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This chapter discusses the problem of coding images using very redundant libraries of waveforms, also referred to as dictionaries. We start with a discussion of the shortcomings of classical approaches based on orthonormal bases. More specifically, we show why these redundant dictionaries provide an interesting alternative for image representation....
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... singularities may be highly orga-nized along embedded submanifolds and this is exactly what happens at image contours for example. Figure 8.1 shows that wavelets are inefficient at representing contours because they cannot deal with the geometrical regularity of the contours themselves. This is mainly due to the isotropic refinement implemented by wavelet basis: the dyadic scaling factor is ap- plied in all directions, where clearly it should be fine along the direction of the local gradient and coarse in the orthogonal direction in order to ef- ficiently localize the singularity in a sparse way. ...
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... advantages offered by both the greedy expansion, and the structured dictionary are used to provide flexibility in image representation. The encoder can be represented as in Figure 8.2. The input image is compared to a redundant library of functions, using a Matching Pur- suit algorithm. ...
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... of the energy of the Gaussians lies in the frequency band taken by the AR fuctions. The biggest scale for these Gaussian atoms has been chosen so that at least 50% of the atom energy lies within the signal space when centered in the image. Lastly, due to isotropy, rotations are obviously useless for this kind of atoms. Sample atoms are shown in Fig. 8.3. ...
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... other two dictionaries are simply built on orthogonal wavelet bases. Figure 8.4 shows the reconstructed quality as a function of the number of iterations in the MP expansion using different types of dictionaries. ...
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... highlight the coding penalty due to anisotropic refinement, the image has also been coded with the same dictionary, built on isotropic atoms, all other parameters staying identical to the proposed scheme. Figure 8.5 illustrates the quality of the MP en- coding of Lena, as a function of the coding rate, with both dictionaries. To perform the comparison, the isotropic and the anisotropic dictionaries are generated with the same generating function and with the same discretiza- tion of the parameters (3 scales per octave and an angle resolution of 10 degrees). ...
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... Figure 8.6 presents a comparison between images compressed with Match- ing Pursuit, and respectively JPEG-2000 2 . It can be seen that the PSNR rating is in favor of JPEG-2000, which is not completely surprising since a lot of research efforts are being put in optimizing the encoding in JPEG- 2000 like schemes. ...
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... coding artifacts are quite different, and the degradations due to Matching Pursuit are less annoying to the Human Visual System, than the ringing due to wavelet coding at low rate. The detailed view of the hat, as illustrated in Figure 8.7, confirms this impression. ...
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... be complete, Figure 8.8 shows the rate-distortion performance of the Matching Pursuit encoder for common test images, at low to medium bit rates. ...
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... this channel only. Once the best atom has been identified, its contribution in the other two channels is also computed and encoded. The reduced complexity algorithm obviously performs in a suboptimal way compared to the maximization of the global energy, but in most of the cases the quality of the approximation does only suffer a minimal penalty (Fig. 8.9 is an example of a Matching Pursuit performed in the most ener- getic ...
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... one index for all channels is minimized. The choice of the RGB color space thus seems very natural. This can also be highlighted by the fol- lowing experiments. The coefficients [c 1 n , c 2 n , c 3 n ] of the MP decomposition can be represented in a cube, where the three axes respectively correspond to the red, green and blue components (see Fig. 8.10(a)). It can be seen that the MP coefficients are interestingly distributed along the diagonal of the color cube, or equivalently that the contribution of MP atoms is very similar in the three color channels. This very nice property is a real ad- vantage in overcomplete expansions, where the coding cost is mainly due to the atom indexes. ...
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... is very similar in the three color channels. This very nice property is a real ad- vantage in overcomplete expansions, where the coding cost is mainly due to the atom indexes. On the contrary, the distribution of MP coefficients, resulting from the image decomposition in the YUV color space, does not seem to present any obvious structure (see Fig. 8.10(b)). In addition, the YUV color space has been shown to give quite annoying color distortions for some particular images (see Fig. 8.9 for ...
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... cost is mainly due to the atom indexes. On the contrary, the distribution of MP coefficients, resulting from the image decomposition in the YUV color space, does not seem to present any obvious structure (see Fig. 8.10(b)). In addition, the YUV color space has been shown to give quite annoying color distortions for some particular images (see Fig. 8.9 for ...
