Color image decomposition using quaternion singular value decomposition

We introduce a new tool for color image decomposition based on quaternion representation of color images and quaternion matrix algebra. Using quaternions to represent color images is done by encoding the three RGB channels into the three imaginary parts of a quaternion. Then, to perform the decomposition of such an image, we introduce the Singular Value Decomposition of a Quaternionic matrix (QSVD). This canonical decomposition for color images allows the identification of any color image as a sum of "rank 1" images, each one carrying information. The truncation of the QSVD is also introduced and gives access; to the best rank a approximation of a color image. Attention is paid to the geometrical interpretation of the decompositon elements. Results of the decomposition on the lena image are presented.