The fundamentals of average local variance--Part II: Sampling simple regular patterns with optical imagery

In this investigation, the characteristics of the average local variance (ALV) function is investigated through the acquisition of images at different spatial resolutions of constructed scenes of regular patterns of black and white squares. It is shown that the ALV plot consistently peaks at a spatial resolution in which the pixels has a size corresponding to half the distance between scene objects, and that, under very specific conditions, it also peaks at a spatial resolution in which the pixel size corresponds to the whole distance between scene objects. It is argued that the peak at object distance when present is an expression of the Nyquist sample rate. The presence of this peak is, hence, shown to be a function of the matching between the phase of the scene pattern and the phase of the sample grid, i.e., the image. When these phases match, a clear and distinct peak is produced on the ALV plot. The fact that the peak at half the distance consistently occurs in the ALV plot is linked to the circumstance that the sampling interval (distance between pixels) and the extent of the sampling unit (size of pixels) are equal. Hence, at twice the Nyquist sampling rate, each fundamental period of the pattern is covered by four pixels; therefore, at least one pixel is always completely embedded within one pattern element, regardless of sample scene phase. If the objects in the scene are scattered with a distance larger than their extent, the peak will be related to the size by a factor larger than 1/2. This is suggested to be the explanation to the results presented by others that the ALV plot is related to scene-object size by a factor of 1/2-3/4.