Digital Halftoning Algorithms Based on Optimization Criteria and Their Experimental Evaluation

Tetsuo ASANO
Thomas ROOS

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E79-A    No.4    pp.524-532
Publication Date: 1996/04/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
discrete algorithm,  combinatiorial optimization,  digital halftoning,  computational complexity,  experimental evaluation,  

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Digital halftoning is a well-known technique in image processing to convert an image having several bits for brightness levels into a binary image consisting only of black and white dots. A great number of algorithms have been presented for this problem, some of which have only been evaluated just by comparison with human eyes. In this paper we formulate the digital halftoning problem as a combinatiorial problem which allows an exact solution with graph-theoretic tools. For this, we consider a d-dimensional grid of n := Nd pixels (d 1). For each pixel, we define a so-called k-neighborhood, k {0,...N - 1}, which is the set of at most (2k + 1)d pixels that can be reached from the current pixel in a distance of k. Now, in order to solve the digital halftoning problem, we are going to minimize the sum of distances of all k-neighborhoods between the original picture and the halftoned one. We show that the problem can be solved in linear time in the one-dimensional case while it looks hopeless to have a polynomial-time algorithm in higher dimension including the usual two-dimensional case. We present an exact algorithm for the one-dimensional case which runs in O(n) time if k is regarded to be a constant. For two-dimensional case we present fast approximation techniques based on space filling curves. An experimental comparison of several implementations of approximate algorithms proves that our algorithms are of practical interest.