The Continuous Wavelet Transform on Conic Sections

Antoine, Jean-Pierre; Bogdanova, Iva; Vandergheynst, Pierre

doi:10.1142/s0219691308002288

Cited by 18 publications

(9 citation statements)

References 31 publications

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“…Given a two-dimensional manifold M, i.e., locally isomorphic to R 2 , authors in [298] The theoretical invertibility of this transform has however to be studied specifically in each case, i.e., given M and Π ξ . Results exist for instance for the two-sheeted hyperboloid and the paraboloid in R 3 [299].…”

Section: Wavelets On General 2-manifoldsmentioning

confidence: 99%

A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity

et al. 2011

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International audienceThe 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 multi-scale 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 de-compositions 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 ". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding

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Section: Wavelets On General 2-manifoldsmentioning

confidence: 99%

A panorama on multiscale geometric representations, intertwining spatial, directional and frequency selectivity

et al. 2011

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“…In fact, there is an infinite family of projections from H 2 + onto the plane z = 0, leading to r(χ) = 1 p sinh pχ, but the case p = 1 2 seems the most natural one. More details may be found in [5,7,9]. The same technique can be used for the one-sheeted hyperboloid H 1,1 , of equation z 2 = −1 + x 2 + y 2 , which has the same tangent null-cone…”

Section: The Conical Projection Of the Hyperboloidmentioning

confidence: 99%

“…A group-theoretical tentative exists [19], based on a homeomorphim from a cylinder onto the pointed paraboloiḋ P 2 := P 2 \ {0}, but it does not produce convincing results, in particular, the formalism does not allow for a local dilation around an arbitrary point. Some additional details may be found in [5].…”

Section: Without Global Isometry Groupmentioning

confidence: 99%

Wavelet transform on manifolds: Old and new approaches

Antoine

Rosca

Vandergheynst

2010

Applied and Computational Harmonic Analysis

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Given a two-dimensional smooth manifold M and a bijective projection p from M on a fixed plane (or a subset of that plane), we explore systematically how a wavelet transform (WT) on M may be generated from a plane WT by the inverse projection p −1 . Examples where the projection maps the whole manifold onto a plane include the two-sphere, the upper sheet of the two-sheeted hyperboloid and the paraboloid. When no such global projection is available, the construction may be performed locally, i.e., around a given point on M. We apply this procedure both to the Continuous WT, already treated in the literature, and to the Discrete WT. Finally, we discuss the case of a WT on a graph, for instance, the graph defined by linking the elements of a discrete set of points on the manifold.

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“…Interesting results have been obtained, but much work still lies ahead. We will report in detail on these nonspherical cases in a further publication [5] and give here only a glimpse.…”

Section: Outcome: the Cwt On Other Conic Sectionsmentioning

confidence: 99%