Approximating filtered scale-variant signals

• Published in: IEEE transactions on image processing Vol. 14; no. 1; pp. 23 - 35
• Main Authors: Bovik, A.C., Raj, R.G.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.01.2005; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Approximation error; Artificial Intelligence; Biological system modeling; Computer Graphics; Computer Simulation; Continuous wavelet transforms; Convolution; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; Filtering; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Information Storage and Retrieval - methods; Information, signal and communications theory; Kernel; Laboratories; Large scale integration; Models, Statistical; Nonlinear filters; Numerical Analysis, Computer-Assisted; Pattern Recognition, Automated - methods; Reproducibility of Results; Sensitivity and Specificity; Signal and communications theory; Signal processing; Signal Processing, Computer-Assisted; Signal, noise; Subtraction Technique; Telecommunications and information theory; Linear filtering; Signal processing; Approximation; Simulation; Image processing; System response
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2004.838696

We develop theorems that place limits on the point-wise approximation of the responses of filters, both linear shift invariant (LSI) and linear shift variant (LSV), to input signals and images that are LSV in the following sense: they can be expressed as the outputs of systems with LSV impulse responses, where the shift variance is with respect to the filter scale of a single-prototype filter. The approximations take the form of LSI approximations to the responses. We develop tight bounds on the approximation errors expressed in terms of filter durations and derivative (Sobolev) norms. Finally, we find application of the developed theory to defoveation of images, deblurring of shift-variant blurs, and shift-variant edge detection.