## A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity (2011)

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Citations: | 7 - 3 self |

### BibTeX

@MISC{Jacques11apanorama,

author = {Laurent Jacques and Laurent Duval and Caroline Chaux and Gabriel Peyré},

title = {A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity},

year = {2011}

}

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### Abstract

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”.

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Citation Context ...ied by additional degradations from which acquired data may suffer such as blur, jitter and noise. Descriptive mathematical models of images combining cartoon and textures become increasingly popular =-=[12, 13]-=- and progressively yield tractable algorithms. We note that there exists a continuum of real-world images between cartoon and textures, ranging from cartoon-ish Yogi bear in Fig. 1(a) to “textural” fi... |

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Citation Context ...es with different orientations. The resulting oriented wavelet dictionary has a small redundancy and is also computationally efficient. The corresponding wavelet is approximately shift invariant, see =-=[113]-=- for more details. It is extended to the M-band setting by Chaux et al. [114] and to wavelet packets in [115, 116]. In Fig. 10, one subband of the wavelet transform (red square in Fig. 7), two subband... |

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Citation Context ...−τ)}m,τ . For a generic basis, this frame has up to N 2 atoms. For a wavelet basis, the frame has O(N log(N)) atoms, and the coefficients are computed with the fast “à trous” algorithm in O(N log(N)) =-=[60, 68]-=-. The translation invariant paradigm additionally draws a connection between the scale-space formalism (Sec. 2.1) [69] and thresholding (Sec. 1.3.2). Several 2-D design described in the next sections ... |

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Citation Context ...ile larger ones persist for a larger duration. Interestingly, this diffusion is equivalently described by a filtering process: the convolution of the image by a Gaussian function Gσ of width σ = √ 2τ =-=[38, 39, 40]-=-. This image unfolding into a scale-space domain has led to many new image processing techniques such as edge, ridge and feature detection [41, 42]. This is illustrated in Fig. 4, where the original i... |