Digital Dilation: Scale Expansion in Discrete Systems

• Digital dilation is a family of discrete operations that expand an operator’s effective support without increasing parameter count.
• It encompasses methods such as dilated convolutions, morphological set expansion, and HDR component growth applied in image processing and geometric analysis.
• Recent designs emphasize controlled dilation rates and learnable patterns to balance context enrichment with artifact minimization.

Searching arXiv for recent and relevant papers on “digital dilation” across its main technical senses. Digital dilation denotes a family of discrete operations that enlarge an operator’s effective support, influencing neighborhood, or sampling footprint without collapsing the problem into a single standardized formalism. Across recent arXiv literature, the term is used for dilated or atrous convolutions in CNNs, connected-component growth in HDR environment maps, support control in convolution finite elements for digital image correlation, differentiable approximations of morphological dilation for structural constraints, wavelet-domain significance propagation in image coding, zero-inserted filter expansion in classical edge detection, and affine-geometric scaling structures on dilation surfaces (Vesal et al., 2018, Debevec et al., 2022, Lu et al., 2023, Riva et al., 2021, Wu et al., 2010, Orhei et al., 2021, Boulanger et al., 2021). The unifying idea is discrete expansion: the underlying object is not necessarily made larger in image space, but the region that contributes to inference, coding, correlation, or dynamics is.

1. Morphological and set-theoretic foundations

In mathematical morphology, dilation is a set operation on a discrete domain. For a binary image represented as a set $A \subset \mathbb{Z}^2$ and a structuring element $B$, dilation is defined by

$$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$

so the support of the foreground expands according to the footprint of $B$. In fuzzy form, used for spatial-relation modeling, a membership function $m : \Omega \to [0,1]$ is dilated by a fuzzy structuring element $w$ as

$$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$

with $t$ a t-norm, often the product t-norm (Riva et al., 2021).

This morphological formulation is central when digital dilation is used to encode spatial relations. A relation $\rho = (k,l,B_{kl})$ is modeled by dilating a source object $k$ with a structuring element $B$0 that encodes semantics such as “close to,” “to the left of,” or “inside.” The resulting relational map is intersected with a target object $B$1, and the normalized overlap yields a satisfaction score. In that setting, dilation is not merely a preprocessing primitive; it becomes a structural prior that can regularize neural networks trained on small datasets (Riva et al., 2021).

A recurring misconception is that all digital dilation in later machine-learning work remains morphological in this strict sense. It does not. Many later uses retain the idea of an expanded neighborhood but replace the supremum or set union by linear convolution, weighted averaging, sparse sampling, or affine transport. The morphological definition therefore serves as a conceptual baseline rather than a universal implementation.

2. Radiometric dilation in HDR lighting and energy redistribution

A distinct use of digital dilation appears in HDR lighting reproduction for virtual production. Here the problem is not object morphology but radiometric range compression: an HDRI environment map contains concentrated light sources whose pixel values exceed the maximum luminance of an LED stage. Direct clipping reduces total light energy, weakens directionality, distorts illumination, and can create chromatic artifacts (Debevec et al., 2022).

The proposed HDR lighting dilation algorithm first identifies saturated pixels,

$B$2

groups them into connected components, then iteratively dilates each component by one pixel while preventing intersection with other components. After each dilation step it computes the solid-angle-weighted average color

$B$3

and stops when no channel of that average exceeds the display threshold. The region is then replaced by its average color, preserving its total energy exactly by construction (Debevec et al., 2022).

This is a specialized digital dilation operator: geometric region growth is coupled to a photometric stopping criterion. Unlike classical morphology, the goal is not shape expansion but energy-preserving redistribution of radiance. The method spreads light as little as possible, maintains approximate directionality, and avoids chromatic fringing by averaging full RGB vectors rather than channels independently. A practical limitation is explicit in the formulation: if the HDRI contains too much global energy, local dilation cannot solve the constraint and global attenuation is required (Debevec et al., 2022).

3. Dilated convolution as digital dilation in neural networks

The most common contemporary use of digital dilation is dilated, or atrous, convolution. In its standard discrete form,

$B$4

the dilation rate $B$5 spaces kernel taps farther apart without increasing the number of trainable weights. In 2D and 3D, this expands the receptive field while preserving feature-map resolution (Salehi et al., 2021, Yu et al., 2017).

