Non-Stationary Deconvolution Using a Multi-Resolution Stack

Abstract

Non-stationary deconvolution is of considerable importance is several areas of medical image processing. Quantitative errors occur in tomography as a result of the partial volume effect which has two components; the loss of ‘recovery’ due to the object being incomplete within the slice thickness, and loss due to sampling observed when the object is comparable in size to the resolution of the system. In SPECT, this second problem, in particular, is non-stationary since the resolution of the detecting system is a function of distance from the detector. This paper described a method for attacking such problems using a multi-resolution stack. A filter is chosen which generates successively the images of a multi-resolution stack, by successive blurring with a Gaussian filter function. The filter function chosen for this purpose should be a smoothing filter with only a slight blurring effect such that the sampling in the ‘blurred direction’ is reasonably fine. Prior to smoothing, a deconvolution filter is used, matched with the Gaussian blurring filters to give the range of deconvolutions required for different positions. Thus, after an initial stationary deconvolution a series of progressively smoother versions of this image are stored in a stack (S). The desired non-stationary filtered image can be shown to be an intersection of some given surface within the processed stack. In the case of the SPECT sampling problem, this intersection take the form of a simple radially symmetrical bowl shaped surface. The coordinates of this surface may be calculated in advance in terms of determining a k (scale space) coordinate in the processed stack corresponding each i, j coordinate in the original image. The third stage of the filtering operation is that of projecting this intersecting surface down to give a processed image I’, such that I’(i,j)=S(i,j,k), where (i,j,k) is a point on the surface within the stack. In most cases this may be performed while only remembering one level of the stack at a time. The whole process then becomes highly efficient, and requires little storage space, being merely n times a compact linear filtering operation where n depends on the scale sampling chosen. It may be demonstrated that small values of n can be used with little distortion of the resulting image. Preliminary results, which could be of value in other types of tomographic instrumentation, have confirmed that such a technique can aid in eliminating non-stationary partial volume effects.