) is an old heritage clubhouse established by a group of Chinese sports enthusiasts in the 1900s. The Victorian style clubhouse was converted from an old heritage mansion and to this date, the club is still active with a restaurant located within the premise that caters generally to Chinese wedding banquets. CRC is located at Jalan Padang Victoria (Victoria Green) in George Town.
George Town Street View: Peranakan Facade
Ancient sculpture adorning the old rooftop that forms part of the heritage facade of the Pinang Peranakan Mansion, in Church Street, George Town.
George Town Street View: Penang Islamic Museum
The Penang Islamic Museum heritage mansion located along Armenian Street in George Town. The museum is dedicated for preserving the history of Islamic culture and heritage in Penang and Malaysia.
Hutton Lane Street View: "Leong Kim Tiew" House
Despite its signboard with the name, this old historical house located at Hutton Lane actually used to belong to the former Penang Chief Minister, Tun Dr. Lim Chong Eu. The house was passed down to Lim Chong Eu from his father, Dr Lim Chwee Leong. It was said that at one point the house was leased to a carpenter friend named Leong Kim Tiew who then created the signboard.


Is your diary always looking full and overwhelming? Are you struggling to find time to do any actual work or enjoy the things you love? It could be that you're saying 'yes' too often, giving away your precious time to others – which can leave you feeling frustrated, frazzled and resentful.
This is where learning to say 'no' will become your time management weapon of choice. You're not a superhero. There are only so many hours in each day. Saying no doesn't make you a bad person. It's about putting yourself first, and valuing your own time, health and happiness.
Even Steve Jobs himself said that "Focusing is about saying no".
First things first, take a step back and look at how you spend your time. Dig out your diary, go through recent events on your work calendar, and get to grips with what you're saying yes to. Do you see any patterns? Who or what are you not turning down? Are you putting too much pressure on yourself? Identify where you're going wrong, so you can say no to these people or things in future.
Before we teach you the art of saying no, understand that you're not doing anything wrong. You're not selfish – it's just that you can't do everything or please everyone. 'No' isn't a dirty word. It can be very healthy and necessary in certain scenarios. By saying no you're actually being true to yourself and respecting your own time and values.
If anyone is making you feel guilty for saying no, then that says more about them than you. No one has the right to control your life. Take your power back and understand that guilt can also be a weapon people use to get what they want.
How do you decide whether you should do something? That can be tough. Often, we feel put on the spot and might panic and say yes when we really want to say no. In which case, the next time someone asks you to do something, you don't have to give an immediate response.
It is okay to simply say: "I'll need to check and get back to you". This buys you some time to decide whether to say yes or no. People should understand that you have commitments and will want to consult your diary before finalising anything.
How do you use the right tone and language to say no without offending anyone? It's not easy, that's for sure. What you don't want to do is nervously ramble, as you might give mixed messages. That's why a polite but firm no will suffice. You don't have to make excuses. You mustn't apologise – you have done nothing wrong. Just get straight to the point, but keep it light and friendly.
Still not sure how to frame your response? Quite simply, don't give the other person any chance to misunderstand. Start off with a compliment, thank the person for the opportunity, say no and then offer some advice on how they might proceed forward. Then change the subject or excuse yourself.
"It's great to think you've asked me for support. I'm not able to help right now, as I've got this project that needs my full attention this week/month. Thanks for coming to me though. It's tough for all of us at the moment, but I know we'll get through this."
"I'm sorry I can't invite you in/talk to you right now. I'm up to my neck in deadlines and really need to buckle down. Can we catch up later when I'm not pulling my hair out? When's a good time for you?"
"I'm so flattered you've come to me for help. I'd love to offer my services. Here's my business card, which includes my website address. If you want to call or email me during office hours, I'd be happy to discuss my rates."
"This sounds like a great piece of work. Would you like me to cost this up for you? Do you have time to get together to discuss in more detail, so I can put together a comprehensive quote?"
"As I said, I really can't find the time right now. I wouldn't want to commit to something I know I can't do. I wish you luck though, and know you'll make this a success."
Saying no is liberating. It can be a positive word that means you're saying yes to other things – like having more time and energy, and a healthier work/life balance. Be true to yourself and your own convictions – not just because time is money, but because time is all we really have.


