iccv iccv2013 iccv2013-161 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Chenglong Bao, Jian-Feng Cai, Hui Ji
Abstract: In recent years, how to learn a dictionary from input images for sparse modelling has been one very active topic in image processing and recognition. Most existing dictionary learning methods consider an over-complete dictionary, e.g. the K-SVD method. Often they require solving some minimization problem that is very challenging in terms of computational feasibility and efficiency. However, if the correlations among dictionary atoms are not well constrained, the redundancy of the dictionary does not necessarily improve the performance of sparse coding. This paper proposed a fast orthogonal dictionary learning method for sparse image representation. With comparable performance on several image restoration tasks, the proposed method is much more computationally efficient than the over-complete dictionary based learning methods.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract In recent years, how to learn a dictionary from input images for sparse modelling has been one very active topic in image processing and recognition. [sent-4, score-0.93]
2 Most existing dictionary learning methods consider an over-complete dictionary, e. [sent-5, score-0.773]
3 However, if the correlations among dictionary atoms are not well constrained, the redundancy of the dictionary does not necessarily improve the performance of sparse coding. [sent-9, score-1.766]
4 This paper proposed a fast orthogonal dictionary learning method for sparse image representation. [sent-10, score-1.139]
5 With comparable performance on several image restoration tasks, the proposed method is much more computationally efficient than the over-complete dictionary based learning methods. [sent-11, score-1.01]
6 It is now well established that the sparse image models are very powerful tools for many image restoration and recognition tasks. [sent-14, score-0.312]
7 Sparse image model assumes that most local image patches can be well approximated by a sparse linear combination of basis elements, the so-called atoms. [sent-15, score-0.236]
8 A fundamental question is then how to find a dictionary under which an input image can be sparsely modelled. [sent-17, score-0.754]
9 Earlier work on designing dictionary for sparse image modelling focuses on the design of fixed orthogonal dictionaries, e. [sent-18, score-1.114]
10 The basic idea is to learn the dictionary adaptive to the target image so as to achieve better sparsity than the fixed ones. [sent-29, score-0.768]
11 Most existing dictionary learning methods consider an over-complete dictionary and formulate the learning process as a minimization problem. [sent-30, score-1.662]
12 Taking the popular K-SVD method [12] for example, the K-SVD method learns an over-complete dictionary from an input image via solving the following minimization model: Dm,{αini}? [sent-31, score-0.927]
13 The iteration scheme for solving (1) is proposed in [12] which alternatively iterates between two modules: sparse coding for {αi} and dictionary updating mforo Dule. [sent-44, score-1.125]
14 In this paper, we proposed a new variational model to learn an adaptive dictionary for sparse image modelling. [sent-61, score-0.849]
15 Different from the K-SVD method, the dictionary learned in our approach is an orthogonal dictionary. [sent-62, score-0.971]
16 The seemingly performance loss on sparse coding when adopting an orthogonal dictionary over an over-complete dictionary indeed has little negative impact on the performance of image restoration. [sent-63, score-1.951]
17 The performance of the proposed orthogonal dictionary learning method is at least comparable to the K-SVD method in several image restoration applications. [sent-64, score-1.229]
18 The gain by using an orthogonal dictionary is very noticeable. [sent-65, score-0.971]
19 In short, the proposed sparsity-based orthogonal dictionary learning method is much faster than the K-SVD method but with comparable performance in image restoration, 1. [sent-67, score-1.067]
20 Motivation and our contributions The main computational issue of the K-SVD method comes from the fact that the dictionary D ∈ Rn×k is a highly rferdoumnd tahent dictionary e(k d =ict 4onna riny [ D12] )∈ ∈w Rhich lacks additional constraints on the correlations among atoms. [sent-69, score-1.529]
