圖像分類的字典學習方法概述

1 字典學習（dictionary learning）

旨在從原始數據中找到一組特殊的稀疏信號，在機器視覺中稱為視覺單詞（visual words），這一組稀疏元素能夠足夠線性表示所有的原始信號。字典學習來源于壓縮感知，后來廣泛用于圖像去噪、去霧、聚類、分類等方面。

兩類：

1）? 直接學習區分性的字典（directly forcing the dictionary discriminative）Track I

2）稀疏化系數，使得到的子弟那具有可區分性（making the sparse coefficients discriminative (usuallythrough simultaneously learning a classifier) to promote the discrimination ofthe dictionary.）Track II

2 字典學習的原理

假設為原始數據，字典學習的目的是通過下面的優化得到可區分性能很好表示原始數據的字典：

（1）??????Track I: Directly Making theDictionary Discriminative

The methods from Track I use the reconstruction error for the finalclassification, thus the learned dictionary ought to be as discriminative aspossible.

1）Meta-face learning

注：SRC（Sparse Representation-Based Classification）最開始用于魯棒性人臉識別，C代表不同人臉個數，為原始訓練樣本，是類的子集樣本。SRC將原始數據當作完備字典，學習規則表達式如下：

SRC對于人臉是被效果很好，對噪聲光照魯棒性強，雖然不是直接涉及字典學習，但是用到了稀疏編碼，開創了字典學習史上的先河，對大量數據處理速度問題有很大的改進和突破。

但是SRC將原始人臉圖像作為完備字典，這種預定的字典會帶有很多原始圖像中的噪聲等污染，而且字典冗余，當訓練圖像位數增加時，計算速度和事件遇到瓶頸問題。為此，提出了Meta-Face Learning方法。學習規則如下：

但是這樣還是會遺漏子類的結構信息，如同類之間有共性，不同類間有差異性，為了使字典的可區分性更強，結構性更明顯，可以再字典約束中加入規則項，極速那類間的關聯度和相似度，體現結構特性。為此提出了Dictionary Learning with StructuredIncoherence方法。

3）Dictionary Learning with Structured Incoherence

Ramirez et al. note that the learned sub-dictionaries may share some common bases, i.e. some visual words from differentsub-dictionaries can be very coherent [8]. Undoubtedly, the coherence of the atomscan be used for reconstructing the query image interchangeably, and thereconstruction error based classifier will fail in identifying some queries. Tocircumvent this problem, they add an incoherence term term to drive thedictionaries associated to different classes as independent as possible.

(2)?Track II: Making the CoefficientsDiscriminative

Track II is different from Track I in the way of discrimination. Contraryto Track I, it forces the sparse coefficients to be discriminative, andindirectly propagates the discrimination power to the overall dictionary. TrackII only need to learn an overall dictionary, instead of class-specificdictionaries.

1)Supervised Dictionary Learning

2) Discriminative K-SVD forDictionary Learning

discriminative K-SVD (D-KSVD) tosimultaneously achieve a desired dictionary which has good representation powerwhile supporting optimal discrimination of the classes. D-KSVD adds a simple linearregression as a penalty term to the conventional DL framework:

前兩項可以合并一起，最后一項可以省略（原理見K-SVD，detailsin [13]).）

3) Label Consistent K-SVD

Jiang et al. propose alabel consistent K-SVD (LC-KSVD) method to learn a discriminative dictionaryfor sparse coding [5]. They introduce a label consistent constraint called“discriminative sparse-code error”, and combine it with the reconstructionerror and the classification error to form a unified objective function asbelow:

4) Fisher DiscriminantDictionary Learning

Yang et al. propose Fisher discrimination dictionary learning (Fisher DL) method based on the Fisher criterion to learn a structured dictionary [11], whose atomhas correspondence to the class label. The structured dictionaryis denoted as D = [D1, . . . ,DC], where Dc isthe class-specific sub-dictionary associated with the cth class.Denote the data set X = [X1, . . . ,XC], where Xc is the sub-set of the trainingsamplesfrom the cth class. Then they solve the following formulation overthe dictionary and the coefficients to derive the desired discriminativedictionary:

3 總結

回顧前面各種字典學習的方法，我們可以總結出字典學習規則的表達式如下：

第一部分是字典規則向，約束重構誤差的；第二項是稀疏系數的可區分性約束，第三項和第四項是拉格朗如乘子的規則項.

References

[1] D. M. Bradley and J. A. Bagnell. Differentiablesparse coding. NIPS, 2008.

[2] B. Cheng, J. Yang, S. Yan, Y. Fu, and T. S.Huang. Learning with ?1-graph for image analysis.

IEEE Trans. Img. Proc., 19(4):858–866, Apr. 2010.

[3] M. Elad and M. Aharon. Image denoising vialearned dictionaries and sparse representation.

CVPR, 2006.

[4] M. Elad, M. Figueiredo, and Y. Ma. On the roleof sparse and redundant representations in

image processing. proceedings of IEEE, 98(6):972–982, 2010.

[5] Z. Jiang, Z. Lin, and L. S. Davis. Learning adiscriminative dictionary for sparse coding via

label consistent k-svd. CVPR,2011.

[6] J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A.Zisserman. Supervised dictionary learning.

NIPS, 2008.

[7] R. Raina, A. Battle, H. Lee, B. Packer, and A.Y. Ng. Self-taught learning: transfer learning

from unlabeled data. ICML,2007.

[8] I. Ramirez, P. Sprechmann, and G. Sapiro.Classification and clustering via dictionary learning

with structured incoherence and shared features. CVPR, 2010.

[9] J. Wright, Y. Ma, J. Mairal, G. Sapiro, T.Huang, and S. Yan. sparse representation for computer vision and patternrecognition. proceedingsof IEEE, 98(6):1031–1044, 2010.

[10] J. Wright, A. Yang, A. Ganesh, S. Sastry, andY. Ma. Robust face recognition via sparse representation. PAMI, 2009.

[11] M. Yang, L. Zhang, X. Feng, and D. Zhang.Fisher discrimination dictionary learning for sparse

representation. ICCV,2011.

[12] M. Yang, L. Zhang, J. Yang, and D. Zhang.metaface learning for sparse representation based

face recognition. ICIP,2010.

[13] Q. Zhang and B. Li. Discriminative k-svd fordictionary learning in face recognition. CVPR,

2010.

關于Fisher判別法原理可以參考其他資料。

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