Adaptive histogram equalization in constant time

Journal of Real-Time Image Processing (2024) 21:93 https://doi.org/10.1007/s11554-024-01465-1 RESEARCH Adaptive histogram equalization in constant time Philipp Härtinger1 · Carsten Steger1 Received: 11 March 2024 / Accepted: 17 April 2024 / Published online: 16 May 2024 © The Author(s) 2024 Abstract Adaptive Histogram Equalization (AHE) and its contrast-limited variant CLAHE are well-known and effective methods for improving the local contrast in an image. However, the fastest available implementations scale linearly with the filter mask size, which results in high execution times. This presents an obstacle in real-world applications, where large filter mask sizes are desired while maintaining low execution times. In this work, we propose an efficient algorithm for AHE that reduces the per-pixel computational complexity to O(1). To the best of our knowledge, this is the first time that a constant-time algorithm is proposed for AHE and CLAHE. In contrast to commonly used fast implementations, our method computes the exact result for each pixel without interpolation artifacts. We benchmark and compare our method to existing algorithms. Our experiments show that our method exhibits superior execution times independent of the filter mask size, which makes AHE and CLAHE fast enough to be usable in real-world applications. Keywords Histogram equalization · Contrast enhancement · Image processing · Computational efficiency 1 Introduction Histogram Equalization (HE) is a classical method for improving the global contrast of an image. It linearizes the gray value histogram, such that the result image uses the full range of the possible gray values. A drawback of this method is, however, that local variations are not taken into account. Adaptive Histogram Equalization (AHE) [4, 5, 11] solves this issue by considering only the gray values within a rectangular filter window around each pixel to compute an individual HE transfer function for the respective pixel. Because AHE can lead to over-amplification of noise, Pizer et al. [12] proposed Contrast-Limited Adaptive Histogram Equalization (CLAHE), which allows to limit the maximum desired contrast in homogeneous regions. Due to its efficacy, CLAHE is still a popular image preprocessing method [9, 14] and has successfully been used to improve the performance of Deep Learning models [2, 13]. The computational complexity of AHE and CLAHE is very high, because they * Philipp Härtinger require computing the histogram of each filter window. Using a naive implementation in which the histogram is recomputed explicitly for each window, the runtime can easily become multiple seconds, minutes, or even hours, depending on the size of the filter window. To circumvent this problem, most publicly available implementations of AHE and CLAHE are based on an approximative algorithm that is fast, but can lead to visible artifacts in the resulting image. This severely impacts the applicability of AHE in many realworld scenarios, where runtimes of just a few milliseconds are required and visual artifacts are not acceptable. Because of this restriction, we only consider methods that implement an exact variant of AHE and CLAHE and thus do not lead to such artifacts. In this work, we propose a fast constant-time algorithm for AHE and CLAHE that is free of visual artifacts and thus suitable for practical applications. Section 2 discusses related work. In Sect. 3, we describe the O(1) algorithm for sliding-window histograms as well as efficient implementations of the AHE and CLAHE transfer functions. In Sect. 4, we benchmark our method against previous algorithms and discuss the results. Finally, Sect. 5 presents our conclusions. Carsten Steger 1 MVTec Software GmbH, Arnulfstr. 205, 80634 Munich, Germany Vol.:(0123456789) 93 Page 2 of 9 Journal of Real-Time Image Processing (2024) 21:93 (a) Input (b) IAHE (c) CLAHE Fig. 1  Interpolation artifacts of IAHE. (a) shows the original input image with a linear gray value ramp. (b) shows the result of IAHE, with banding artifacts due to the interpolation. (c) shows the smooth result of exact CLAHE. The artifacts on the left and right are due to border treatment and thus inevitable 2 Related work In the past, many improvements for AHE and CLAHE have been proposed. Pizer et al. [12] suggest an approximative approach in which the HE transfer function is computed only for a subset of all pixels and interpolated between these points. This method is also referred to as Interpolated Adaptive Histogram Equalization (IAHE). It is a popular choice in many computer vision libraries due to its low execution time. However, the interpolation can lead to artifacts, as shown in Fig. 1. In practice, filtering an image translated by several pixels can lead to a significantly different result than filtering the original image, meaning that the interpolation-based approach is not shift-equivariant. This poses a problem, e.g., in industrial inspection, where the artifacts might be interpreted as defects, leading to falsely rejected parts. Our work differs fundamentally from IAHE in that we compute the exact transfer function for each pixel and thus avoid the interpolation artifacts shown in Fig. 1b. Kim et al. [7] propose Block-Overlapped Histogram Equalization (BOHE), which applies Huang’s O(n) sliding-window histogram method [3] to AHE. The core idea in [3] is that when the window slides one pixel to the right, one can simply update the histogram computed for the previous pixel instead of fully recomputing it. Sund et al. [15, 16] propose SlidingWindow Adaptive Histogram Equalization (SWAHE), which applies the O(n) algorithm from [3] to CLAHE. Wang and Tao [17] also apply [3] to AHE and propose several small improvements to speed up the computation of the HE transfer function. Kong and Ibrahim [8] propose Multiple Layers Block-Overlapped Histogram Equalization (MLBOHE), which also applies Huang’s O(n) algorithm [3] to AHE. Furthermore, they employ an optimized zig-zag sliding order to avoid recomputation of the window histogram at the beginning of each row. Kim et al. [6] propose Partially Overlapped Sub-Block Histogram Equalization (POSHE), where HE is applied to overlapping sub-blocks of the image and the results are accumulated. Due to the large stride of half the window size, the result must be filtered to reduce blocking artifacts. Fu et al. [1] propose POSHE-based Optimum Clip-Limit Contrast Enhancement (POSHEOC), which combines POSHE [6] with CLAHE. Our work mainly follows [7] and [15–17] in that we improve the computational complexity of the AHE and CLAHE algorithms. In contrast to the existing linear-time methods, our proposed method has a constant per-pixel runtime that is independent of the filter window size. Contrary to [1, 6, 12], we do not use approximative algorithms. This means that our method reproduces the exact same results as the original algorithms for AHE and CLAHE and does not introduce visu (...truncated)