Theoretical Bounds of Direct Binary Search Halftoning

Abstract

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.

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“…DBS [23] is a pixel-based processing method which changes the regional 3×3 pattern centered at the current processing position iteratively. For the DBS swap operator, it swaps the current mean value with the mean value of its eight nearest neighbors and calculate the effects of all the trial changes.…”

“…Thus, this study adopts a halftone image optimization technology called Direct Binary Search (DBS). DBS is widely used instead of the heuristic approach to optimize halftone images by using a human visual system to minimize the total squared perceived error between a continuous tone image and a halftone image [23,24]. In this study, we adopt the Swap operator from the DBS framework and use this operator to optimize the stego image that is compressed and contains the hidden data.…”

Fast halftoning policy network with error feedback

“…Halftoning techniques aim for reproducing continuous-tone images c∈[0,1]N with binary pixels h∈{0,1}N, where N denotes the number of pixels. In addition to classic approaches like ordered dithering, 1 – 4 error diffusion, 5 – 12 and search-based methods, 13 – 18 recently, deep learning-based solutions 19 – 25 are showing their abilities in rendering decent halftones with reversibility 21 or less computational complexity 23 . Specifically, convolutional neural networks (CNNs) are trained to project white Gaussian noise maps into halftone pixels conditioning on the continuous-tone image [illustrated in Fig.…”