This mechanism is used in several distinct ways. In left atrial segmentation from 3D gadolinium-enhanced MRI, a modified 3D U-Net places four parallel $B$6 convolutions with dilation rates $B$7 through $B$8 in the bottleneck. Their outputs are summed and passed to the decoder. The design was motivated by a specific failure mode of conventional U-Net: the lowest level has a small receptive field and may misclassify regions with similar local appearance because it lacks non-local anatomical context. The dilated model improved validation Dice from $B$9 to $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$0, test Dice from $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$1 to $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$2, and reduced parameter count from $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$3M to $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$4M relative to the 3D U-Net baseline (Vesal et al., 2018).

In dense prediction, DDCNet uses dilation as the primary mechanism for shaping the effective receptive field while maintaining high spatial resolution. A 30-layer comparison in the paper showed that a network with linearly increasing dilation rates attains a much larger effective receptive field than a non-dilated network of the same depth. DDCNet-B0 employs dilation rates $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$5 followed by $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$6, while DDCNet-B1 uses a more compact schedule with even dilation steps and a feature-refiner module. Reported model sizes are about $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$7M parameters for B0 and $$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$8M for B1, substantially smaller than FlowNet-Simple ($$A \oplus B = \{ x \in \mathbb{Z}^2 \mid (\hat{B})_x \cap A \neq \emptyset \},$$9M) and FlowNet2 ($B$0M), while remaining competitive on Sintel, KITTI, and Middlebury within the lightweight regime (Salehi et al., 2021).

Dilated Residual Networks pursue the same goal in classification and downstream dense tasks: remove late-stage stride, keep high-resolution feature maps, and recover receptive field with dilation. DRN-A replaces the last two stride-2 reductions of ResNet with dilation-2 and dilation-4 convolutions, reducing the final downsampling factor from $B$1 to $B$2. DRN-C further introduces “degridding” by removing early max pooling, adding layers with decreasing dilation, and removing residual connections in the last added blocks. On ImageNet, DRN-A and DRN-C improve on corresponding ResNets; on Cityscapes, DRN-C-42 reaches mean IoU $B$3 versus $B$4 for the comparable ResNet-101 baseline, and on weakly supervised localization DRN-C-26 attains top-1 localization error $B$5 (Yu et al., 2017).

The main controversy in this line of work is not whether dilation enlarges receptive fields, but how it does so. Several papers emphasize that aggressive or poorly scheduled dilation produces gridding artifacts: the network samples disjoint subsets of pixels and loses dense spatial coverage. This is why linear or mild dilation growth, multi-rate fusion, and late low-dilation refinement are repeatedly preferred over uncontrolled exponential spacing (Yu et al., 2017, Salehi et al., 2021).

4. Learnable and heterogeneous dilation

A major research direction replaces fixed dilation schedules by learned ones. In semantic segmentation of street scenes, one approach assigns each input channel its own learnable dilation factor $B$6 and samples feature maps at fractional positions using bilinear interpolation. The forward pass becomes

$B$7

with gradients propagated through the interpolation weights. Integrated into DeepLab-LargeFOV, DeepLab-v2, and PSPNet, this strategy improves Cityscapes and CamVid performance; for example, DeepLab-LargeFOV rises from $B$8 mean IoU for the best fixed conv5 dilation setting to $B$9 when conv5 and fc6 dilations are learned, and PSPNet improves from $m : \Omega \to [0,1]$0 to $m : \Omega \to [0,1]$1 on Cityscapes validation (He et al., 2017).

Generalized Dilation Neural Networks take a more abstract route. Instead of fixing an equidistant dilation pattern, they represent a large receptive field with a sparse binary mask $m : \Omega \to [0,1]$2 and define

$m : \Omega \to [0,1]$3

The mask is relaxed by sigmoids and optimized end-to-end with barrier terms enforcing sparsity constraints. In that formulation, standard dilated convolution is a special case of masked convolution with a regular zero-insertion pattern; the generalization allows non-equidistant and anisotropic support patterns without interpolation at fractional image locations (Chadha et al., 2019).

“Inception convolution” extends this idea in a channel-wise and axis-wise manner. Each output channel $m : \Omega \to [0,1]$4 receives its own dilation pair $m : \Omega \to [0,1]$5, with

$m : \Omega \to [0,1]$6

The search space contains $m : \Omega \to [0,1]$7 channel-wise configurations per layer. Efficient Dilation Optimization searches this space statistically from a pre-trained supernet. Replacing the $m : \Omega \to [0,1]$8 standard convolutions of a ResNet-50 Faster R-CNN backbone with inception convolution raises MS COCO AP from $m : \Omega \to [0,1]$9 to $w$0; on ImageNet, top-1 accuracy improves from $w$1 to $w$2 for ResNet-50 (Liu et al., 2020).

