SPARSE APPROXIMATION WITH ADAPTIVE DICTIONARY FOR IMAGE PREDICTION Mehmet T¨urkan, Christine Guillemot INRIA/IRISA - University of Rennes 1 Campus Universitaire de Beaulieu, 35042 RENNES, France
[email protected] ABSTRACT
introduced as a heuristic method which aims at finding approximate solutions to the sparse signal approximation problem with tractable complexity [3]. It has then been improved to give at each iteration the linear span of atoms which would give the best signal approximation in the sense of minimizing the residue of the new approximation. This approach is known as orthogonal matching pursuit (OMP) [4]. In the experiments reported here, the OMP algorithm has been used. The first sparse spatial prediction approach with static dictionaries formed by pre-defined DCT or DFT waveforms has been proposed in [5]. These dictionaries are particularly well suited for predicting periodic texture patches. However, the prediction fails for more complex non-periodic structures with discontinuities. The technique described here considers instead a dynamic and locally adaptive dictionary formed by atoms derived from texture patches present in a causal neighborhood of the block to be predicted. The principle of the approach is to first search for the linear combination of basis functions which best approximates known sample values in a causal neighborhood, and keep the same linear combination of basis functions to approximate the unknown sample values in the block to be predicted. The best approximation support among a set of seven pre-defined modes is selected according to a given criterion. The considered criteria are the mean square error (MSE) of the predicted signal and a rate-distortion cost function when the prediction is used in a coding scheme. Note that, the OMP algorithm stops when the residual error energy of selected atoms is under a pre-specified threshold. The set of coefficients selected by the iterative algorithm leads to a good representation of the causal neighborhood. However, the algorithm does not put any constraint on the values of the block to be predicted. Therefore, the last iteration may not lead to the best predicted signal or to the best prediction in a rate-distortion sense when the prediction is included in a complete coding algorithm. To optimize the prediction according to a given criterion, one keeps track of the sparse vectors computed at each iteration of the algorithm and keeps the one leading which will satisfy the chosen criterion, that is either minimize residual error energy on the predicted block or minimize a rate-distortion cost function (as for the choice of the approximation support). The corresponding it-
The paper presents a dictionary construction method for spatial texture prediction based on sparse approximations. Sparse approximations have been recently considered for image prediction using static dictionaries such as a DCT or DFT dictionary. These approaches rely on the assumption that the texture is periodic, hence the use of a static dictionary formed by pre-defined waveforms. However, in real images, there are more complex and non-periodic textures. The main idea underlying the proposed spatial prediction technique is instead to consider a locally adaptive dictionary, A, formed by atoms derived from texture patches present in a causal neighborhood of the block to be predicted. The sparse spatial prediction method is assessed against the sparse prediction method based on a static DCT dictionary. The spatial prediction method is then assessed in a complete image coding scheme where the prediction residue is encoded using a coding approach similar to JPEG. Index Terms— Texture prediction, image coding, sparse approximations, adaptive dictionary, matching pursuits 1. INTRODUCTION Closed-loop spatial prediction is a key component of image and video coding algorithms. E.g., in H.264, there are two intra prediction types called Intra-16x16 and Intra-4x4 respectively [1]. The Intra-16x16 type supports four intra prediction modes while the Intra-4x4 type supports nine modes. Each 4x4 block is predicted from prior encoded samples from spatially neighboring blocks. In addition to the so-called “DC” mode which consists in predicting the entire 4x4 block from the mean of neighboring pixels, eight directional prediction modes are specified. The prediction is done by simply “propagating” the pixel values along the specified direction. This approach is suitable in presence of contours, when the directional mode chosen corresponds to the orientation of the contour. However, it fails in more complex textured areas. An alternative prediction method based on so-called template matching has been described in [2]. Here, methods based on sparse signal approximation are considered. The matching pursuit (MP) algorithm has been
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eration number then needs to be transmitted to the decoder. The quality of the predicted signal (visually and in terms of peak signal-to-noise ratio (PSNR)) is significantly improved compared with static waveform (e.g. DCT) dictionaries. The prediction method has then been assessed in an image coding algorithm in which the residue is coded with an algorithm similar to JPEG. The iteration number and the approximation support type are Huffman coded. The PSNR-bitrate performance curves show a gain of up to 2.5-3 dB when compared with the JPEG image coding standard.
In this case, the value of the coefficient is added to the previous one. The algorithm proceeds until the stopping criterion 2 (Axk − b ≤ ρ) is satisfied, where ρ is a tolerance parameter which controls the sparseness of the representation. The MP algorithm yields an approximation error which decreases with each iteration. However, it is sub-optimal. At any iteration step, the newly obtained residual is orthogonal only to the immediately selected atom, but it may not be orthogonal to all the atoms selected at the previous steps. As a result, some atoms selected at an earlier iteration may get selected again. This causes slow convergence. The OMP removes this drawback by updating the coefficients of all previously selected atoms so that the newly derived residual is orthogonal to not only the immediately selected atom, but also all the atoms selected at previous iterations. As a consequence, once an atom is selected, it is never selected again in subsequent iterations. Like the MP, at kth iteration the algorithm first identifies the atom ajk having the maximum correlation with the approximation error. Let Ak denote the matrix containing all the atoms selected in the previous iterations. One then projects b onto the subspace spanned by the columns of Ak , i.e. one solves
2. SPARSE REPRESENTATION ALGORITHMS Given A ∈ Rm×n and b ∈ Rm with m
