Theoretical Bounds of Direct Binary Search Halftoning

Direct binary search (DBS) produces the images of the best quality among half-toning algorithms. The reason is that it minimizes the total squared perceived error instead of using heuristic approaches. The search for the optimal solution involves two operations: (1) toggle and (2) swap. Both operations try to find the binary states for each pixel to minimize the total squared perceived error. This error energy minimization leads to a conjecture that the absolute value of the filtered error after DBS converges is bounded by half of the peak value of the autocorrelation filter. However, a proof of the bound's existence has not yet been found. In this paper, we present a proof that shows the bound existed as conjectured under the condition that at least one swap occurs after toggle converges. The theoretical analysis also indicates that a swap with a pixel further away from the center of the autocorrelation filter results in a tighter bound. Therefore, we propose a new DBS algorithm which considers toggle and swap separately, and the swap operations are considered in the order from the edge to the center of the filter. Experimental results show that the new algorithm is more efficient than the previous algorithm and can produce half-toned images of the same quality as the previous algorithm.