Gulf Front Beauty! 22 feet to the sand. 180 degrees unobstructed beach view.
Sundial Suite K303 is a beautiful 2BR/2BA plus den, third floor (2 units from elevator) beach front unit that overlooks the pristine white sand beaches of Sanibel Island. Just 22 feet to the sand. The sliding doors open wide to allow you to enjoy the unobstructed panoramic Gulf of Mexico vista from the large screened lanai and living room. All Sundial condos are privately owned. We chose to handle the rentals ourselves and that way we can save our guests money. Also, because we are in charge of the rentals, we keep improving the unit and are always on top of it's maintenance. There are only a few restrictions that you have regarding the Sundial resort when you rent through the owner and not the Sundial resort directly. You cannot use the resort pool (we have a beautiful, heated pool next to our building for your use) You cannot sit in the lounge chairs by the pool, nor can you use the workout facility. (There is a great workout facility on Sanibel that you can use for a fee) Otherwise, you're free to purchase drinks at the Tiki bar, sundries and groceries at Baileys, dining in any of the restaurants and use of the tennis and pickleball courts for a fee. All these amenities accept cash and credit cards.
We rent in multiples of 7 nights beginning on Saturdays.
There is a washer and dryer in the unit. We offer Free Wi-Fi, Free Bikes, Beach Chairs, Beach Umbrellas and more!
CHECK REVIEWS: Luxury Suite K303 is a 5 Star unit. We have furnished and equipped K303 with every convenience to make your vacation perfect. There are 3 Flat Screen HD Cable TVs and 3 DVD players.
The kitchen has all appliances, including dishwasher and microwave. All cookware and dishware are provided. There is a washer and dryer in the unit. There is central air conditioning/heat as well as ceiling fans throughout the condo.
The Master Bedroom Suite has a King bed, private access to the screened lanai, large walk-in shower, and double lavs. The 2nd bedroom has 1 Queen Bed and 1 Twin bed plus its own private bath with tub and shower. The living room has a queen sleeper sofas.
I'm retired from many years in the sales profession. I've personally been coming to Sanibel with my family for almost 30 years and decided I loved it so much that I had to have a piece of this island paradise called Sanibel. If you've visited here, you'll know what I mean, if it's your first visit, you'll find that just like the rest of us, you'll leave with a smile and always have Sanibel on your mind. Enjoy!
Sundial building K is as close to the beach as one can get. It's just 22 feet to the sand and I knew that that is what vacationers to Sanibel wanted. My unit K303 is on the 3rd floor and offers a full panoramic view of the beach and Gulf of Mexico waters. K303 is just two condos over from the elevator and is ultra-convenient to both our beautiful beachfront heated pool and to the amenities at the Sundial resort.
Sanibel has been awarded national recognition as a 'Silver City Biking' community. It's not surprising with all the miles and miles of bike trails on the island. It's a fun and healthy activity to try when you're on the island, and because we offer 3 bikes, you'll save money when you rent with us. Of course, shelling is extremely popular on the island , too. The fact that there are no stop- lights allows one to escape the hub-bub of city life. Sanibel doesn't allow fast food restaurants except the 2 that have been grandfathered in, and so you can enjoy all our unique "islandy" eateries. If you like boating, you have the chance to rent any type of boat nearby for pleasure boating, fishing, or dolphin chasing.
toilet, shower, The master bathroom has a walk-in shower.
toilet, combination tub/shower, The guest bath has a shower/tub combo.
There is a tub/shower in the 2nd bathroom and a lovely walk-in shower in the master bedroom bath with 2 safety grips.. The master bedroom has a dual sink vanity.
Our condo is within 20 feet from the white sand beach. You have a full panoramic view of the Gulf of Mexico from the screened lanai. How much more relaxing and convenient can one get?
No obstacles to viewing the Gulf of Mexico from our condo. You get a 180 degree view.
Fully air conditioned throughout the condo.
The condo has central heating throughout. However, it is usually not needed.
All bedding linens provided as well as , kitchen, and bath towels. You may want to bring beach towels, since we do not provide them.
Washer and dryer combo in the condo.
Bath towels, wash cloths and hand towels supplied. We do not supply beach towels.
Huge fitness facility available daily or weekly at the Sanibel Recreation Dept. on SanCap road about 4 miles from condo.
All new furniture featuring a comfortable sleeper sofa and side chair.
Full size electric range with self-cleaning oven and cooktop.
There is an automatic coffeemaker.
Full set of dishes and utensils as well as glassware.
Our dining room looks directly out to the Gulf of Mexico and the beautiful white sand beach.
Cable TV provided in Living Room and both Master and Guest Bedrooms.
A small library of books is available.
Plenty of puzzles, and board games for all ages are provided.
There are plenty of beach toys for kids of all ages - sand toys, beach floats, tennis rackets, beach umbrellas, beach chairs and more including 3 adult bikes.
We provide several 'family-friendly' videos.
A great place to rent a kayak or canoe is at Tarpon Bay Nature Area which is about 3 miles from Sundial Resort. Sundial Resort also rents water equipment.
We offer 3 bikes which are free for you use. You can also rent bikes nearby.
There are numerous places to rent boats on the island.
You can rent windsurfing boards and jet skis on the island.
The Sanibel Recreation Center has 4 very nice hard courts and you can also rent a clay court at Sundial or the Dunes.
Beachview Country Club is within 1/2 mile of Sundial and they offer daily fee golfing (18 holes). The Dunes Country Club is about 2 miles away and they also offers daily fee golf (18 Holes).
The condo is the second over from the elevator. The master bedroom has a walk-in shower.
A lovely large free heated Beach-Front pool is located within 150 feet of the condo.
Sanibel Island has over 30 miles of biking/hiking trails, a variety of boating and biking rentals, and world reknown shelling on Sanibel beaches. You can walk seemingly endlessly along the white sand beaches and partake in one of the island's favorite pasttimes - shelling. Live theater is nearby as well as many wonderful restaurants. If you feel like eating in, there's a gas bbq grill just outside the condo building for your use. Tennis and golf are available. Sanibel Recreation center offers a fully equipped gym as well as a lap pool, and kiddie pool. The rec center offers daily or weekly memberships which include various aerobic, yoga and water aerobic classes.
Sanibel Island has over 30 miles of walking/biking trails. Phone me at 1-630-287-0205 with any questions or if you want to book my condo. Nearby there is golf, as well as a world-reknown National Wildlife Sanctuary. Tennis & Pickleball are on-site (fee). Sanibel has wonderful Shopping, and Live Theater. Boat rental and all varieties of water sports are available nearby. Fishing and shelling excursions with captains/guides are available. World famous shelling on all Sanibel beaches. Sight seeing tram, canoe, kayak, and small fishing boat rentals are available at Tarpon Bay (Ding Darling National Wildlife Refuge) just a few miles from Sundial. There is a Shell Museum on the Island. Two thirds of the island is a wildlife refuge (tropical jungle). There are no fast food restaurants, no high rises, and no stress! ENJOY!!!


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This list was generated on Tue Apr 23 20:34:16 2019 MYT.