Multiscale Analysis for Images on Riemannian Manifolds

Calderero, Felipe; Caselles, Vicent

doi:10.1137/130923142

Cited by 5 publications

(13 citation statements)

References 67 publications

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“…The proof follows the same lines of the corresponding result in [1], particularly section 3.1, Theorem 1 (see also Theorem 3.1 in [5]), and so we shall omit it.…”

Section: Let T T Be a Multiscale Analysis Satisfying The Recursivitymentioning

confidence: 82%

“…The multiscale similarity measure C(t, x, y) = R N g t (z)C(0, x + Ah, y + Bh) dh, where g t is the Gaussian of scale t, and C(0, x, y) = (I(x) − J(y)) 2 , satisfies (4.26). Let us finally say that from the mathematical point of view the basic ingredients are the papers [1,8,7,5], and our results are an extension of them. This paper contains mostly the theoretical results that define multiscale analysis for image comparison.…”

mentioning

confidence: 82%

“…The following results were proved in [1] (section 3.2, Theorem 2) for multiscale analysis on images and extended to images on manifolds in [5] (particularly in Theorem 3.2). We follow the presentation in [5].…”

Section: Let T T Be a Multiscale Analysis Satisfying The Recursivitymentioning

confidence: 98%

“…We follow the presentation in [5]. Recall that we have denoted G = (G 1 , G 2 ) and Γ = Γ (1) ⊗ Γ (2) .…”

Section: Let T T Be a Multiscale Analysis Satisfying The Recursivitymentioning

confidence: 99%

“…In the case of comparing image patches defined on Riemannian manifolds, there will appear a large family of possibilities, derived from the axiomatic approach. As in [1,5], the set of axioms will include architectural axioms and a comparison principle that permit us to define multiscale analyses as solutions of a degenerate parabolic PDE. Further specification can be attained by including linear or morphological assumptions.…”

mentioning

confidence: 99%

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Multiscale Analysis of Similarities between Images on Riemannian Manifolds

Ballester¹,

Calderero²,

Caselles³

et al. 2014

Multiscale Model. Simul.

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Abstract. In this paper we study the problem of comparing two patches of an image defined on a Riemannian manifold, which can be defined by the image domain with a suitable metric depending on the image. The size of the patch will not be determined a priori, and we identify it with a variable scale. Our approach can be considered as a nonlocal extension (comparing two points) of the multiscale analyses defined using the axiomatic approach byÁlvarez et al. [Arch. Ration. Mech. Anal., 123 (1993), pp. 199-257]. Following this axiomatic approach, we can define a set of similarity measures that appear as solutions of a degenerate partial differential equation. This equation can be further specified in the linear case, and we observe that it contains as a particular instance the case of using weighted Euclidean distances as comparison measures. Finally, we discuss the case of some morphological scale spaces that exhibit a higher complexity. 1. Introduction. Our purpose in this paper is to compare two patches of an image defined on a Riemannian manifold, which can be defined by the image domain with a suitable metric depending on the image. The size of the patch will not be determined a priori, and we identify it with a variable scale. Our approach can be considered as a nonlocal extension (comparing two points) of the multiscale analyses defined using the axiomatic approach in [1].Let us review the fundamentals of that approach. A multiscale analysis represents a given image at different scales of smoothing, the scale being related to the size of the neighborhood which is used to give an estimate of the brightness of the picture at a given point. It is a basic preprocessing step for shape recognition [27] (see [16,6] and references therein).The systematic study of multiscale analyses for images was the purpose of the axiomatic approach proposed in [1]. Based on a series of axioms which define the structure of the multiscale space and a set of geometric and photometric invariants, multiscale analyses were defined in terms of (viscosity) solutions of a parabolic equation. In the case of linear multiscale analysis they obtained the Gaussian scale space (already proposed and studied in [28,24,25,49,18,17,26,48], using also an axiomatic approach in some of those papers). In addition to the Gaussian scale space, classification covers many of the classical models that were proposed in the literature, such as the Perona-Malik equation [34] (see also [9]), the Rudin-Osher-Fatemi model

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“…The proof follows the same lines of the corresponding result in [1], particularly section 3.1, Theorem 1 (see also Theorem 3.1 in [5]), and so we shall omit it.…”

Section: Let T T Be a Multiscale Analysis Satisfying The Recursivitymentioning

confidence: 82%

mentioning

confidence: 82%

Section: Let T T Be a Multiscale Analysis Satisfying The Recursivitymentioning

confidence: 98%

“…We follow the presentation in [5]. Recall that we have denoted G = (G 1 , G 2 ) and Γ = Γ (1) ⊗ Γ (2) .…”

Section: Let T T Be a Multiscale Analysis Satisfying The Recursivitymentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Multiscale Analysis of Similarities between Images on Riemannian Manifolds

Ballester¹,

Calderero²,

Caselles³

et al. 2014

Multiscale Model. Simul.

Self Cite

View full text Add to dashboard Cite

show abstract

Anisotropic tempered diffusion equations

2020

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Keypoint Detection in RGBD Images Based on an Anisotropic Scale Space

Karpushin

Valenzise

Dufaux

2016

IEEE Trans. Multimedia

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The increasing availability of texture+depth (RGBD) content has recently motivated research towards the design of image features able to employ the additional geometrical information provided by depth. Indeed, such features are supposed to provide higher robustness than conventional 2D features in presence of large changes of camera viewpoint. In this paper we consider the first stage of RGBD image matching, i.e., keypoint detection. In order to obtain viewpoint-covariant keypoints, we design a filtering process, which approximates a diffusion process along the surfaces of the scene, by means of the information provided by depth. Next, we employ this multiscale representation to find keypoints through a multiscale keypoint detector. The keypoints obtained by the proposed detector provide substantially higher stability to viewpoint changes than alternative 2D and RGBD feature extraction approaches, both in terms of repeatability and image classification accuracy. Furthermore, the proposed detector can be efficiently implemented on a GPU.

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Multiscale Analysis for Images on Riemannian Manifolds

Cited by 5 publications

References 67 publications

Multiscale Analysis of Similarities between Images on Riemannian Manifolds

Multiscale Analysis of Similarities between Images on Riemannian Manifolds

Anisotropic tempered diffusion equations

Keypoint Detection in RGBD Images Based on an Anisotropic Scale Space

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