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... on the diagonal of the cube, S (Saturation) is the distance of the coefficient to the diagonal and H (Hue) is the direction perpendicu- lar to the diagonal, where the RGB coefficient is located. The HSV values of the MP coefficients present the following characteristics distributions. The Value distribution is Laplacian, centered in zero (see Fig. 8.11(c)), Saturation presents an exponential distribution (see Fig. 8.11(b)), and a Laplacian-like distribution with two peaks can be observed for Hue val- ues 8.11(a). Finally, once the HSV coefficients have been calculated from the available RGB coefficients, the quantization of the parameters is per- formed as ...
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... coefficient to the diagonal and H (Hue) is the direction perpendicu- lar to the diagonal, where the RGB coefficient is located. The HSV values of the MP coefficients present the following characteristics distributions. The Value distribution is Laplacian, centered in zero (see Fig. 8.11(c)), Saturation presents an exponential distribution (see Fig. 8.11(b)), and a Laplacian-like distribution with two peaks can be observed for Hue val- ues 8.11(a). Finally, once the HSV coefficients have been calculated from the available RGB coefficients, the quantization of the parameters is per- formed as ...
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... coefficients and indexes are entropy coded, along the same tech- nique used herebefore for grayscale images. Compression performances of this algorithm are illustrated on Figure 8.12, where a comparison with JPEG-2000 is also provided. ...
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... simple spatial adaption procedure is illustrated in Fig. 8.13, where the encoded image of size 256×256 has been re-scaled with irrational factors Table 8.1 clearly shows that our scheme offers results competitive with respect to state-of-the-art coders like JPEG-2000 for octave-based down- sizing. In addition it allows for non-dyadic spatial resizing, as well as easy up-scaling. The quality of ...
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... simple rate-adption, or filtering operation is equivalent to dropping the last iterations in the MP expansion, focusing on the highest energy atoms. Figure 8.14 illustrates the rate adaptivity performance of the MP en- coder. ...
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... rate scalability is also almost automatic for the color image stream. Fig. 8.15 shows the effects of truncating the MP expansion at different number of coefficients. It can be observed again that the MP algorithm will first describe the main objects in a sketchy way (keeping the colors) and then it will refine the ...
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Citations
... Due to the presence of the stego-message, when the decoder applies again the MP algorithm it may select from the redundant basis a different set of elements and in a different Order hence making it impossible for the decoder to correctly extract the hidden message [9]. Even though the subset of elements of the basis (and their order) is fixed, a change to one coefficient of the decomposition usually results in a variation of all the coefficients of the decomposition when the MP is applied to the modified image. ...
A new secret writing using DES and DEFLATE algorithm is explored by improving the previous attempts in this sense in terms of security and compression. A Matching Pursuit algorithm using redundant basis decomposition technique is used to securely hide a message. The stability and computational complexity problems are solved by introducing new selection and update rules working entirely in the integer domain. Image decomposition is randomized in several ways thus improving the stego-message undetectability, and making the hidden message undetectable by targeted steganalyzers. A refinement decomposition step is applied to the three color bands to increase the stego-message payload. Due to the ever-increasing security threats, there is a need to develop algorithms with more complexity and advanced features for a secured message transfer. A comparative study is performed between Lempel-Ziv-Welch (LZW) data compression algorithm and deflate loss-less compression algorithm which proves the efficiency of deflate algorithm in both rate of compression and its compression speed.
... They are generally created to provide a very rich and dense family of functions built from the geometrical features of the analyzed image. They have applications in image and video coding [217], multi-modal signal analysis (e.g., video plus audio) [218], and also for signal decomposition on non-Euclidean spaces [219]. ...
