Neural Disjunctive Normal Forms for Learning Exact Rank Filters
Authors: Diego Marcondes, Wonder A. L. Alves, Ronaldo F. Hashimoto
Organizations: Mathematical Sciences Institute and France-Australia Mathematical Sciences and Interactions ANU-CNRS International Research Lab, The Australian National University · PPGI, Universidade Nove de Julho, São Paulo, Brazil · IME, Universidade de São Paulo, São Paulo, Brazil
Gradient-based learning of morphological operators is hindered by locally sparse derivatives with respect to the structuring elements. We develop a method for learning rank filters that retains exact operator evaluation while distributing the derivatives across the window. Using threshold decomposition, we construct a neural disjunctive normal form representation and establish a sufficient condition for exactness with integer-valued images and structuring elements. We then derive a smooth surrogate from the unique root of a smooth threshold equation. Implicit differentiation gives closed-form derivatives with normalised local weights, whose distribution is controlled by a smoothing parameter. These derivatives are used in a straight-through procedure: the forward pass evaluates the exact rank filter, while the backward pass uses the surrogate. For fixed ranks, the procedure learns real-valued structuring elements of any rank filter, including the median, and extends to compositions, infima, and suprema of rank filters. Experiments on image denoising using MNIST and Fashion-MNIST and retinal vessel segmentation show that, starting from a generic random initialisation, the proposed approach consistently improves performance over training with exact subgradients in every setting. In image denoising, our best network also outperforms smooth morphological networks in all settings. On vessel segmentation, the learned erosions recover structuring elements that fit the vessels, as theoretically expected, whereas most of those trained with exact subgradients remain close to their initialisation.
Figures & tables
Figure 1: Threshold decomposition for m=8 , q=2 and fW−a=(6,3,7,5,5) , with W={y1,…,y5} .
Figure 2: The neural DNF layer with vy=1 and δ=1/2 computes the q=2 order statistic of X∈{−1,+1}W , ∣W∣=5 .
Figure 3: Training an exact rank filter. The forward pass evaluates the exact filter ψa,q , so the loss is that of the exact operator. The backward pass replaces its sparse subgradient by the derivative of the root τ⋆ of the smooth DNF threshold equation.
Network
Training
1%
5%
10%
25%
50%
MNIST
ASF
Ours
33.03±1.41
29.56±0.60
26.88±0.73
22.79±0.38
18.78±0.09
Exact subgradient
31.47±1.56
27.58±0.77
25.98±0.77
22.30±0.46
18.50±0.30
Median
Ours
29.70±1.42
28.16±1.16
26.87±1.01
22.20±0.60
15.20±0.12
Exact subgradient
28.87±1.99
27.09±1.73
25.39±1.47
21.35±0.80
14.93±0.24
Median × 2
Ours
28.73±1.05
27.67±0.78
26.73±0.52
23.95±0.30
18.80±0.19
Table 1: Test PSNR (dB) of the denoising networks, mean ± standard deviation over 10 seeds. For each dataset and noise level, the best mean is in bold. ASF: four alternating erosion and dilation layers.
Training
AUC
F1
Accuracy
Sensitivity
Specificity
Ours
0.9279±0.0002
0.6415±0.0004
0.9303±0.0003
0.6765±0.0025
0.9561±0.0005
Exact subgradient
0.9096±0.0015
0.6129±0.0024
0.9248±0.0008
0.6459±0.0086
0.9531±0.0016
Top-hat input, no learning
0.9042
0.5834
0.9166
0.6336
0.9453
Table 2: Test metrics of the supremum of 256 erosions on CHASE_DB1, computed on the foreground, mean ± standard deviation over 10 seeds, and of the top-hat input thresholded in the same way. The best mean of each metric is in bold.
Figure 4: Retinal vessels, seed 0 . Top: a test image of CHASE_DB1, its black top-hat input, and the output of the supremum of 256 erosions trained by our method and by the exact subgradient, in greyscale and thresholded at the value that maximises the F1 on the training images, on the foreground. Bottom: structuring elements of the 10 erosions that most often attain the supremum on the vessel pixels of the training images detected by the network (labelled as vessel and above the threshold), with the share of these pixels below each. Each element a is shown as a−maxa , from white ( 0 ) to black ( ≤−1 ).
A spatial-local image-denoising filter is proposed, and its performance metrics are evaluated in comparison with baseline filtering algorithms, including the median, adaptive median, Gaussian, bilateral, Wiener, anisotropic diffusion, and non-local means. The developed filter is based on aligning the intensity value of the central pixel in a 3x3 window with the statistical majority intensity of one of the two clusters formed by optimal Otsu's partitioning of a pixel set sorted by intensity and trimmed to seven elements. This is followed by a fuzzy fusion of the calculated value with the median intensity of the pixels within the window. The proposed filter demonstrates the highest robustness to variations in image noise levels, particularly when processing mixed noise consisting of salt-and-pepper impulse noise and additive Gaussian noise in various proportions
The convolution operator is the fundamental building block of modern convolutional neural networks (CNNs), owing to its simplicity, translational equivariance, and efficient implementation. However, its structure as a fixed, linear, locally-averaging operator limits its ability to capture structured signal properties such as low-rank decompositions, adaptive basis representations, and non-uniform spatial dependencies. This paper presents a systematic taxonomy of operators that extend or replace the standard convolution in learning-based image processing pipelines. We organise the landscape of alternative operators into five families: (i) decomposition-based operators, which separate structural and noise components through singular value or tensor decompositions; (ii) adaptive weighted operators, which modulate kernel contributions as a function of spatial position or signal content; (iii) basis-adaptive operators, which optimise the analysis bases together with the network weights; (iv) integral and kernel operators, which generalise the convolution to position-dependent and non-linear kernels; and (v) attention-based operators, which relax the locality assumption entirely. For each family, we provide a formal definition, a discussion of its structural properties with respect to the convolution, and a critical analysis of the tasks for which the operator is most appropriate. We further provide a comparative analysis of all families across relevant dimensions -- linearity, locality, equivariance, computational cost, and suitability for image-to-image and image-to-label tasks -- and outline the open challenges and future directions of this research area.
We propose the effective rank (ERank) of the channel covariance of an image's deep feature map as a per-sample, label-free measure of visual richness, computed from a single forward pass through a frozen pretrained encoder. ERank counts how many decorrelated channel directions an image activates, and we characterize its properties, including its behavior under noise. Empirically, ERank orders images from plain to visually rich, correlates with codec bitrate, sharpness, and edge density, and correlates with human complexity annotations on IC9600 with r=0.72. As a data-selection criterion, removing low-ERank samples improves super-resolution and removing high-ERank samples improves OCR, in both pretraining and finetuning, while selection does not help classification, segmentation, or denoising. ERank is thus a cheap richness signal, useful exactly when task difficulty is governed by input richness.
Maksim Smirnov, Grigory Kononov, Anastasiia Linich +2