cvpr cvpr2013 cvpr2013-392 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Simon Hawe, Matthias Seibert, Martin Kleinsteuber
Abstract: Many techniques in computer vision, machine learning, and statistics rely on the fact that a signal of interest admits a sparse representation over some dictionary. Dictionaries are either available analytically, or can be learned from a suitable training set. While analytic dictionaries permit to capture the global structure of a signal and allow a fast implementation, learned dictionaries often perform better in applications as they are more adapted to the considered class of signals. In imagery, unfortunately, the numerical burden for (i) learning a dictionary and for (ii) employing the dictionary for reconstruction tasks only allows to deal with relatively small image patches that only capture local image information. The approach presented in this paper aims at overcoming these drawbacks by allowing a separable structure on the dictionary throughout the learning process. On the one hand, this permits larger patch-sizes for the learning phase, on the other hand, the dictionary is applied efficiently in reconstruction tasks. The learning procedure is based on optimizing over a product of spheres which updates the dictionary as a whole, thus enforces basic dictionary proper- , ties such as mutual coherence explicitly during the learning procedure. In the special case where no separable structure is enforced, our method competes with state-of-the-art dictionary learning methods like K-SVD.
Reference: text
sentIndex sentText sentNum sentScore
1 se ibert Abstract Many techniques in computer vision, machine learning, and statistics rely on the fact that a signal of interest admits a sparse representation over some dictionary. [sent-3, score-0.312]
2 Dictionaries are either available analytically, or can be learned from a suitable training set. [sent-4, score-0.037]
3 While analytic dictionaries permit to capture the global structure of a signal and allow a fast implementation, learned dictionaries often perform better in applications as they are more adapted to the considered class of signals. [sent-5, score-1.103]
4 In imagery, unfortunately, the numerical burden for (i) learning a dictionary and for (ii) employing the dictionary for reconstruction tasks only allows to deal with relatively small image patches that only capture local image information. [sent-6, score-1.485]
5 The approach presented in this paper aims at overcoming these drawbacks by allowing a separable structure on the dictionary throughout the learning process. [sent-7, score-1.075]
6 On the one hand, this permits larger patch-sizes for the learning phase, on the other hand, the dictionary is applied efficiently in reconstruction tasks. [sent-8, score-0.772]
7 The learning procedure is based on optimizing over a product of spheres which updates the dictionary as a whole, thus enforces basic dictionary proper- , ties such as mutual coherence explicitly during the learning procedure. [sent-9, score-1.772]
8 In the special case where no separable structure is enforced, our method competes with state-of-the-art dictionary learning methods like K-SVD. [sent-10, score-1.082]
9 Introduction Exploiting the fact that a signal s ∈ Rn has a sparse represeEnxtpaltoioitni over some dictionary D s ∈∈ R Rn×d is the backrbeosneen oatfi many rsu scomceess dfuicl signal D rec ∈ons Rtruction and data analysis algorithms. [sent-12, score-1.03]
10 Having a sparse representation means that s is the linear combination of only a few columns of D, referred to as atoms. [sent-13, score-0.123]
11 Formally, this reads as s = Dx, (1) where the transform coefficient vector x ∈ Rd is sparse, iw. [sent-14, score-0.063]
12 mreo thste o trfa intssf oernmtri ceos are zero or rsm xall ∈ i nR magnitude. [sent-16, score-0.123]
13 de , For the performance of algorithms exploiting this model, it is crucial to find a dictionary that allows the signal of interest to be represented most accurately with a coefficient vector x that is as sparse as possible. [sent-18, score-0.806]
14 Basically, dictionaries can be assigned to two classes: analytic dictionaries and learned dictionaries. [sent-19, score-0.896]
15 Analytic dictionaries are built on mathematical models of a general type of signal they should represent. [sent-20, score-0.459]
16 They can be used universally and allow a fast implementation. [sent-21, score-0.062]
