iccv iccv2013 iccv2013-114 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Mehrtash Harandi, Conrad Sanderson, Chunhua Shen, Brian Lovell
Abstract: Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graphembedding Grassmann discriminant analysis.
Reference: text
sentIndex sentText sentNum sentScore
1 In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. [sent-3, score-0.367]
2 To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. [sent-4, score-0.682]
3 Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. [sent-5, score-0.506]
4 Introduction Linear subspaces of Rd can be considered as the core of many inference algorithms in computer vision and machine learning. [sent-8, score-0.169]
5 For example, several state-of-the-art methods for matching videos or image sets model given data by subspaces [9, 11, 24]. [sent-9, score-0.194]
6 Auto regressive and moving average models, which are typically employed to model dynamics in spatio-temporal processing, can also be expressed as linear subspaces [24]. [sent-10, score-0.244]
7 More applications of linear subspaces in computer vision include, but are not limited to, chromatic noise filtering [23], biometrics [20], and domain adaptation [8]. [sent-11, score-0.246]
8 , the set of all reflectance functions produced by Lambertian objects lies in a linear subspace), subspaces lie on a special type of Riemannian manifold, namely the Grassmann manifold, which makes their analysis very challenging. [sent-14, score-0.22]
9 This paper tackles and provides efficient solutions to the following two fundamental problems for learning on Grassmann manifolds: 1. [sent-15, score-0.114]
10 Given a signal X and a dictionary D = {Di}iN=1 with N elements (also known as atoms), Dwh =ere { DX and Di are linear subspaces, how X can be approximated by a combination of a “few” atoms in D ? [sent-17, score-0.407]
11 Given a set of measurements {Xi}im=1, how can a dictionary D = {Di}iN=1 be {leXarn}ed to, represent { aX di}icimt=io1 sparsely =? [sent-20, score-0.282]
12 Our main motivation here is to develop new methods for analysing video data and image sets. [sent-21, score-0.064]
13 This is inspired by the success of sparse signal modelling that suggests natural signals like images (and hence video and image sets as our concern here) can be efficiently approximated by superposition of atoms of a dictionary, where the coefficients of superposition are usually sparse (i. [sent-22, score-0.566]
14 We generalise the traditional sparse coding, which operates on vectors, to sparse coding on subspaces. [sent-25, score-0.344]
15 Sparse encoding with the dictionary of subspaces can then be seamlessly used for categorising video data. [sent-26, score-0.505]
16 Before we present our main results, we want to highlight that the proposed algorithms outperform state-of-the-art methods on various recognition tasks and in particular has achieved the highest reported accuracy in classifying dynamic textures. [sent-27, score-0.13]
17 While significant steps have been taken to develop the theory of the sparse coding and dictionary learning in Euclidean spaces, similar problems on non33 111203 Euclidean geometry have received comparatively little attention [10, 15]. [sent-29, score-0.596]
18 To our best knowledge, among a handful of solutions devised on Riemannian manifolds, none is specialised for the Grassmann manifolds which motivates our study. [sent-30, score-0.468]
19 In [10], the authors addressed sparse coding and dictio- nary learning for the Riemannian structure of Symmetric Positive Definite (SPD) matrices or tensors. [sent-31, score-0.329]
20 The solution was obtained by embedding the SPD manifold into Reproducing Kernel Hilbert Space (RKHS) using a Riemannian kernel. [sent-32, score-0.189]
21 Another approach to learning a Riemannian dictionary is by exploiting the tangent bundle of the manifold, as for example in [15] for the manifold of probability distributions. [sent-33, score-0.525]
22 Since the sparse coding has a trivial solution in this approach, an affine constraint has to be added to the problem [15]. [sent-34, score-0.269]
23 While having an affine constraint along with sparse coding is welcome in specific tasks (e. [sent-35, score-0.341]
24 , clustering [2]), in general, the resulting formulation is restrictive and no longer addresses the original problem. [sent-37, score-0.031]
25 Also, working in successive tangent spaces, though common, values only a first-order approximation to the manifold at each step. [sent-38, score-0.209]
26 Furthermore, switching back and forth to the tangent spaces of a Grassmann manifold (as required by this formulation) can be computationally very demanding for the problems that we are interested in (e. [sent-39, score-0.386]
27 This in turns makes the applicability of such school of thought limited for the Grassmann manifolds arising in vision tasks. [sent-42, score-0.375]
28 In light of the above discussion, in this paper, we introduce an extrinsic method for learning a Grassmann dictionary. [sent-44, score-0.126]
29 To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by a diffeomorphism that preserves Grassmann projection distance (a special class of distances on Grassmann mani- folds). [sent-45, score-0.536]
30 We show how sparse coding can be accomplished in the induced space and devise a closed-form solution for updating a Grassmann dictionary atom by atom. [sent-46, score-0.655]
31 Furthermore, in order to accommodate non-linearity in data, we propose a kernelised version of our dictionary learning algorithm. [sent-47, score-0.535]
32 We propose an extrinsic dictionary learning algorithm for data points on Grassmann manifolds by embedding the manifolds into the space of symmetric matrices. [sent-49, score-1.082]
33 We derive a kernelised version of the dictionary learning algorithm which can address the non-linearity in data. [sent-51, score-0.506]
34 We apply the proposed Grassmannian dictionary learning methods to several computer vision tasks where the data are videos or image sets. [sent-53, score-0.36]
35 Our proposed algorithms outperform state-of-the-art methods on a wide range ofclassification tasks, including face recognition from image sets, action recognition and dynamic texture classification. [sent-54, score-0.122]
36 Background Before presenting our algorithms, we review some concepts of Riemannian geometry of Grassmann manifolds, which provide the grounding for the proposed algorithms. [sent-56, score-0.109]
37 Details on Grassmann manifolds and related topics can be found in [1]. [sent-57, score-0.325]
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