Scale-Space for Discrete Signals

We address the formulation of a scale-space theory for discrete signals. In one dimension it is possible to characterize the smoothing transformations completely and a n exhaustive treatment is given, answering the following two main questions: 1) Which linear transformations remove structure in the sense that the number of local extrema (or zero-crossings) in the output signal does not exceed the number of local extrema (or zero-crossings) in the original signal? 2) How should one create a multiresolution family of representations with the property that a signal a t a coarser level of scale never contains more structure than a signal a t a finer level of scale? We propose that there is only one reasonable way to define a scalespace for 1-D discrete signals comprising a continuous scale parameter, namely by (discrete) convolution with the family of kernels T ( n ; t ) = e ' l n ( t ) , where I . are the modified Bessel functions of integer order. Similar arguments applied in the continuous case uniquely lead to the