21 The purpose using such a high redundant dictionary is for maximizing the sparsity of the code {αj } by having more atoms imn zthineg dictionary. [sent-70, score-0.984]
22 Hofo twheev ceord, although a highly o rerdeu antodamnst dictionary allows the existence of more sparse codes, accurately estimating these code becomes less computationally feasible with the increased redundancy. [sent-71, score-0.874]
23 One well-known measure on the quality of dictionary for sparse coding is the so-called mutual incoherence μ(D) ([11]) defined as μ(D) = mi? [sent-72, score-1.016]
24 It is known in compressed sensing literatures that the mutual in- coherence constant μ(D) need to be small enough to guarantee the performance of sparse coding when using matching pursuit methods (see e. [sent-79, score-0.288]
25 However, the constant μ of the redundant dictionary obtained via the K-SVD method and its variations usually is not small, as no constraints on its mutual incoherence are imposed during dictionary updating. [sent-82, score-1.616]
26 In other words, the sparse coding using a redundant dictionary with large μ(D) becomes not only computationally demanding, but also may not be optimal. [sent-83, score-1.055]
27 The negative impact of the dictionary with large μ(D) has been noticed in various sparse coding based recognition systems; see e. [sent-84, score-0.956]
28 One solution is to simultaneously minimize the term μ(D) when update the dictionary which leads to a complex non-convex minimization problem. [sent-87, score-0.864]
29 Moreover, the ideal atoms of the learned dictionary should be those represents repetitive local image patterns. [sent-88, score-0.894]
30 For example, it is shown in [22] that the dimension of the dictionary learned by the K-SVD method for face images can be reduced by half without causing much performance loss. [sent-91, score-0.727]
31 In summary, we argue that when learning a dictionary for sparsity-based image restoration, a highly redundant dictionary often is not necessary for having a good sparse approximation. [sent-92, score-1.696]
32 Instead, a dictionary with little redundancy and with very small μ(D) could perform as effi- × cient as the redundant ones. [sent-93, score-0.824]
33 For example, when using patch size of 8 8, the dictionary size of K-SVD is four times of tshizaet ooff t8h ×e proposed orthogonal dictionary. [sent-94, score-1.06]
34 Based on the above discussions, we propose to use an orthogonal dictionary for sparsity-based dictionary learning in image restoration, which leads to the following minimization model: D∈sR. [sent-97, score-1.86]
35 0, (2) where {gi} ⊂ Rn denotes the set of image patches colwlehceterde f{rgom} t⊂he input image, αi denotes the code of the patch gi. [sent-106, score-0.25]
36 D = {Di}jr=1 denotes the set of r atoms of the dictionary f =or learning, A ∈ Rn×n−r denotes the set of n r taitoomnars ye fitohrer le farronmin experiences or from other sources (nA − −is r a altloowmesd e titoh e ber empty). [sent-107, score-0.939]
37 eTrhiee adoption oomf an orthogonal dictionary will greatly simplify the computation of both dictionary updating and sparse coding. [sent-108, score-1.869]
38 Indeed, we will show in the main body that both sparse coding and dictionary updating in our model have explicit solutions. [sent-109, score-1.032]
39 Related Work This section roughly categorizes the sparsity-based dictionary design. [sent-112, score-0.727]
40 The small image patches are projected onto the space spanned by the atoms of the given transform to yield a set of sparse coefficients. [sent-117, score-0.44]
41 The widely used transforms include both the orthogonal ones (e. [sent-118, score-0.244]
42 In recent years, the concept of the adaptivity has been exploited to design the dictionary specifically optimized for the target image, the so-called dictionary learning. [sent-126, score-1.454]
43 The earlier works [23, 15] learn the dictionary from the statistics of image features or patches to obtain a sparser representation of natural images. [sent-127, score-0.816]
44 The pioneering K-SVD method [12] learns an over-complete dictionary as well as the sparse representations of the patches under that dictionary in an alternating minimization framework. [sent-128, score-1.805]