17 It is well known that learned dictionaries yield a sparser representation than analytic ones. [sent-23, score-0.576]
18 Given a set of representative training signals, dictionary learning algorithms aim at finding the dictionary over which the training set admits a maximally sparse representation. [sent-24, score-1.493]
19 , xm] ∈ Rd×m contain the corresponding m sparse transform co]e ∈ffic Rient vectors, then learning a dictionary can be stated as the minimization problem minXim,Dizeg(X) subject to ? [sent-31, score-0.79]
20 (2) Therein, g : Rd×m → R is a function that promotes sparsity, ? [sent-35, score-0.033]
21 reflects the no→ise R power, nanctdio Cn tish some predefined admissible set of solutions. [sent-36, score-0.036]
22 Common dictionary learning approaches employing optimization problems related to (2) include probabilistic ones like [11, 14, 26], and clustering based ones such as K-SVD [3], see [20] for a more comprehensive overview. [sent-37, score-0.748]
23 The dictionaries produced by these techniques are unstructured matrices that allow highly sparse representations of the signals of interest. [sent-38, score-0.716]
24 However, the dimension of the signals which are sparsely represented and, consequently, the possible dictionaries’ dimensions are inherently restricted by limited memory and limited computational resources. [sent-39, score-0.183]
25 Furthermore, when used within signal reconstruction algorithms where many matrix vector multiplications have to be performed, those dictionaries are computationally expensive to apply. [sent-40, score-0.539]
26 In this paper, we present a method for learning dictio- 444333668 naries that are efficiently applicable in reconstruction tasks. [sent-41, score-0.108]
27 The crucial idea is to allow the dictionary to have a separable structure, where separable means that the dictionary D is given by the Kronecker product of two smaller dictionaries A ∈ Rh×a and B ∈ Rw×b, i. [sent-42, score-2.321]
28 (3) The relation between a signal s ∈ Rhw and its sparse reprTehseen retaltaiotino x ∈tw Reeanb as given i ∈n R(1) is accordingly s = (reBs e⊗nt aAti)oxn = x ve ∈c( RAvec−1(x)B? [sent-45, score-0.17]
29 ), where the vector space isomorphism vec : ARav×ebc → Rab is defined as the operation that stacks the columns on top ofeach other. [sent-46, score-0.161]
30 Employing this separable structure instead of a full, unstructured dictionary clearly reduces the computational costs of both the learning algorithm and the reconstruction tasks. [sent-47, score-1.218]
31 More precisely, for a separation with h, w √∼ the computational burden freodru ac seesp afrroamtio On w(ni)th t oh Ow(√ ∼n). [sent-48, score-0.083]
32 We will refer to this new learning approach as SeDiL (Separable Dictionary Learning). [sent-49, score-0.06]
33 However, we will focus on signals that have an inherently two dimensional structure such as images. [sent-51, score-0.26]
34 However, it is worth mentioning that SeDiL can straightforwardly be extended to signals with higher dimensional structure, such as volumetric 3D-signals, by employ- √n, ×× ing multiple Kronecker products. [sent-52, score-0.249]
35 To fix the notation for the rest of this work, if A and B are as above, the two dimensional signal S ∈ Rh×w has the sparse representation mXe n∈s iRona×abl, s i g. [sent-53, score-0.242]
36 T∈h Re proposed dictionary learning scheme SeDiL is based on an adaption of Problem (2) to a product of unit spheres. [sent-57, score-0.768]
37 Furthermore, it incorporates a regularization term that allows to control the dictionary’s mutual coherence. [sent-58, score-0.112]
38 For the general separable case, the method is able to learn dictionaries for large patch dimensions where conventional learning techniques fail while if we define B = 1 SeDiL yields a new algorithm for learning standard unstructured dictionaries. [sent-60, score-0.939]
39 A denoising experiment is given that shows the performance of both a separable and a non-separable dictionary learned by SeDiL on (8 8)-dimensional image patches. [sent-61, score-0.986]
40 From this experiment (it8 can b)e-d seen tshioatn tahle i separable dictionary outperforms eintst analytic counterpart, the overcomplete discrete cosine transform, and the non-separable one achieves similar performance as state-of-the-art learning methods like K-SVD. [sent-62, score-1.199]
