Scale Space - Scale Selection

Scale Selection

The theory presented so far describes a well-founded framework for representing image structures at multiple scales. In many cases it is, however, also necessary to select locally appropriate scales for further analysis. This need for scale selection originates from two major reasons; (i) real-world objects may have different size, and this size may be unknown to the vision system, and (ii) the distance between the object and the camera can vary, and this distance information may also be unknown a priori. A highly useful property of scale-space representation is that image representations can be made invariant to scales, by performing automatic local scale selection based on local maxima (or minima) over scales of normalized derivatives

where is a parameter that is related to the dimensionality of the image feature. This algebraic expression for scale normalized Gaussian derivative operators originates from the introduction of -normalized derivatives according to

and

It can be theoretically shown that a scale selection module working according to this principle will satisfy the following scale invariance property: if for a certain type of image feature a local maximum is assumed in a certain image at a certain scale, then under a rescaling of the image by a scale factor the local maximum over scales in the rescaled image will be transformed to the scale level .