Geodesic Methods in Computer Vision and Graphics

Peyré, Gabriel; Péchaud, Mickaël; Keriven, Renaud; Cohen, Laurent D.

doi:10.1561/0600000029

Cited by 95 publications

(37 citation statements)

References 254 publications

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“…The geodesic map satisfies the following Eikonal equation [26] ∇u(τ) G −1 τ = 1 for τ ∈ T \{e}, and u(e) = 0,…”

Section: Invariance Score Computationmentioning

confidence: 99%

“…We only provide here a brief description of FM due to space constraints, and focus on the case where the manifold T is two-dimensional (i.e., p = 2). The extension to arbitrary dimensions is straightforward, and we refer to [26,28] …”

Section: Invariance Score Computationmentioning

confidence: 99%

“…The complexity of Manitest is O (N log(N)), where N is the number of visited nodes if a min-heap structure is used [26] (for constant p, and constant cost for evaluation of f ). It is important to note however that the complexity of the algorithm has an exponential dependence on the dimension p since our method involves the enumeration of simplices in dimension p; this is however not a big limitation as our main focus goes to low-dimensional transformation groups (e.g., p ≤ 6 for affine transformations).…”

Section: Invariance Score Computationmentioning

confidence: 99%

See 2 more Smart Citations

Manitest: Are classifiers really invariant?

Fawzi¹,

Frossard²

2015

Procedings of the British Machine Vision Conference 2015

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Invariance to geometric transformations is a highly desirable property of automatic classifiers in many image recognition tasks. Nevertheless, it is unclear to which extent state-of-the-art classifiers are invariant to basic transformations such as rotations and translations. This is mainly due to the lack of general methods that properly measure such an invariance. In this paper, we propose a rigorous and systematic approach for quantifying the invariance to geometric transformations of any classifier. Our key idea is to cast the problem of assessing a classifier's invariance as the computation of geodesics along the manifold of transformed images. We propose the Manitest method, built on the efficient Fast Marching algorithm to compute the invariance of classifiers. Our new method quantifies in particular the importance of data augmentation for learning invariance from data, and the increased invariance of convolutional neural networks with depth. We foresee that the proposed generic tool for measuring invariance to a large class of geometric transformations and arbitrary classifiers will have many applications for evaluating and comparing classifiers based on their invariance, and help improving the invariance of existing classifiers.

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“…The geodesic map satisfies the following Eikonal equation [26] ∇u(τ) G −1 τ = 1 for τ ∈ T \{e}, and u(e) = 0,…”

Section: Invariance Score Computationmentioning

confidence: 99%

Section: Invariance Score Computationmentioning

confidence: 99%

Section: Invariance Score Computationmentioning

confidence: 99%

See 1 more Smart Citation

Manitest: Are classifiers really invariant?

Fawzi¹,

Frossard²

2015

Procedings of the British Machine Vision Conference 2015

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show abstract

“…A line is both the shortest, and the straightest path between two points, but in curved space the two criteria do not coincide [12]. In computer graphics and discrete geometry, discrete geodesics as shortest path between two given point have been studied extensively [13,14]. This is not the case of geodesics as straightest path given an initial point and velocity-with the noticeable exception of [15], in the framework of simplicial complexes.…”

Section: Introductionmentioning

confidence: 99%

Discrete Geodesics and Cellular Automata

Arrighi

Dowek

2015

Theory and Practice of Natural Computing

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Abstract. This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation-as shown by computing the perihelion shift of a Mercury-like planet. It is consistent, in the continuum limit, with the standard notion of timelike geodesics in a pseudo-riemannian manifold. Whether the algorithm fits within the framework of cellular automata is discussed at length.

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“…It often leads to disconnection and irregularity on segments' shape and size. In order to avoid the above flaws, the similarity measure in the proposed algorithm is defined by the geodesic distance [27]. The aforementioned density function forms the basis of the geodesic distance, namely that the distance increment at a particular image point becomes large if the local density is high, and vice versa.…”

Section: Introductionmentioning

confidence: 99%

Structure-Sensitive Superpixels via Geodesic Distance

et al. 2012

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Over-segments (i.e. superpixels) have been commonly used as supporting regions for feature vectors and primitives to reduce computational complexity in various image analysis tasks. In this paper, we describe a structuresensitive over-segmentation technique by exploiting Lloyd's algorithm with a geodesic distance. It generates smaller superpixels to achieve lower under-segmentation in structuredense regions with high intensity or color variation, and produces larger segments to increase computational efficiency in structure-sparse regions with homogeneous appearance. We adopt geometric flows to compute the geodesic distances amongst pixels, and in the segmentation procedure, the density of over-segments is automatically adjusted according to an energy functional that embeds color homogeneity, structure density and compactness constraints. Comparative experiments with the Berkeley database show that the proposed algorithm outperforms prior arts while offering a comparable computational efficiency with fast methods, such as TurboPixels.

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Geodesic Methods in Computer Vision and Graphics

Cited by 95 publications

References 254 publications

Manitest: Are classifiers really invariant?

Manitest: Are classifiers really invariant?

Discrete Geodesics and Cellular Automata

Structure-Sensitive Superpixels via Geodesic Distance

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