From the Superderivative to a Relative-Gradient Edge Operator

Alejandro Garnung Menéndez

eess.IVOct 28, 2026

From the Superderivative to a Relative-Gradient Edge Operator

Authors: Alejandro Garnung Menéndez

Organizations: University of Oviedo, Gijón Polytechnic School of Engineering, Asturias, España

Abstract

First-derivative edge detectors such as Sobel, Prewitt, Roberts and Scharr measure absolute grey-level differences. We revisit the log-gradient magnitude, a relative-gradient edge operator, and motivate it by analogy with the superderivative, a multiplicative derivative in which the independent variable is perturbed multiplicatively rather than additively. In one and two dimensions the logarithm of the superderivative is a radial logarithmic derivative; its local, origin-free, isotropic analogue is the magnitude of the logarithmic gradient. The resulting operator measures relative rather than absolute change, giving an edge-strength map that is invariant under global intensity rescaling. We present the theory, show how the operator can be implemented in two steps---a classical gradient kernel plus a pointwise division---and compare it with standard first-derivative filters on synthetic images with ground-truth edges and on the official BSDS500 benchmark. We also study a parametric family that interpolates between the absolute gradient and the log-gradient; empirically, the best members are small intermediate exponents, which edge the classical kernels by a small but statistically detectable margin in the per-image metric and, with the same Gaussian smoothing as the single- $σ$ Canny, match it in ODS and lead it in OIS and AP, while the tuned Canny remains ahead in ODS. The single-threshold margin over the kernels does not replicate on the held-out test split, and the fully relative endpoint is never the best member, its ranking depending on the edge-map normalisation. The contribution is the construction and the invariance it buys, not a new state of the art.

Figures & tables

Operator	$G_{x}$ / masks	Reference
Roberts	$G_{x}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}$ , $G_{y}=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}$	[ 15 ]
Prewitt	$\begin{bmatrix}-1&0&1\\ -1&0&1\\ -1&0&1\end{bmatrix}$	[ 16 ]
Sobel	$\begin{bmatrix}-1&0&1\\ -2&0&2\\ -1&0&1\end{bmatrix}$	[ 17 ]
Frei–Chen / Isotropic	$\begin{bmatrix}-1&0&1\\ -\sqrt{2}&0&\sqrt{2}\\ -1&0&1\end{bmatrix}$	[ 18 ]
Scharr	$\begin{bmatrix}-3&0&3\\ -10&0&10\\ -3&0&3\end{bmatrix}$	[ 19 ]

TABLE I: Common first-derivative kernels.

G_{x}

estimates the horizontal and

G_{y}

the vertical derivative. Roberts uses the two

2\times 2

masks shown; the remaining operators define

G_{y}

as the transpose of

G_{x}

Fig. 1: Comparison of the canonical radial score $L=(xI_{x}+yI_{y})/I$ , its normalised version $|L|/r$ , and the isotropic superderivative $E_{\mathrm{SD}}=\|\nabla I\|/I$ (Eq. ( 13 )). The retinal fundus is radially structured about the image centre (radial illumination falloff, macula at the centre, vessel arcade sweeping around it), which is the regime the radial forms assume; the astronaut has no such symmetry, and the isotropic form is the recommended operator for general edge detection.

Fig. 2: Alternative superderivative formulations, one row per image and one column per formulation: original, Sobel, the log-gradient $p=1$ , the parametric $p=0.5$ , the additively regularised $\alpha=0.05$ and the local-mean denominator. The lunar image is included because its dark sky is where an unregularised division by $I$ is worst behaved, which is what the regularised and local-mean variants are for. Maps are scaled to their 99th percentile, so a single saturated pixel in the dark sky cannot blacken the rest of the panel.

Image	$E_{0.25}$ (Sobel)	$\|L\|/r$	Sobel	Scharr	Prewitt	Roberts	Canny
Circles	0.986	0.985	0.986	0.988	0.984	0.979	0.980
Square	1.000	1.000	0.998	0.998	0.998	0.998	0.996
Checkerboard	1.000	0.942	0.998	0.998	0.998	1.000	0.998
Triangle	1.000	1.000	1.000	1.000	1.000	0.871	0.998
Ramp	0.996	0.188	0.996	0.996	0.996	1.000	0.996

TABLE II: Best F1 on synthetic ground-truth edges under a threshold sweep with one-pixel localisation tolerance (Section VI-F ). Highest score per image is in bold . Most first-derivative methods are within about

0.02

of each other; the exceptions are Roberts on the triangle and the radial score on the ramp and checkerboard, where their respective definitions are a genuine handicap.

Fig. 3: Edge-strength maps for synthetic images. Rows correspond to concentric annuli, square, checkerboard, triangle and intensity ramp; columns show the original image, ground truth, and the outputs of the superderivative operator ( $p=0.25$ ), the radial score, Sobel, Scharr, Prewitt, Roberts and Canny. The radial score on the ramp collapses to a single spot because the gradual slope contributes only a weak, coordinate-weighted response, while the additive step produces a strong localised peak that dominates the normalised colour scale.

Fig. 4: Intensity-scale invariance. Rows correspond to global scalings $c=0.3,1.0,3.0$ of the same image. Both edge maps are displayed on a common absolute scale, without per-image normalisation. Sobel’s absolute response grows linearly with $c$ ; the superderivative absolute response remains unchanged (up to numerical round-off). The displayed image for $c=3$ is saturated, but the operators process the unclipped values.

