In order to apply a multi-dimensional linear transform, over an arbitrarily shaped support, the usual practice is to fill out the support to a hypercube by zero padding. This does not however yield a satisfactory definition for transforms in two or more dimensions. The problem that we tackle is: how do we redefine the transform over an arbitrary shaped region suited to a given application? We present a novel iterative approach to define any multi-dimensional linear transform over an arbitrary shape given that we know its definition over a hypercube. The proposed solution is (1) extensible to all possible shapes of support (whether connected or unconnected) (2) adaptable to the needs of a particular application. We also present results for the Fourier transform, for a specific adaptation of the general definition of the transform which is suitable for compression or segmentation algorithms.
Related Publications:
- K. Ratakonda and N. Ahuja, Discrete Multi-Dimensional Linear Transforms over Connected Arbitrarily Shaped Supports, IEEE International Conference on Acoustics, Speech and Signal Processing, Vol. 4, Munich, Germany, April 1997, 3041-3044. Processing, Geneva, Switzerland, September 1996, 81-84.
- R. Dugad and N. Ahuja, A Fast Scheme for Down-sampling and Up-sampling in the DCT Domain, International Conference on Image Processing, Kobe, Japan, Oct. 1999, II-909-913.
- R. Dugad and N. Ahuja, A Fast Scheme for Altering Resolution in the Compressed Domain, EEE Conference on Computer Vision and Pattern Recognition, Ft. Collins, CO, June 1999, I-213-218.
- R. Dugad and N. Ahuja, A Fast Scheme for Image Size Change in the Compressed Domain, IEEE Trans. on Circuits and Systems for Video Technology, Vol. 11, No. 4, April 2001, 461-474.