TAILIEUCHUNG - The Essential Guide to Image Processing- P10

The Essential Guide to Image Processing- P10:We are in the middle of an exciting period of time in the field of image processing. Indeed, scarcely a week passes where we do not hear an announcement of some new technological breakthrough in the areas of digital computation and telecommunication. | Weighted Median Smoothers and Filters 273 decomposition of x amounts to decomposing this vector into 2M binary vectors x-M 1 . x0 . xM where the th element of xm is defined by xm Tm x 1 if x m 1 if Xj m where Tm is referred to as the thresholding operator. Using the sign function the above can be written as xm sgn xj m where m represents a real number approaching the integer m from the left. Although defined for integer-valued signals the thresholding operation in can be extended to noninteger signals with a finite number of quantizationlevels. The threshold decomposition of the vector x 0 0 2 2 1 1 0 1 1 T with M 2 for instance leads to the 4 binary vectors x2 -1 -1 1 -1 -1 -1 -1 -1 1 T x1 -1 -1 1 -1 1 1 -1 -1 1 T x0 1 1 1 -1 1 1 1 -1 -1 T x 1 1 1 -1 1 1 1 1 1 T. Threshold decomposition has several important properties. First threshold decomposition is reversible. Given a set of thresholded signals each of the samples in x can be exactly reconstructed as M E 1 2 m x x m M 1 Thus an integer-valued discrete-time signal has a unique threshold signal representation and vice versa T D x TD xm T .D. where denotes the one-to-one mapping provided by the threshold decomposition operation. The set of threshold decomposed variables obey the following set of partial ordering rules. For all thresholding levels m f it can be shown that x xf. In particular if x 1 then xf 1 for all f m. Similarly if xf 1 then x 1 for all m partial order relationships among samples across the various thresholded levels emerge naturally in thresholding and are referred to as the stacking constraints 18 . Threshold decomposition is of particular importance in WM smoothing since they are commutable operations. That is applying a WM smoother to a 2M 1 valued signal is equivalent to decomposing the signal to 2M binary thresholded signals processing each binary signal separately with the corresponding WM smoother and then adding the binary outputs together to obtain .

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