Wavelet crosstalk matrix and its application to assessment of shift-variant imaging systems

The objective assessment of image quality is essential for design of imaging systems. Barrett and Gifford introduced the Fourier crosstalk matrix and use it to analyze cone-beam tomography. Fourier crosstalk matrix is a powerful technique for discrete imaging systems that are close to shift invariant because it is diagonal for continuous linear shift-invariant imaging systems. However, for a system that is intrinsically shift-variant, Fourier techniques are not particularly effective. Since Fourier bases have no spatial localization property, the shift-variance of the imaging system cannot be shown by the response of individual Fourier bases; rather, it is shown in the correlation between the Fourier coefficients. This makes the analysis and optimization quite difficult. In this paper, we introduce a wavelet crosstalk matrix based on wavelet series expansions. The wavelet crosstalk matrix allows simultaneous study of the imaging system in both the frequency and spatial domains. Hence, it is well suited for shift-variant systems. We compared the wavelet crosstalk matrix with the Fourier crosstalk matrix for several simulated imaging systems, namely the interior and exterior tomography problems, a dual-planar positron emission tomograph, and a rectangular geometry positron emission tomograph. The results demonstrate the advantages of the wavelet crosstalk matrix in analyzing shift-variant imaging systems.