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Steerable Discrete Cosine Transform Politecnico di Torino Porto Institutional Repository [Proceeding] Steerable Discrete Cosine Transform Original Citation: Fracastoro, G.; Magli, E. (2015). Steerable Discrete Cosine Transform. In: 2015 IEEE International Workshop on Multimedia Signal Processing, 2015. pp. 1-6 Availability: This version is available at : http://porto.polito.it/2638898/ since: April 2016 Publisher: IEEE Published version: DOI:10.1109/MMSP.2015.7340827 Terms of use: This article is made available under terms and conditions applicable to Open Access Policy Article ("Public - All rights reserved") , as described at http://porto.polito.it/terms_and_conditions. html Porto, the institutional repository of the Politecnico di Torino, is provided by the University Library and the IT-Services. The aim is to enable open access to all the world. Please share with us how this access benefits you. Your story matters. (Article begins on next page) Steerable Discrete Cosine Transform Giulia Fracastoro #1, Enrico Magli #2 # Department of Electronics and Telecommunications, Politecnico di Torino, Italy 1 [email protected] 2 [email protected] Abstract—Block-based separable transforms tend to be inef- the use of symmetry to reduce the number of transform ficient when blocks contain arbitrarily shaped discontinuities. matrices needed [7]–[9]. However, the main problem of this For this reason, transforms incorporating directional information methods is that training sets must be processed to obtain are an appealing alternative. In this paper, we propose a new approach to this problem, designing a new transform that can transforms that are optimal for a given mode. be steered in any chosen direction and that is defined in a Recently, a new approach in image and video coding is rigorous mathematical way. This new steerable DCT allows to emerging. Any image can be viewed as a graph, where rotate in a flexible way pairs of basis vectors, enabling precise each pixel is a node of the graph and the edges describe matching of directionality in each image block, and thereby the connectivity relations among the pixels, e.g. in terms achieving improved coding efficiency. We tested the proposed transform on several images and the results show that it provides of similarity [10]. This graph representation allows one to a significant performance gain compared to the DCT. Moreover, design an edge-aware transform in an elegant and effective the mathematical framework on which the steerable DCT is way. Block-based method using graph transform have been based allows to generalize the transform to more complex steering proposed in [11], [12], but they all reported unsatisfactory patterns than a single pure rotation. results on natural images that are not piece-wise smooth. For the specific case of texture images, a new class of transforms I. INTRODUCTION called graph template transforms was proposed to approximate The 2D Discrete Cosine Transform (DCT) is by far the most the KLT by exploiting a priori information known about widespread transform used for block-based image and video signals represented by a graph-template [13]. However, one of compression [1]. The 2D DCT is implemented by two separa- the main drawbacks of graph-based compression techniques ble 1D transforms along the vertical and horizontal directions. lies in the cost required to represent and encode the graph, Thus, the conventional DCT can efficiently represent images in which may outweigh the coding gain provided by the edge which horizontal or vertical edges are dominating. Instead, one adaptive transform. For this reason, recently some graph-based of the main drawbacks of the DCT is that it becomes inefficient compression methods that require a small overhead have been when a block contains arbitrarily shaped discontinuities. presented [14], [15], however, even if their performances are To overcome this problem, various solutions have been competitive compared to the DCT, the construction of the proposed in the past and different approaches have been transform matrix has a high computational cost. explored. Several researchers have proposed to modify the In this paper, we present a new framework for directional implementation of the separable 2D DCT in order to incor- transforms. Starting from the graph transform of a grid graph, porate directional information into the transform [2]–[4]. The we design a new transform, the Steerable DCT (SDCT), that Directional DCT (DDCT) presented in [2] is the first attempt can be obtained simply by rotating the 2D DCT basis. In this in this sense. The authors developed a new separable transform way, the proposed transform can be oriented