Image decomposition via the combination of sparse representations and a variational approach

By Jean-Luc Starck, Michael Elad, David L. Donoho. IEEE Transactions on Image Processing, 2005

1,055 citations recorded by the source corpus.

The separation of image content into semantic parts plays a vital role in applications such as compression, enhancement, restoration, and more. In recent years, several pioneering works suggested such a separation be based on variational formulation and others using independent component analysis and sparsity. This paper presents a novel method for separating images into texture and piecewise smooth (cartoon) parts, exploiting both the variational and the sparsity mechanisms. The method combines the basis pursuit denoising (BPDN) algorithm and the total-variation (TV) regularization scheme. The basic idea presented in this paper is the use of two appropriate dictionaries, one for the representation of textures and the other for the natural scene parts assumed to be piecewise smooth. Both dictionaries are chosen such that they lead to sparse representations over one type of image-content (either texture or piecewise smooth). The use of the BPDN with the two amalgamed dictionaries leads to the desired separation, along with noise removal as a by-product. As the need to choose proper dictionaries is generally hard, a TV regularization is employed to better direct the separation process and reduce ringing artifacts. We present a highly efficient numerical scheme to solve the combined optimization problem posed by our model and to show several experimental results that validate the algorithm's performance.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Signal Processing

View Paper: https://www.semanticscholar.org/paper/de305eb90504fd32805094847bf352de57c1ac4c

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Citations in the corpus

Sparse Representation for Signal Classification (2006, methodology): “Research has focused on three aspects of the sparse representation: pursuit methods for solving the optimization problem, such as matching pursuit [1], orthogonal matching pursuit [2], basis pursuit [3], LARS/homotopy methods [4]; design of the dictionary, such as the K-SVD method [5]; the applicati”
Morphological Component Analysis: An Adaptive Thresholding Strategy (2007, methodology): “In [1] and [2], it was proposed to estimate the components y1 andy2 by solving the following constrained optimization problem :”
Compressed Sensing MRI (2008, methodology): “Even if another sparsifying transform is intended, it is often useful to include a TV penalty as well [49]–[51] Such a combined objective seeks image sparsity both in the transform domain and the finite-differences domain, simultaneously.”
Remote Sensing Image Fusion via Sparse Representations Over Learned Dictionaries (2013, neutral): “One approach focuses on the mathematical model, such as undecimated wavelet [23] and discrete cosine transformation [24].”
Graph Regularized Sparse Coding for Image Representation (2011, background): “It has been proven useful in many applications [1]–[3].”
Collaborative Sparse Regression for Hyperspectral Unmixing (2014, background): “Another library organization was tested in [37], in which the dictionary is organized in two separate blocks, one accounting for texture representation and one dealing with natural scene parts assumed to be piecewise smooth.”
Image Segmentation Using Deep Learning: A Survey (2020, methodology): “Numerous image segmentation algorithms have been developed in the literature, from the earliest methods, such as thresholding [4], histogram-based bundling, regiongrowing [5], k-means clustering [6], watershed methods [7], to more advanced algorithms such as active contours [8], graph cuts [9], cond”
Removing Rain from a Single Image via Discriminative Sparse Coding (2015, methodology): “[23] separates the texture region and cartoon region using both wavelet ﬁlter dictionary and local DCT dictionary.”
Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients (2009, neutral): “Here we choose the special case where each dictionary is an orthonormal basis, for example, two MDCT bases with two diﬀerent sizes for the windows, and apply the Block Coordinate Relaxation method [4] (BCR for short) which inspired the Morphological Component Analysis (MCA) algorithms [17].”
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images (2009, methodology): “In fact, there have been several successful trials of this idea, ﬁrst in acoustic and later in image processing [122, 148, 147, 7, 8].”
Fast Solution of $\ell _{1}$ -Norm Minimization Problems When the Solution May Be Sparse (2008, background): “ideas; examples include [23], [11], [47], [48], [27]; related ideas”
Sparse MRI: The application of compressed sensing for rapid MR imaging (2007, background): “As a result we currently possess a library of diverse transformations that can sparsify many different type of images (23).”
Split Bregman Methods and Frame Based Image Restoration (2009, neutral): “Here again we focus on the analysis based approach, and the other approaches can be found in [11,43, 63].”
Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) (2005, methodology): “In previous papers we presented an alternative approach to layer decomposition, optimizing the sparsity of each layer's representation [25] [26].”
Quantitative Robust Uncertainty Principles and Optimally Sparse Decompositions (2004, background): “These ideas have recently been tested in practical settings, with spectacular success [23] (see also [19] for earlier and related ideas).”
Group-Sparse Representation With Dictionary Learning for Medical Image Denoising and Fusion (2012, neutral): “For the standard sparse representation, two approaches about dictionary construction are popularly used: the analytic approach (such as wavelet [20]) and the learning-based approach (such as K-SVD [21]).”