45 Starting from the set of overlapping image patches collected from the input image, the K-SVD method alternatively iterates between two subproblems: sparse coding and dictionary updating. [sent-129, score-1.096]
46 Both subproblems in [12] are based on heuristic greedy methods: the 33337858 sparse coding under the overcomplete dictionary is solved via orthogonal matching pursuit (OMP) and the dictionary is estimated via column-wise sequentially SVD updates. [sent-130, score-2.082]
47 An efficient implementation of the K-SVD method is developed in [26] which uses the Batch-OMP to accelerate the sparse coding and use two simple matrix-vector product to replace the SVD operation. [sent-132, score-0.229]
48 Fore recognition, the term μ(D) is usually included in the minimization model when updating a dictionary to lower its mutual incoherence. [sent-141, score-0.924]
49 The non-local approach such as × BM3D [6] is another representative patch-based image restoration approach which groups the similar patches into a 3D array and filters the 3D array. [sent-146, score-0.301]
50 For example, based on the groups of similar patches, the K-SVD method is used in [17], the local PCA-based method is used in [9, 29] and the PCA-based dictionary learning is used [10] for image denoising. [sent-148, score-0.773]
51 Fast orthogonal dictionary learning Throughout the paper, the following mathematical notations are adopted for discussion. [sent-150, score-1.017]
52 Now we consider the sparse approximation prob- lem for the set G under an orthogonal dictionary Dˆ := [A, D] ∈ Rn×n whose columns refers to dictionary atoms. [sent-180, score-1.82]
53 T[Ahe, dictionary has two sub-dictionaries in our implementation: one is A ∈ Rn×n−r which contains the input orthogotnioaln a:too mnes i skn Aow ∈n as good ones from other sources; the other is D ∈ Rn×r which denotes the set of atoms need to be ilsea Drned ∈ f rRom the input image. [sent-181, score-0.958]
54 The orthogonal constraint on the dictionary says that Dˆ? [sent-182, score-0.971]
55 We propose to learn the orthogonal dictionary D via solving the following minimization model D∈sR. [sent-187, score-1.144]
56 More specifically, let D(0) be the initial dictionary to start (e. [sent-201, score-0.727]
57 sparse tco ddicitniogn: given thheen orthogonal dictionary −D 1(,k) , find the sparse code V(k) via solving V(k) := aVr ∈gRmn×inm? [sent-208, score-1.272]
58 dictionary updating: given the sparse code V(k) , update the dictionary via solving the minimization: D(k+1) :=argmin ? [sent-213, score-1.654]
59 In the next, we show that both the minimization (4) for sparse coding and (5) for dictionary update are trivial to solve. [sent-220, score-1.093]
60 , where associated with A and VD(k) asso- For dictionary update, let V(k) VA(k) denotes the codes = ciated with D(k). [sent-239, score-0.762]
61 Let PA denote projection the orthogonal operator from Rn. [sent-240, score-0.244]
62 , where P and Q denote the orthogonal matrices defined by the following SVD (In − PA)GVD? [sent-252, score-0.244]
63 There is no need for solving any minimization problem when doing the sparse coding and dictionary updating. [sent-257, score-1.101]
64 The sparse coding is done via a hard thresholding operation and the dictionary updating is done via a single SVD. [sent-258, score-1.087]
65 Algorithm 1 Online orthogonal dictionary learning Algorithm 1Online orthogonal dictionary learning Input: image patches G, input orthogonal atoms A Output: learned dictionary D Main procedure: 1. [sent-260, score-3.263]
66 1 In this section, we give a detailed analysis on the computational complexity of Algorithm 1for sparsity-based orthogonal dictionary learning. [sent-278, score-0.996]
67 The OMP used for sparse coding is known to be slow. [sent-289, score-0.229]
68 The dictionary update of the K-SVD method need to call SVD operators for 4n times. [sent-290, score-0.748]
69 Thus, a fast approximate K-SVD method is developed in [26] which use batch-OMP for sparse coding and replacing SVD by matrix-vector multiplication. [sent-291, score-0.229]
70 The analysis of the approximate K-SVD method (the dimension of dictionary is set 4n by default), together with ours are listed in table 1. [sent-292, score-0.755]
71 Applications in image restoration The sparsity-based online orthogonal dictionary learning Algorithm 1is very simple to implement and also very computationally efficient. [sent-297, score-1.232]
