Geometric Transformations involve spatial transformations to modify image geometry and grey level interpolation to assign pixel values in the spatially transformed image. The spatial transformation is defined by a geometric mapping grid (see Appendix G). Within each grid cell, a pair of bilinear equations maps from the spatially transformed space to the original (input) space:
Xoriginal = a0 + a1x + a2y + a3xy
Yoriginal = b0 + b1x + b2y + b3xy
where x and y are pixel locations in the spatially transformed (output) space and a0, a1, a2, a3 and b0, b1, b2, b3 are coefficients defined by each geometric mapping grid cell.
The resample function applies this transformation both directly using a one-pass, two-dimensional algorithm and also with a multi-pass method involving three one-dimensional passes through the data.
Pixel values in the original image are defined at integer locations. Since the pair of bilinear equations were often given map to noninteger locations in the original image, a method to determine pixel grey-level values at noninteger locations, based on surrounding pixel values, is needed. Resampling methods available in the LAS 7.0 include nearest neighbor, bilinear interpolation (a 2 x 2 kernel), parametric cubic convolution, (a 4 x 4 kernel) and a user-entered resampling weight table with a kernel up to 16 x 16. Resampling methods are accurate to 1/32 of a pixel, except nearest neighbor, which is accurate to 1/2 a pixel.
Resampling kernels which are separable into horizontal and vertical dimensions can be represented in the userentered resampling weight table file. Each kernel dimension has 33 entries, corresponding to the 32 increments between two pixels and both end points. Resampling weight table files are generated by the rtable module. Rtable generates a sin(x)/x kernel of user-specified dimensions and also generates resampling kernels from an inverse point spread function. Presently, LAS cannot generate an inverse point spread function; it must be done externally. This is the manner in which resampling kernels for the restoration process are generated.
References:
Park, S. K., and R. A. Schowengerdt, "Image Reconstruction by Parametric Cubic Convolution," Computer Vision, Graphics, and Image Processing 23, Academic Press Inc., 1983 pp., 258-272.
Friedman, D. E., "Two Dimensional Resampling of Line Scan Imagery by One-Dimensional Processing," Photogrammetric Engineering and Remote Sensing, Vol 47, No 10, pp. 1459-1467, October 1981.