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					<h1>FIRLS</h1>
					<p><strong><a href="http://cchen156.web.engr.illinois.edu">Chen Chen</a>,
					  <a href="http://ranger.uta.edu/~huang/">Junzhou Huang</a>,
					  Lei He
					  and Hongsheng Li</strong></p>

					<p>Fast Iteratively Reweighted Least Squares Algorithms for Analysis-Based Sparsity Learning. <strong><a href="code/FIRLS.zip">[MATLAB code]</a></strong></p>


					<div id="Abstract">
					<h3 align="left">Abstract</h3>
					<p align="left">In this work, we propose a novel method for analysis-based sparsity reconstruction. It can solve the generalized
					problem by structured sparsity regularization with an orthogonal basis and total variation regularization. The proposed algorithm
					is based on the iterative reweighted least squares (IRLS) model, which is further accelerated by the preconditioned conjugate
					gradient method. The convergence rate of the proposed algorithm is almost the same as that of the traditional IRLS algorithms,
					that is, exponentially fast. Moreover, with the specifically devised preconditioner, the computational cost for each iteration is
					significantly less than that of traditional IRLS algorithms, which enables our approach to handle large scale problems. In addition
					to the fast convergence, it is straightforward to apply our method to standard sparsity, group sparsity, overlapping group sparsity
					and TV based problems. Experiments are conducted on a practical application: compressive sensing magnetic resonance
					imaging. Extensive results demonstrate that the proposed algorithm achieves superior performance over 14 state-of-the-art
					algorithms in terms of both accuracy and computational cost.
					</p>
					</div>

					<div id="Problem">

							<h3 align="left">Problem</h3>
							<p align="left">
							FIRLS solves the following analysis-based sparsity regularization problems:</p>
							min (1/2)||Ax - b||<span style="vertical-align: sub">2</span><span style="vertical-align: super">2</span> + &lambda;||&Psi; x||<span style="vertical-align: sub">2,1</span> <br>
							<p align="left">
							where A is an m-by-n matrix with m << n, b is the measurement vector, &Psi; is a sparsifying basis,
							the solution x (or its representation &Psi;x) is supposed to be (approximately) sparse. </p>
					</div>

					<div id="Examples">

							<h3 align="left">Examples</h3>
							<p align="left">
							Here are some examples:
							<table border="0" style="width:800px">
							  <tr>
								<td align="left">Standard Sparsity</td>
								<td align="left">min (1/2)||Ax - b||<span style="vertical-align: sub">2</span><span style="vertical-align: super">2</span> + &lambda;||&Phi; x||<span style="vertical-align: sub">1</span></td>
							  </tr>
							  <tr>
								<td align="left">Non-overlapping Group Sparsity</td>
								<td align="left">min (1/2)||Ax - b||<span style="vertical-align: sub">2</span><span style="vertical-align: super">2</span> + &lambda;||&Phi; x||<span style="vertical-align: sub">2,1</span></td>
							  </tr>
							  <tr>
								<td align="left">Overlapping Group Sparsity</td>
								<td align="left">min (1/2)||Ax - b||<span style="vertical-align: sub">2</span><span style="vertical-align: super">2</span> + &lambda;||G &Phi; x||<span style="vertical-align: sub">2,1</span></td>
							 </tr>
							 <tr>
								<td align="left">Total Variation</td>
								<td align="left">min (1/2)||Ax - b||<span style="vertical-align: sub">2</span><span style="vertical-align: super">2</span> + &lambda;||D<span style="vertical-align: sub">1</span> x||<span style="vertical-align: sub">1</span> + &lambda;||D<span style="vertical-align: sub">2</span> x||<span style="vertical-align: sub">1</span></td>
							  </tr>
							  <tr>
								<td align="left"></td>
								<td align="left">min (1/2)||Ax - b||<span style="vertical-align: sub">2</span><span style="vertical-align: super">2</span> + &lambda;||D<span style="vertical-align: sub">1</span> x, D<span style="vertical-align: sub">2</span> x||<span style="vertical-align: sub">2,1</span></td>
							  </tr>
							<tr>
								<td align="left">Joint Total Variation</td>
								<td align="left">min (1/2)||AX - B||<span style="vertical-align: sub">2</span><span style="vertical-align: super">2</span> + &lambda;||D<span style="vertical-align: sub">1</span> X, D<span style="vertical-align: sub">2</span> X||<span style="vertical-align: sub">2,1</span></td>
							  </tr>
							</table>
							</p>


					</div>

					<div id="Preconditioning">

							<h3 align="left">Preconditioning</h3>
							<p align="left">
							Compared with existing Iteratively Reweighted Least Squares (IRLS) Algorithms, our algorithms are accelerated with preconditioning. When the system matrix has a diagonally-dominate structure (e.g., in CSMRI, compressive sensing, image inpainting), our preconditioning can significantly reduce the computational cost.

							</p>


							<img src="researchimages/PCG.png" alt="" height="256px" width="300px"/>  <br>

							<p align="left"><strong>Reference:</strong> <br>
							[1] Chen Chen, Junzhou Huang, Lei He and Hongsheng Li, "Preconditioning for Accelerated Iteratively Reweighted Least Squares in Structured Sparsity Reconstruction", IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2014.<br>

							[2] Chen Chen, Junzhou Huang, Lei He and Hongsheng Li, ""Fast Iteratively Reweighted Least Squares Algorithms for Analysis-Based Sparsity Reconstruction", Medical Image Analysis, Volume 49, pp. 141-
152, 2018. <br></p>

					</div>

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