72 To evaluate its performance in image restoration in terms of recovery quality and computational efficiency, we applied Algorithm 1 on two sample image restoration tasks: image denoising and image in painting. [sent-298, score-0.518]
73 Thus, we fix a low-pass filter in the dictionary and only learn n 1high-pass filters from the input image. [sent-302, score-0.776]
74 The patches for denoising are the patches uniformly selected with overlaps. [sent-320, score-0.291]
75 Using a dictionary D generated from wavelet frame filters, Cai et al. [sent-326, score-0.782]
76 Algorithm 2 Denoising via orthogonal dictionary learning Algorithm 2 Denoising via orthogonal dictionary learning Input: noisy image g Output: denoised image g∗ Main procedure: 1. [sent-332, score-2.134]
77 Learning a dictionary D∗ using Algorithm 1with input G and A = [a0] . [sent-336, score-0.754]
78 Experiments In this section, we evaluate the performance of the proposed orthogonal diction learning on image denoising and image in-painting. [sent-343, score-0.403]
79 The ini- ×× tial dictionary is generated by the local DCT transform: ther 8 8 or 16 16. [sent-346, score-0.727]
80 , K, (a) learning a dictionary D(k) using one iteration of Algorithm 1 with input G(k) and A = [a0] ; (b) synthesizing the image h(k+1) from the denoised patch matrix G∗ := (c) defining g(k+1) 3. [sent-358, score-1.009]
81 widely used over-complete dictionary learning: the K-SVD algorithm [12] and its fast version, the approximated KSVD algorithm [26] with the implementations from the original authors 1. [sent-366, score-0.752]
82 7λ as the thresholding value for the dictionary learning process. [sent-399, score-0.799]
83 Our results are compared against two fixed transform based thresholding methods: linear spline framelet [8] and 8 8 DCT, the PCA-based non-local hierfarracmhieclaelt m [8e]tah nodd [89 ×] a 8n dD CtheT ,K th-Se PVDCA denoising nm-letohcoald h[i1e2r-] with patch size of both 8 8 and 16 16. [sent-400, score-0.284]
84 The number of dictionary atoms is 256 fTrhoem s tizhee oKf- SatoVmDs sm iset 8h×od8 a. [sent-406, score-0.869]
85 In the first example, the results are compared to the classic in-painting method [2], and two dictionary learning based methods derived from the K-SVD method ([27]). [sent-412, score-0.773]
86 The main difference between two dictionary learning methods lies in the choice of sparsity promoting functional: one uses the ? [sent-413, score-0.814]
87 Algorithm 3 and two dictionary learning methods [27] are applied to recover the missing pixel values. [sent-419, score-0.803]
88 Discussion and conclusion In this paper, we proposed an orthogonal dictionary learning for image restoration, as an replacement of the widely used K-SVD method. [sent-426, score-1.017]
89 The performance of the proposed orthogonal dictionary learning method is comparable to the K-SVD method, but it runs much faster than the KSVD method. [sent-427, score-1.067]
90 In future, we would like to study how to effectively combine the non-local scheme and the proposed orthogonal dictionary learning method to develop better image restoration methods. [sent-429, score-1.228]
91 K-svd: An algorithm for designing of overcomplete dictionaries for sparse representation. [sent-443, score-0.233]
92 (a) image with overlapped texts; (b-e) correspond to the results from [2], two over-complete dictionary learning method with ? [sent-465, score-0.773]
93 (a) Original image; (b) corrupted image; (c-e) the results from from over-complete dictionary learning method with ? [sent-469, score-0.773]
94 Sparsity-based image denoising via dictionary learning and structured clustering. [sent-523, score-0.914]
95 Optimally sparse representation in general (non-orthogonal) dictionary via ? [sent-528, score-0.877]
96 Image denoising via sparse and redundant representations over learned dictionaries. [sent-534, score-0.337]
97 Learning a dicscriminative dictionary for sparse coding via label consistent K-SVD. [sent-547, score-0.984]
98 Efficient implementation of the k-svd algorithm using batch orthogonal matching pursuit. [sent-625, score-0.244]
99 Dˆ By the fact that = In, the minimization (6) is the equivalent to the following minimization mVin ? [sent-662, score-0.232]
100 which is orthogonal nton ethde b space spanned by IA−. [sent-712, score-0.297]
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