Quaternion principal component analysis of color images

In this paper, we present quaternion matrix algebra techniques that can be used to process the eigen analysis of a color image. Applications of Principal Component Analysis (PCA) in image processing are numerous, and the proposed tools aim to give material for color image processing, that take into account their particular nature. For this purpose, we use the quaternion model for color images and introduce the extension of two classical techniques to their quaternionic case: Singular Value Decomposition (SVD) and Karhunen-Lòeve Transform (KLT). For the quaternionic version of the KLT, we also introduce the problem of EigenValue Decomposition (EVD) of a quaternion matrix. We give the properties of these quaternion tools for color images and present their behavior on natural images. We also present a method to compute the decompositions using complex matrix algebra. Finally, we start a discussion on possible applications of the proposed techniques in color images processing. 1. QUATERNIONS AND COLOR IMAGES Quaternions were discovered in 1843 by W.R. Hamilton [1] and are a specific class of hypercomplex numbers [2]. A quaternion q is made of one real and three imaginary parts: q = a+ bi+ cj+ dk, (1) where i = j = k = ijk = −1, ij = k , ji = −k, (2) and where a, b, c and d are real numbers. The conjugate q of a quaternion is: q = a − bi − cj − dk. A quaternion with no real part (a = 0) is said to be pure. The norm of a quaternion is |q| = √ qq = √ qq = √ a2 + b2 + c2 + d2 and its inverse is q−1 = q / |q|. Quaternions are one of the four existing division algebras (with real, complex and octonions). A characteristic property of quaternions is their non-commutativity under multiplication (q1q2 6= q2q1). For a complete review of quaternion properties, see [2]. Quaternion model for color image was first given by Pei [3] and led to the definition of powerful tools for color image processing such as Fourier transforms [4], correlation [5], edge detection [6] and color filters [7]. According to this model, the Red, Green and Blue values of each pixel of a color image are represented as a single pure quaternion valued pixel. A N ×M color image is then represented as a pure quaternion image: S(x, y) = r(x, y)i+ g(x, y)j+ b(x, y)k (3) where r(x, y), g(x, y) and b(x, y) are respectively the red green and blue components of the pixel at position (x, y) in the image S(x, y). Thus, the color image is a matrix of size N ×M over the quaternion field (restricted to pure quaternions). Using this model, it is possible to analyze any color image using the PCA technique over the quaternion field H. Then, a color image of N lines and M columns is an element of HN×M . We now review the matrix decompositions needed for this purpose. 2. QUATERNION PRINCIPAL COMPONENT ANALYSIS In this section, we introduce some matrix algebra tools over the field of quaternions. Matrices considered here are pure quaternion valued, but the extension to images with four components could be derived from what is said here. Two decompositions of quaternion matrices as well as a PCA technique based on image correlation (the Quaternion KLT) are presented. The interested reader can find an almost complete overview and state of the art in the (still open) research field of matrices of quaternions in [8]. 0-7803-7750-8/03/$17.00 ©2003 IEEE. ICIP 2003 2.1. Quaternion Singular Value Decomposition Any matrix S ∈ HN×M admits a Singular Value Decomposition [8] given as: S = U ( Σr 0 0 0 ) V (4) where / is the conjugate-transposition operator and where U ∈ HN×N and V ∈ HM×M are unitary quaternion matrices. So, S (UU) = S (VV) = I and V (UU) = V (VV) = O (with S(.) and V (.) denoting respectively the scalar and pure quaternion parts). In the proposed notation, I is the identity matrix and matrix O contains only zeros. These matrices contain the left and right quaternionic singular vectors of S. Σr is a real diagonal matrix (Σr ∈ Rr×r), where r is the rank of S (i.e. the number of nonnull singular values). The values on the diagonal of Σr, λn (with 1 ≤ n ≤ r), are the singular values of the quaternion matrix, arranged in decreasing magnitude order along the diagonal. The way to obtain this decomposition of a quaternion matrix by means of classical complex algorithms can be found in [9]. It is based on the definition and SVD computation of the complex adjoint matrix [8, 10]. This complex matrix is linked to the original quaternion matrix through the isomorphism: HN×M → C2N×2M . Some of the most significant properties of the QSVD, when applied to color images, are listed below: • Invariance to spatial rotation (also true in the case of grayscale images with SVD). • Invariance to spatial shift (vectors in U and V are shifted by the same amount) • Image decomposed as a sum of rank 1 [9] color images • Invariance to color space rotation An application of the QSVD has been recently proposed in vector-sensor seismic signal processing [9]. It was there used to extend the wave separation technique based on subspace method [11] to polarized waves case. 2.2. Quaternion Eigen Value Decomposition Due to non-commutativity of the quaternion product, there exist two kinds of eigenvalue problem over the quaternion field. Consequently, two types of eigenvaleus exist for a quaternion matrix A ∈ HN×N : the left and right eigenvalues (λl and λr). These are defined as: Right Eigenvalues: Axr = xrλr Left Eigenvalues: Axl = λlxl where xr and xl are the associated eigenvectors. In this paper, we will restrict ourselves to the right eigenvalue problem, due to theoretical problems with left eigenvalues still unresolved at that time [8, 12, 13]. Every quaternion matrix A ∈ HN×N can be decomposed as: A = WΩW (5) where W ∈ HN×N is a unitary matrix that contains the eigenvectors ofA andΩ ∈ CN×N is a diagonal matrix with eigenvalues on its diagonal. IfA is hermitian (i.e. A = A), then its eigenvalues are real valued (Ω ∈ RN×N ). The computation of the right spectrum (right eigenvalues) can be done through the isomorphism specified in the QSVD section. A direct decomposition of color images using QEVD is not presented here (but could be considered in future work). Application of the QEVD is now presented in the Quaternionic version of the Karhunen-Lòeve transform. 2.3. Quaternion Karhunen-Lòeve Transform The Karhunen-Lòeve transform is a well-known technique in image and signal processing, and is based on the EVD of the covariance matrix of the different lines (or columns) of the original image. The study of hypercomplex correlation (and covariance) for color images has been initiated by Sangwine [5, 7]. Given an image S of size N × M , one can build the covariance matrix of si, the ith column of S as: Γi = s̃is̃i (6) where s̃i = si−m [si] is the ith centered (corrected form its mean color value) column of S. Then, the mean covariance matrix is given as: