Steerable Discrete Cosine Transform

Abstract

In image compression, classical block-based separable transforms tend to be inefficient when image blocks contain arbitrarily shaped discontinuities. For this reason, transforms incorporating directional information are an appealing alternative. In this paper, we propose a new approach to this problem, namely a discrete cosine transform (DCT) that can be steered in any chosen direction. Such transform, called steerable DCT (SDCT), allows to rotate in a flexible way pairs of basis vectors, and enables precise matching of directionality in each image block, achieving improved coding efficiency. The optimal rotation angles for SDCT can be represented as solution of a suitable rate-distortion (RD) problem. We propose iterative methods to search such solution, and we develop a fully fledged image encoder to practically compare our techniques with other competing transforms. Analytical and numerical results prove that SDCT outperforms both DCT and state-ofthe-art directional transforms.

A linear programming approach to sparse linear regression with quantized data

“…This assumption naturally occurs in a number of applications, such as localization problems [23], image processing [24], and power allocation [25]. However, extensions to more general classes are possible and will be studied in future work.…”

Transformada Discreta do Cosseno Manobrável em Três Dimensões

“…Logo, a GFT para um sinal sobre um grafo caminhoé equivalenteà DCT. No caso bidimensional, Fracastoro et al [8] relacionaram a GFT para um sinal sobre um grafo grade P N × P N com a 2D-DCT, cujos vetores de base formam uma possível autobase para L(P N × P N ). Uma vez que a 2D-DCT nãoé á unica autobase para L(P N × P N ), pode-se encontrar todas as possíveis autobases a partir da rotação dos referidos vetores, obtendo uma nova transformada denominada transformada discreta do cosseno manobrável (SDCT, steerable discrete cosine transform).…”

“…E-mail: juliano bandeira@ieee.org. grafos, outras transformadas podem ser definidas no referido domínio [8], [9].…”

“…Neste trabalho, propõe-se uma transformada direcional obtida a partir da relação entre a transformada discreta do cosseno (DCT, discrete cosine transform) e a GFT sobre um grafo caminho. A transformada propostaé denominada transformada discreta do cosseno manobrável em três dimensões (3D-SDCT, 3D-steerable discrete cosine transform) e dá sequênciaà ideia originalmente proposta em [8]. Emprega-se o princípio de que os vetores de base da transformada discreta do cosseno em três dimensões (3D-DCT, 3D-discrete cosine transform) constituem uma possível autobase para o Laplaciano do produto de três grafos caminho.…”

Steerable Discrete Fourier Transform

“…In [2], the theory of graph signal processing [13], and particularly the relationship between the graph Fourier transform and grid graphs, has been exploited to define a new directional 2D-DCT [14] that can be steered in a chosen direction. In this letter, we extend this concept and present a new generalization of the DFT, called steerable discrete Fourier transform (SDFT).…”