Steerable Discrete Fourier Transform

Fracastoro, Giulia; Magli, Enrico

doi:10.1109/lsp.2017.2657889

Cited by 19 publications

(10 citation statements)

References 29 publications

(32 reference statements)

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“…Due to the circulant structure, the DFT is one of the GFTs of Gc. The family of all GFTs of Gc is called the steerable DFTs [43]. Note that, for any n, G is φ = (n, n − 1, .…”

Section: Steerable Dftsmentioning

confidence: 99%

Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

Ortega

2019

IEEE Trans. Signal Process.

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The graph Fourier transform (GFT) is an important tool for graph signal processing, with applications ranging from graph-based image processing to spectral clustering. However, unlike the discrete Fourier transform, the GFT typically does not have a fast algorithm. In this work, we develop new approaches to accelerate the GFT computation. In particular, we show that Haar units (Givens rotations with angle π 4) can be used to reduce GFT computation cost when the graph is bipartite or satisfies certain symmetry properties based on node pairing. We also propose a graph decomposition method based on graph topological symmetry, which allows us to identify and exploit butterfly structures in stages. This method is particularly useful for graphs that are nearly regular or have some specific structures, e.g., line graphs, cycle graphs, grid graphs, and human skeletal graphs. Though butterfly stages based on graph topological symmetry cannot be used for general graphs, they are useful in applications, including video compression and human action analysis, where symmetric graphs, such as symmetric line graphs and human skeletal graphs, are used. Our proposed fast GFT implementations are shown to reduce computation costs significantly, in terms of both number of operations and empirical runtimes.

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“…Due to the circulant structure, the DFT is one of the GFTs of Gc. The family of all GFTs of Gc is called the steerable DFTs [43]. Note that, for any n, G is φ = (n, n − 1, .…”

Section: Steerable Dftsmentioning

confidence: 99%

Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

Ortega

2019

IEEE Trans. Signal Process.

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“…De (9), segue que os autovalores de L são todas as possíveis somas entre os autovalores de cada um dos grafos caminho, istoé, λ k, ,m = 4 sen 2 (ak) + sen 2 (a ) + sen 2 (am) ,…”

Section: A Uma Autobase Para O Laplaciano Do Produto De Três Grafos unclassified

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

Section: Introductionunclassified

Transformada Discreta do Cosseno Manobrável em Três Dimensões

Lima¹,

Madeiro²,

Lima³

2018

Anais De XXXVI Simpósio Brasileiro De Telecomunicações E Processamento De Sinais

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Resumo-Este trabalho propõe uma nova transformada direcional, a transformada discreta do cosseno manobrável em três dimensões (3D-SDCT), queé obtida a partir da relação entre a transformada discreta do cosseno (DCT) e a transformada de Fourier sobre grafos de um sinal sobre um grafo caminho; utiliza-se o fato de que os vetores de base da 3D-DCT constituem uma possível autobase para o Laplaciano do produto entre três desses grafos. A transformada proposta emprega uma versão rotacionada da base da 3D-DCT. Mostra-se que, escolhendô angulos de rotação específicos,é possível obter uma 3D-SDCT em que 2/3 dos coeficientes são nulos, o que sugere uma maior compactação de energia quando comparadaàquela obtida pelo uso da 3D-DCT. Palavras-Chave-Processamento de sinais sobre grafos, transformada de Fourier sobre grafos, transformada discreta do cosseno.

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“…Though a digital image contains pixels that reside on a regularly sampled 2D grid, one can nonetheless interpret an image (or an image patch) as a signal on a graph, with edges that connect each pixel to its neighborhood of pixels. By choosing an appropriate graph that reflects the intrinsic image structure, a spectrum of graph frequencies can be defined through eigen-decomposition of the graph Laplacian matrix [2], and notions like transforms [3]- [8], wavelets [9]- [11], smoothness [12]- [16] etc can be correspondingly derived. Then a target image (or image patch) can be decomposed and analyzed spectrally on the chosen graph using developed GSP tools-analogous to frequency decomposition of square pixel blocks via known transforms like discrete cosine transform (DCT).…”

Section: Introductionmentioning

confidence: 99%

“…For image compression, a Fourier-like transform for graphsignals called graph Fourier transform (GFT) [1] and many variants [5]- [8], [18], [19] have been used as adaptive transforms for coding of piecewise smooth (PWS) and natural images. Because the underlying graph used to define GFT can be different for each code block, the cost of describing the graph as well as the cost of coding GFT coefficients to represent the signal must both be taken into consideration.…”

Section: Introductionmentioning

confidence: 99%

Graph Spectral Image Processing

et al. 2018

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Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation.

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Steerable Discrete Fourier Transform

Cited by 19 publications

References 29 publications

Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

Transformada Discreta do Cosseno Manobrável em Três Dimensões

Graph Spectral Image Processing

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