Abstract
Two-dimensional principal component analysis (2DPCA) has been widely used to extract image features. As opposed to PCA, 2DPCA directly treats 2D matrices to extract image features instead of transforming 2D matrices into vectors. However, the classical 2DPCA based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> -norm square is sensitive to noise. To handle this problem, 2DPCAs based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> -norm, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{p}$ </tex-math></inline-formula> -norm, and other norms have been studied. In this paper, as a further development, 2DPCA based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{T}\ell _{1}$ </tex-math></inline-formula> criterion is proposed, referred as 2DPCA- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{T}\ell _{1}$ </tex-math></inline-formula> . Notice that, different from some norms used before, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{T}\ell _{1}$ </tex-math></inline-formula> criterion is bounded and Lipschitz-continuous. So it can be expected that our 2DPCA- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text{T}\ell _{1}$ </tex-math></inline-formula> should be more robust. In fact, the experimental results have shown that its performance is superior to that of classical 2DPCA, 2DPCA-L1, 2DPCAL1-S, N-2-DPCA, G2DPCA, and Angle-2DPCA.
Highlights
Principal component analysis (PCA) [1], [2] is a popular dimensionality reduction and feature extraction method
Μ1 (t ), which shows that transformed 1 (T 1) criterion interpolates 0- and 1-norm. It seems that T 1 criterion with the parameter a ∈ (0, ∞) is similar to p-norm with a parameter p ∈ (0, 1), but they have significant difference
EXPERIMENTS we evaluate the performance of 2DPCA-T 1 on three human face databases Yale [37], ORL [38], Jaffe [39] and one object database COIL-20 [40], where the block noise with black and white dots is added to examine the robustness
Summary
Principal component analysis (PCA) [1], [2] is a popular dimensionality reduction and feature extraction method. In addition to PCA, linear discriminant analysis (LDA) [10], [11] and locally preserving projection (LPP) [12] are the representative dimensionality reduction methods The former extracts the most discriminating features, the latter, as the linear approximation of locally linear embedding (LLE) [13], characterizes the local geometric structure. They are markedly different since T 1 criterion has two properties: boundedness and Lipschitz-continuity, where Lipschitz-continuity measures relative changes in the objective function with respect to the input These two properties make the T 1 criterion to be a suitable distance metric for PCA, for robustness, due to its stronger suppression of noise.
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