IEEE Trans Image Process - The Roadmaker's algorithm for the discrete pulse transform.

Tópicos

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Resumo

The discrete pulse transform (DPT) is a decomposition of an observed signal into a sum of pulses, i.e., signals that are constant on a connected set and zero elsewhere. Originally developed for 1-D signal processing, the DPT has recently been generalized to more dimensions. Applications in image processing are currently being investigated. The time required to compute the DPT as originally defined via the successive application of LULU operators (members of a class of minimax filters studied by Rohwer) has been a severe drawback to its applicability. This paper introduces a fast method for obtaining such a decomposition, called the Roadmaker's algorithm because it involves filling pits and razing bumps. It acts selectively only on those features actually present in the signal, flattening them in order of increasing size by subtracing an appropriate positive or negative pulse, which is then appended to the decomposition. The implementation described here covers 1-D signal as well as two and 3-D image processing in a single framework. This is achieved by considering the signal or image as a function defined on a graph, with the geometry specified by the edges of the graph. Whenever a feature is flattened, nodes in the graph are merged, until eventually only one node remains. At that stage, a new set of edges for the same nodes as the graph, forming a tree structure, defines the obtained decomposition. The Roadmaker's algorithm is shown to be equivalent to the DPT in the sense of obtaining the same decomposition. However, its simpler operators are not in general equivalent to the LULU operators in situations where those operators are not applied successively. A by-product of the Roadmaker's algorithm is that it yields a proof of the so-called Highlight Conjecture, stated as an open problem in 2006. We pay particular attention to algorithmic details and complexity, including a demonstration that in the 1-D case, and also in the case of a complete graph, the Roadmaker's algorithm has optimal complexity: it runs in time O(m), where m is the number of arcs in the graph.

Resumo Limpo

discret puls transform dpt decomposit observ signal sum puls ie signal constant connect set zero elsewher origin develop d signal process dpt recent general dimens applic imag process current investig time requir comput dpt origin defin via success applic lulu oper member class minimax filter studi rohwer sever drawback applic paper introduc fast method obtain decomposit call roadmak algorithm involv fill pit raze bump act select featur actual present signal flatten order increas size subtrac appropri posit negat puls append decomposit implement describ cover d signal well two d imag process singl framework achiev consid signal imag function defin graph geometri specifi edg graph whenev featur flatten node graph merg eventu one node remain stage new set edg node graph form tree structur defin obtain decomposit roadmak algorithm shown equival dpt sens obtain decomposit howev simpler oper general equival lulu oper situat oper appli success byproduct roadmak algorithm yield proof socal highlight conjectur state open problem pay particular attent algorithm detail complex includ demonstr d case also case complet graph roadmak algorithm optim complex run time om m number arc graph

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