in any possible in which the first 1D DCT may choose to follow a direction direction and can achieve better compression performance than other than the vertical or horizontal one, then the coefficients the DCT. The SDCT is defined in a rigorous mathematical produced by all directional transforms in the first step are way and can be easily computed starting from the 2D DCT. rearranged so that the second transform can be applied to the In addition, we show that the proposed framework is very coefficients that are best aligned with each other. However, this general and it is open to many further extensions that could method faces several issues [5]: 1D DCTs of various lengths lead to significant improvements. are needed, some of them are very short and their lengths are The paper is organized as follows. Section 2 presents some not always a power of 2. Moreover, the second DCT may not theoretical results that will be needed in the following sections always be applied to coefficients of similar AC frequencies. of the paper. We then introduce the construction method for In the specific case of intra-frame video coding, another ap- the Steerable DCT basis in Section 3. In Section 4, the results proach to design a directional transform has been investigated. of our experimental tests are presented. A final discussion on In [6], the authors proposed to derive mode-dependent direc- the proposed method is conducted in Section 5. tional transforms (MDDT) from Karhunen-Loeve transform (KLT) using prediction residuals from training video data. II. GRAPH FOURIER TRANSFORM AND DCT Using this approach, various follow-up works have presented We start our analysis by briefly reviewing the graph Fourier several variations and enhancements for the MDDT exploiting transform and highlighting the link between this transform and the DCT. A. Laplacian matrix and graph Fourier transform (a) A graph can be denoted as G = (V; E), where V is the set of vertices (or nodes) with jV j = N and E ⊂ V × V is the set of edges. It is possible to represent a graph by its N×N adjacency matrix A 2 R , where Aij = 1 if there is an edge between node i and j, otherwise Aij = 0. The graph Laplacian is defined as L = D − A, where D is a diagonal matrix whose ith diagonal element di is equal to the number of edges incident to node i. We can define a signal f on the vertices of the graph and it can be represented as a vector f 2 RN , where the ith component of f represents the signal value at the ith vertex in V . (b) In the graph domain, it is possible to define an equivalent of the Fourier transform, i.e. the graph Fourier transform [10]. ^ N Fig. 1. Two graph models: (a) the path graph P4, (b) the square grid graph The graph Fourier transform f of any signal f 2 R is defined P4 × P4. as ^f = Uf; multiplicity of the eigenvalues in (2) is always 1, the 1D DCT where U is the matrix whose rows are the eigenvectors of basis is the unique eigenbasis for the Laplacian of a path graph, the graph Laplacian L. The inverse graph Fourier transform is therefore the graph transform for a signal represented by a path then given by graph is equivalent to the 1D DCT transform. f = U T ^f: It is easy to see that a grid graph as the one shown in Fig. B. Product graph and its Laplacian 1(b) is the product of two path graphs. If the two path graphs have the same number of vertices n, their product graph is a Given two graphs G1 = (V1;E1) and G2 = (V2;E2), the square grid graph. It has been proved that the basis vectors of product of G1 and G2 is the graph G = G1 ×G2 whose vertex the 2D DCT form an eigenbasis of the Laplacian matrix of set is the Cartesian product V1 × V2. Suppose v1, v2 2 V1 and the square grid graph [19]. u1, u2 2 V2. Then (v1; u1) and (v2; u2) are adjacent in G if and only if one of the following conditions are satisfied [16]: III. STEERABLE DCT • v1 = v2 and fu1; u2g 2 E2; Starting from the theoretical results presented in the previ- • fv1; v2g 2 E1 and u1 = u2. ous section, we build a new transform that can be oriented in Theorem [16], [17]. Let G1 and G2 be graphs on n1 and n2 any direction. vertices, respectively. Then the eigenvalues of L(G ×G ) are 1 2 A. Preliminaries all possible sums of λi(G1) + λj(G2), with 0 ≤ i ≤ n1 − 1 (i) and 0 ≤ j ≤ n2 − 1. Moreover, if v is an eigenvector Using the theorem presented in the previous section and (j) of G1 corresponding to λi(G1), v an eigenvector of G2 equations (1) and (2), we can compute the eigenvalues and corresponding to λj(G2) and ⊗ is the Kronecker product, then the corresponding eigenvectors of the Laplacian of the square (i) (j) v ⊗ v is an eigenvector of G corresponding to λi(G1) + grid graph λj(G2).
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