Similarly, multi-scale blob detectors at any given fixed scale can be obtained from local maxima and local minima of either the Laplacian operator (also referred to as the Laplacian of Gaussian) or the determinant of the Hessian matrix In an analogous fashion, corner detectors and ridge and valley detectors can be expressed as local maxima, minima or zero-crossings of multi-scale differential invariants defined from Gaussian derivatives. The algebraic expressions for the corner and ridge detection operators are, however, somewhat more complex and the reader is referred to the articles on corner detection and ridge detection for further details. Scale-space operations have also been frequently used for expressing coarse-to-fine methods, in particular for tasks such as image matching and for multi-scale image segmentation. 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 scale-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 covariance 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 . Scale invariant feature detection Following this approach of gamma-normalized derivatives, it can be shown that different types of scale adaptive and scale invariant feature detectors can be expressed for tasks such as blob detection, corner detection, ridge detection, edge detection and spatio-temporal interest point detection (see the specific articles on these topics for in-depth descriptions of