Fig. 5: Edge-strength maps for natural images (brick, camera, chelsea, coffee, coins, moon). Columns show the original image and the outputs of the superderivative operator ( $p=0.25$ ), Sobel, Scharr, Prewitt, Roberts and Canny.

Fig. 6: Official BSDS500 results on the validation split. Left: ODS for each method. Right: ODS and OIS as a function of the parametric exponent $p$ .

Method	ODS	OIS	AP
Canny (tuned $\sigma$ )	0.679	0.674	0.613
Canny (single $\sigma$ )	0.658	0.653	0.652
Parametric $p=0.25$	0.591	0.643	0.651
Prewitt	0.587	0.638	0.647
Sobel	0.586	0.637	0.647
Scharr	0.582	0.637	0.645

TABLE III: Official BSDS500 boundary benchmark on the 100 validation images (ODS/OIS/AP; five human maps used separately; 99 thresholds; thresholded prediction thinned to one pixel; optimal-assignment matching within

0.0075

of the image diagonal; outlier cost

100

; edge maps scaled to

[0,1]

by their minimum and maximum). Our own detector implementations, so the values are not directly comparable to published results for other detectors. Higher is better; best in bold .

Method	ODS	OIS	AP
$E_{0.25}$ , $\sigma=2.0$	0.659	0.684	0.713
Canny (single $\sigma$ )	0.658	0.653	0.652
Sobel, $\sigma=2.0$	0.650	0.679	0.707
$E_{0.25}$ , $\sigma=1.0$	0.631	0.667	0.686
Sobel, $\sigma=1.0$	0.620	0.662	0.680
$E_{0.25}$ , $\sigma=0.5$	0.609	0.650	0.662

TABLE IV: Effect of Gaussian pre-smoothing on the same BSDS500 validation split (min-max normalisation). A single

\sigma

is chosen by ODS for each method, as for the single-

\sigma

Canny. The unsmoothed kernels are compared against Canny, but once both are smoothed the family matches the single-

\sigma

Canny in ODS and leads it in OIS and AP, while the tuned Canny remains ahead in ODS. Higher is better; best in bold .

Method	ODS	OIS	AP
Canny (tuned $\sigma$ )	0.679	0.673	0.611
Canny (single $\sigma$ )	0.653	0.627	0.619
Parametric $p=0.5$	0.617	0.648	0.631
Regularised $p=0.5$	0.617	0.648	0.632
Parametric $p=0.75$	0.613	0.645	0.629
Parametric $p=0.25$	0.611	0.647	0.638

TABLE V: Same official BSDS500 benchmark as Table III but with each edge map scaled to

[0,1]

by its 99th percentile instead of its minimum and maximum. The interior optimum of the family and its margin over the kernels survive the change; the min-max collapse of the larger exponents does not. Higher is better; best in bold .

	Validation (100)			Test (200)
Method	ODS	OIS	AP	ODS	OIS	AP
Canny (tuned $\sigma$ )	0.679	0.674	0.613	0.681	0.674	0.617
Parametric $p=0.25$	0.591	0.643	0.651	0.588	0.637	0.646
Prewitt	0.587	0.638	0.647	0.591	0.633	0.642
Sobel	0.586	0.637	0.647	0.589	0.633	0.642
Scharr	0.582	0.637	0.645	0.586	0.631	0.641

TABLE VI: Same protocol as Table III on the 100 validation and the 200 held-out test images (min-max normalisation). The exponent

p=0.25

was chosen on validation only. The per-image OIS and AP advantages over the kernels replicate on the held-out split; the ODS margin over Sobel and Prewitt does not. Our own detector implementations; higher is better; best in bold .

Fig. 7: Parametric family $E_{p}=\|\nabla I\|/I^{p}$ for $p=0,0.25,0.5,0.75,1,1.5$ on natural images. $p=0$ is the absolute gradient; $p=1$ is the superderivative/log-gradient.

Fig. 8: Robustness to additive Gaussian noise. Rows correspond to increasing noise standard deviation; columns show the noisy input and the outputs of the superderivative ( $p=0.5$ ), Sobel, Scharr and Prewitt operators.

Fig. 9: Edge detection under normal (top row) and simulated low-light (bottom row) conditions. Low light is a gamma compression $I^{2.5}$ followed by additive sensor noise $\sigma=0.02$ , not the global rescaling $I\mapsto cI$ of the invariance proposition, under which the maps are identical to machine precision (Fig. 4 ). What degrades the response is the noise, not the loss of light. Figures are means over ten noise realisations with a fixed seed (42); the spread is $\pm 0.002$ for $p=0.5$ and the regularised variant and $\pm 0.021$ for $p=1$ .

Fig. 11: Controlled checks of the relative change on analytic images. Top row: reflectance $R$ , smooth illumination field $L$ and the image $I=R\cdot L$ ; bottom row: Sobel, $E_{\mathrm{SD}}$ ( $p=1$ ) and $E_{0.25}$ on $I$ , all scaled to their 99th percentile. Sobel also responds to the illumination gradient, whereas the relative operators are close to the reflectance edges.

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Neural Disjunctive Normal Forms for Learning Exact Rank Filters

Diego Marcondes, Wonder A. L. Alves, Ronaldo F. Hashimoto

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

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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.

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