The richness of natural images makes the quest for optimal representations in
image processing and computer vision challenging. The latter observation has
not prevented the design of image representations, which trade off between
efficiency and complexity, while achieving accurate rendering of smooth regions
as well as reproducing faithful contours and textures. The most recent ones,
proposed in the past decade, share an hybrid heritage highlighting the
multiscale and oriented nature of edges and patterns in images. This paper
presents a panorama of the aforementioned literature on decompositions in
multiscale, multi-orientation bases or dictionaries. They typically exhibit
redundancy to improve sparsity in the transformed domain and sometimes its
invariance with respect to simple geometric deformations (translation,
rotation). Oriented multiscale dictionaries extend traditional wavelet
processing and may offer rotation invariance. Highly redundant dictionaries
require specific algorithms to simplify the search for an efficient (sparse)
representation. We also discuss the extension of multiscale geometric
decompositions to non-Euclidean domains such as the sphere or arbitrary meshed
surfaces. The etymology of panorama suggests an overview, based on a choice of
partially overlapping "pictures". We hope that this paper will contribute to
the appreciation and apprehension of a stream of current research directions in
image understanding.
... For the above reason it may be argued that, from a steganographic point of view, it is preferable to embed the steganographic message at the highest possible semantic level, e.g. by modifying structural elements of the host signal like lines, edges or flat areas in the case of still images. Following a similar need arising from image compression applications [1], a new class of image representation methods has been recently developed that rely on redundant bases decomposition. In practice a dictionary with a large number of elementary signals (called atoms) is built, trying to ensure that, for each image (or image block), a subset of few atoms exists that permits to represent the image efficiently. ...
... Discreteor real-valued atoms can be used and atoms can be generated manually or by means of a generating function. In classical MP techniques, applied to still images [1], the dictionary is 1 The solutions reported below are inherited from the system described in [3]. We included their description in the current paper for the sake of clarity. ...
The possibility of using redundant basis expansion to securely hide a message within a cover color image is explored by improving previous attempts in this sense in terms of security and payload. The stability and computational complexity problems of previous works are solved by introducing new selection and update rules working entirely in the integer domain, and by fully exploiting the availability of three color bands in such a way that all the available atoms in the three color bands are used to convey the stego-message. Image decomposition is randomized in several ways thus improving the stego-message undetectability, and making the hidden message undetectable by targeted steganalyzers explicitly developed to exploit the weaknesses of the MPSteg algorithm. The security of the new scheme is also evaluated by testing it against blind steganalyzers and compared to that of plusmn1 embedding algorithm applied in the pixel domain.
In this paper we propose a new steganographic algorithm based on Matching Pursuit image decomposition. Many modern approaches to detect the presence of hidden messages are based on statistical analysis, preferably on the analysis of higher-order statistical regularities. The idea behind this work is to adaptively choose the elements of a redundant basis to represent the host image. In this way, the image is expressed as the composition of a set of structured elements resembling basic image structures such as lines, corners, and flat regions. We argue that embedding the watermark at this, more semantic, level results in a lower modification of the low-level statistical properties of the image, and hence in a lower detectability of the presence of the hidden message.
A new steganographic algorithm for color images (MPSteg-color) is presented based on Matching Pursuit (MP) decomposition of
the host image. With respect to previous works operating in the MP domain, the availability of three color bands is exploited
to avoid the instability of the decomposition path and to randomize it for enhanced security. A selection and an update rule
working entirely in the integer domain have been developed to improve the capacity of the stego channel and limit the computational
complexity of the embedder. The system performance are boosted by applying the Matrix Embedding (ME) principle possibly coupled
with a Wet Paper Coding approach to avoid ambiguities in stego-channel selection. The experimental comparison of the new scheme
with a state-of-the-art algorithm applying the ME principle directly in the pixel domain reveals that, despite the lower PSNR
achieved by MPSteg-color, a classical steganalyzer finds it more difficult to detect the MP-stego messages.
We introduce greedy basis pursuit (GBP), a new algorithm for computing sparse signal representations using overcomplete dictionaries. GBP is rooted in computational geometry and exploits equivalence between minimizing the l<sup>1</sup>-norm of the representation coefficients and determining the intersection of the signal with the convex hull of the dictionary. GBP unifies the different advantages of previous algorithms: like standard approaches to basis pursuit, GBP computes representations that have minimum l<sup>1</sup>-norm; like greedy algorithms such as matching pursuit, GBP builds up representations, sequentially selecting atoms. We describe the algorithm, demonstrate its performance, and provide code. Experiments show that GBP can provide a fast alternative to standard linear programming approaches to basis pursuit.