41 Besides that, to show that a learned separable dictionary is able to extract and to recover the global information contained in the training data, a separable dictionary is learned on a face database with each face image having a resolution of 64 64 pixels. [sent-63, score-2.001]
42 This dictionary is then applied in a face inpainting experiment dwichetiroen large missing regions are re- covered solely based on the information contained in the dictionary. [sent-64, score-0.665]
43 Structured Dictionary Learning Instead of learning dense unstructured dictionaries, which are costly to apply in reconstruction tasks and are unable to deal with high dimensional signals, techniques exist that aim at learning dictionaries which bypass these limitations. [sent-66, score-0.807]
44 In the following, we shortly review some existing techniques that focus on learning efficiently applicable and high dimensional dictionaries, followed by introducing our approach. [sent-67, score-0.194]
45 Related Work In [17] and [24], two different algorithms have been proposed following the same idea of finding a dictionary such that the atoms themselves are sparse over some fixed analytic base dictionary. [sent-70, score-0.907]
46 The algorithm proposed in [17] enforces each atom to have a fixed number of non-zero coefficients, while the one suggested in [24] imposes a less restrictive constraint by enforcing sparsity over the entire dictionary. [sent-71, score-0.173]
47 However, both algorithms employ optimization problems that are not capable of finding a large dictionary for high dimensional signals. [sent-72, score-0.708]
48 In [2] an alternative structure for dictionaries has been proposed. [sent-73, score-0.383]
49 The so called signature dictionary is a small image itself, where every patch at varying locations and size is a possible dictionary atom. [sent-74, score-1.272]
50 The advantages of this structure include near-translation- invariance, reduced overfitting, and less memory and computational requirements, compared to unstructured dictionary approaches. [sent-75, score-0.825]
51 However, the small number of parameters in this model also makes this dictionary more restrictive than other structures. [sent-76, score-0.687]
52 Hierarchical frameworks for tackling high dimensional dictionary learning are presented in [13] and [23]. [sent-78, score-0.796]
53 The latter work uses this framework in conjunction with a screening technique and random projections. [sent-79, score-0.043]
54 Proposed Approach We aim at learning a separable dictionary D = B ⊗ A fromW a given s leeta ronfi training samples cSti =n (rSy1 D , . [sent-83, score-1.038]
55 We den)ot ∈e the collection of the m sparse representations by X = t(hXe1 , . [sent-87, score-0.061]
56 , Xtiomn) aonfd t measure airtss eov reerparlel sparsity vsia b ? [sent-90, score-0.091]
57 1 (4) where xklj is the (k, l)-entry of Xj ∈ Ra×b and ρ > 0 is a weighting factor. [sent-102, score-0.097]
58 We impose the following regularization 444333779 on the dictionary. [sent-103, score-0.03]
59 Constraint (i) is commonly employed in various dictionary learning procedures to avoid the scale ambiguity problem, i. [sent-106, score-0.696]
60 the entries of D tend to infinity, while the entries of X ti. [sent-108, score-0.098]
61 d t htoe zero as tfh Dis ties nthde t global my,iwn ihmilizee thr eo fe nthtrei unconstrained sparsity measure g(X). [sent-110, score-0.168]
62 Matrices with normalized csotrlauinmends sapdamrsiitt a mmeaansifuoreld g structure, rkicnoewsn w as tnhoer product of spheres, which we denote by S(n, d) := {D ∈ Rn×d| ddiag(D? [sent-111, score-0.084]
63 (5) Here, ddiag(Z) forms a diagonal matrix with the diagonal entries of the square matrix Z, and Id is the (d d)-identity matrix. [sent-113, score-0.049]
64 Consequently, we require thaits tAh ei s( an edl)e-mideenntt toyf S(h, a) and that B is an element of S(w, b). [sent-114, score-0.04]
65 The soft constraint (ii) of requiring a moderate mutual coherence of the dictionary is a well known regularization procedure in dictionary learning, and is motivated by the compressive sensing theory. [sent-115, score-1.487]
66 Roughly speaking, the mutual coherence of D measures the similarity between the dictionary’s atoms, or, ”a value that exposes the dictionary’s vulnerability, as [. [sent-116, score-0.222]
67 ] two closely related columns may confuse any pursuit technique. [sent-119, score-0.091]
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