We present a method for multiclass thresholding of histograms whose modes may overlap so strongly that they merge into a unimodal distribution. The histogram is approximated by a nonparametric kernel density (KD) estimate, a mixture of Gaussians with one component per bin and a common bandwidth, whose parameters are determined by Expectation-Maximization (EM) iterations. Because the estimate is a Gaussian mixture, the semigroup property of Gaussian convolution yields its scale-space representation in closed form, at bandwidths both larger and smaller than the fitted one. The algorithm increases or decreases the bandwidth until the number of local minima of the estimate equals the number of requested classes minus one, and returns the positions of those minima as the threshold values. We validate the method on synthetic histograms with known thresholds, and we apply it to the histogram of X-ray computed tomography images of a natural porous sample in which the three phases strongly overlap. The porosity computed from the resulting segmentation agrees with the direct experimental measurement to within 4%, whereas K-means clustering misestimates it by 52%.
Figures & tables
Figure 1: Schematic representation of a mixture model with (a) well-separated components and (b) highly overlapping components. The blue solid line shows the mixture model while dashed lines correspond to components of the mixture. The black vertical line shows the position of the component intersection points and the red vertical line shows the closest local minimum of the mixture.
Figure 2: Evolution of the KD estimate of the histogram across the iterations of Algorithm 1 , where the blue dots show the histogram. The orange curves represent the separated modes while the red curve corresponds to the sum of the modes.
Figure 3: KD estimation of a noisy histogram whose underlying distribution is a mixture of three modes. The blue solid line shows the histogram, the red solid line the KD estimate, the dashed orange lines the modes of the underlying distribution, the solid black line the KD estimate at the scale at which the local minima are resolved, and finally the vertical dashed gray lines the threshold values extracted from the minima of the KD estimate.
Figure 4: Comparison between the predicted ( x axis) and reference ( y axis) threshold values for synthetic histograms with (a) 100 bins and (b) 1000 bins. The blue and red markers refer to cases where the number of local minima of the KD estimate at the fitted scale is greater (i.e., #M>C−1 ) or smaller (i.e., #M<C−1 ) than the number of requested threshold values, respectively. The green line shows the function y=x . The filled markers correspond to threshold values that lie within the band ∣x−y∣<1 .
Figure 5: (a) A representative XCT image of sediment column C6197A. (b) Zoomed-in views of the solid phase (red square), the unresolved porous phase (green square), the void phase (black square) and a mixture of the three phases (blue square).
Figure 6: The combined intensity histogram (blue line) (a) at the original scale, and (b) at the scale at which the minima are resolved. The vertical red and black lines indicate the threshold values extracted by the proposed method and by K-means clustering, respectively.
Multilevel Image thresholding is an important preprocessing algorithm in computer vision applications nowadays. Since most common thresholding methods take the desired count of thresholds as input by the user, thresholding methods that automatically determines a suitable count of thresholds from the input image itself are advantageous. In this article, a novel thresholding method based on a dynamic programming algorithm and a modification of Minimum Error Thresholding (MET) criterion is thoroughly presented. An empirical statistical study is performed to pinpoint why this proposed method is superior. Moreover, an extended comparison between this proposed method and other state-of-the-art methods is performed on a comprehensive set of natural, satellite and medical test images. The numerical results show that the proposed MET-DP method takes much less time than traditional dynamic programming thresholding methods when the number of thresholds is high. The proposed method can detect a suitable count of thresholds for most of tested images of different types. However, traditional methods that take the count of thresholds as input produce thresholded images of higher structural similarity index measure (SSIM) and peak signal-to-noise ratio (PSNR) values than MET-DP. Source code can be found on https://w3id.org/met-dp/article1-code
Eslam Hegazy, Mohamed Gabr
Computer Science and Engineering Department German University in Cairo, New Cairo, 11865, Cairo, Egypt
Multi-class density estimators can count objects in crowded scenes, but their class-wise outputs offer no indication of how reliably density mass has been allocated. We introduce a training-free confidence proxy based on local competition between the strongest class channels. The resulting prominence measure requires only the predicted density maps, with no retraining or access to model internals. We test whether residual competition is diagnostic of cross-class leakage by attenuating contested density at inference time. On DSACA predictions for the Hicks et al. dataset, this reduces overall MAE from 0.920 to 0.846, suggesting that class competition contains useful information about count reliability.
Villanelle O'Reilly, Georgios Leontidis, Marc Hanheide
Lincoln Centre for Autonomous Systems, University of Lincoln, UK · Applied Vision and Artificial Intelligence Lab, University of Lincoln, UK · UiT The Arctic University of Norway, Norway
Multilevel image thresholding is widely used for segmentation in applications ranging from medical imaging to remote sensing. Classical objective functions, such as Otsu's between-class variance and Kapur's entropy, are often optimized using metaheuristic algorithms, with performance evaluated via metrics like Structural Similarity Index (SSIM) and Peak Signal-to-Noise Ratio (PSNR). These evaluations implicitly assume that SSIM and PSNR provide unbiased measures of segmentation quality. In this study, we examine this assumption by analyzing the correlation between thresholding objective functions and quality metrics across all possible thresholds for images in the BSDS500 dataset. Results show that Otsu's criterion consistently exhibits high correlation with both SSIM and PSNR, while Kapur's entropy demonstrates weaker and more variable correlation. Otsu outperforms Kapur in correlation with PSNR for all images and with SSIM for over 91%. Our findings reveal an inherent metric-objective-function bias. This work highlights the need for more neutral evaluation frameworks and motivates extending the analysis to additional thresholding criteria and domains. Source code of this paper can be found at https://w3id.org/met-dp/icpr26-95