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... A. How can we wisely choose A to perform well on the signals we have in mind? One line of work considered choosing preconstructed dictionaries, such as undecimated wavelets [149], steerable wavelets =-=[145, 37, 136]-=-, contourlets [38, 39, 40, 70, 71], curvelets [146, 12], and others [22, 123]. These are generally suitable for stylized “cartoon-like” image content, assumed to be piecewise smooth and with smooth bo...

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...e “9-7” filters from [31, 174]. Another interesting example is based on the original LP filter suggested in [18], which is especially popular for applications in vision, together with its dual filter =-=[5]-=-. Tables 2.1 and 2.2 list the filter coefficients. Note that these filters are symmetric and very close to being orthogonal. As a result, the proposed reconstruction in Figure 2.3 is very close to the...

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... S{Wy}, and finally, an inverse wavelet transform to produce the outcome x. Such a direct solution stands as a refreshing alternative to the iterative and slow methods mentioned above. This shrinkage algorithm becomes indeed the optimal solver of (1). This optimality depends strongly on the `2-norm used in evaluating the distance x−y, and this has direct roots in the white Gaussianity assumptions on the noise. Also, crucial to the optimality of this method is the orthogonality of W. A new trend of recent years is the use of overcomplete transforms, replacing the traditional unitary ones (see [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] as representative works). This trend was partly motivated by the growing realization that orthogonal wavelets are weak in describing the singularities found in images. Another driving force in the introduction of redundant representations is the sparsity in the representation it can provide, which many applications find desirable. Finally, we should mention the desire to obtain shift-invariant transforms, again calling for redundancy in the representation. In these methods the transform is defined via a non-square full rank matrix T ∈ IRL×N , with L > N . Such redundant methods, like the un-d...

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...ximate and design orientation-selective feature detectors [5], [6], [7], [8]. The concept of steerablity was also applied successfully in other areas of image processing such as texture analysis [9], =-=[10]-=- and image denoising [11]. In this paper, we propose to reconcile the two methodologies—computational approach and steerable filterbanks—by presenting a general strategy for the design of 2D steerable...

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...on the noise. Also, crucial to the optimality of this method is the orthogonality of W. A new trend of recent years is the use of overcomplete transforms, replacing the traditional unitary ones – see =-=[7, 8, 9, 10, 11, 12]-=- for representative works. This trend was partly motivated by the growing realization that orthogonal wavelets are weak in describing the singularities found in images. Another driving force in the in...

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...acterized by statistical interactions within local neighborhoods [23]. Recently, a new framework for statistical signal processing based on wavelet-domain hidden Markov models has been proposed [24], =-=[25]-=-. It provides an attractive modeling of both the non-Gaussian statistics and the property of persistence across scales in a wavelet decomposition. In this paper, we proceed by grouping the wavelet sub...

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...y result in certain dependencies across the three DWT subbands. It was also shown in [15] that the dependencies across subbands are useful for wavelet-based texture characterization. Specifically, in =-=[36, 37]-=-, a vector wavelet-domain HMM was proposed which 3sincorporates multivariate Gaussian densities. The statistical dependencies across DWT subbands can be captured by the non-diagonal entries in the cov...

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...tions and rotations of a single function, and a ſlter at any orientation can be computed as a linear combination of the basis ſlters. This property can be used in rotation invariant texture retrieval =-=[9]-=-. The main advantage of the complex wavelet decomposition is that it can produce texture features which are more robust to translation of the image [8]. Recently, the complex directional ſlter bank is...

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...e images based on their visual contents. Earlier efforts in CBIR research have been focused on effective feature representations for images. The visual features of images, such as color [32], texture =-=[5]-=-[9], and shape features [18][38] have been extensively explored to represent and index image contents, resulting in a collection of research prototypes and commercial systems [6][7]. There are also so...