Dilated Convolution with Learnable Spacings goes further by learning the positions of a small set of non-zero kernel elements inside a large support. A constructed kernel

$w$3

is obtained by placing each weight $w$4 at continuous coordinates $w$5 through bilinear interpolation. On ConvNeXt, replacing depthwise $w$6 convolutions by DCLS with a $w$7 support improves ImageNet-1k top-1 accuracy from $w$8 to $w$9 for ConvNeXt-T, from $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$0 to $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$1 for ConvNeXt-S, and from $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$2 to $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$3 for ConvNeXt-B, while also improving ADE20K and COCO downstream performance. By contrast, classic fixed-rate dilated convolution degrades both ResNet50 and ConvNeXt in the same study (Khalfaoui-Hassani et al., 2021).

A practical conclusion suggested by these works is that “digital dilation” in CNNs is no longer synonymous with a hand-picked integer rate. It has become a learnable design variable governing receptive-field geometry, anisotropy, and channel specialization.

5. Dilation as support control in correlation, coding, and classical filtering

Outside CNNs, digital dilation often acts as a support or window parameter. In convolution finite element digital image correlation, the dilation parameter $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$4 controls the window size, or influencing domain, of kernel functions in the convolution patch $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$5. With radial basis kernels $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$6, the kernel is non-zero only for $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$7, so $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$8 sets the support radius in parametric coordinates. The method couples $$\delta_w m(x) = \sup_{y \in \Omega} t\big(w(x-y), m(y)\big),$$9 with polynomial order $t$0 and patch size $t$1; the authors report that $t$2, $t$3, $t$4 provides overall good performance, and that the C-FE approach outperformed standard FE in all tested DIC results (Lu et al., 2023).

In wavelet image coding, morphological dilation becomes a significance-propagation mechanism rather than a pixel-domain shape operator. The image is decomposed by a 5-level 9/7 Daubechies wavelet transform; significant coefficients at a given bitplane act as seeds, and dilation tests their 8-connected neighbors for significance. The contribution of the cited work is to control this dilation with context weights and to combine it with variable-length group test coding, thereby reducing the output of insignificant coefficients and coding high-probability events with few bits. Reported average gains are about $t$5 dB over SPECK and $t$6 dB over SPIHT without arithmetic coding, with larger gains against several baselines when arithmetic coding is included (Wu et al., 2010).

In classical edge detection, digital dilation means inserting zeros between the coefficients of a standard filter, thereby enlarging its geometric footprint while keeping the same non-zero weights and arithmetic count. A $t$7 kernel becomes a sparse $t$8 or $t$9 kernel with gaps, and convolution is effectively performed on a more widely spaced sampling grid. This construction was applied systematically to Sobel, Prewitt, Kirsch, Laplacian, Canny, Marr–Hildreth, Shen–Castan, and Edge Drawing. The reported findings are not uniform: many first-order operators benefit, especially within Canny and several gradient-based pipelines, but pure Laplacian and Shen–Castan generally do not, and large dilation can miss thin structures (Orhei et al., 2021).

These cases illustrate a broader pattern. Digital dilation is often introduced to enlarge an operator’s footprint at fixed local complexity, but the benefit depends on the algorithm’s error mode. Where the main problem is limited context or noisy local evidence, dilation tends to help; where the main problem is already aliasing, double-edge formation, or over-smoothed zero crossings, it may not.

6. Geometric dilation surfaces and dynamical meaning

A mathematically distinct meaning of digital dilation appears in the theory of dilation surfaces. A dilation surface is a marked surface equipped with charts whose transition maps lie in

$\rho = (k,l,B_{kl})$0

so the local geometry is glued by positive real dilations and translations rather than pure translations (Boulanger et al., 2021).

This structure defines directional foliations $\rho = (k,l,B_{kl})$1 for $\rho = (k,l,B_{kl})$2, because straight lines of a fixed direction are preserved by maps $\rho = (k,l,B_{kl})$3 with $\rho = (k,l,B_{kl})$4. A closed geodesic is a periodic leaf of one of these foliations. The principal result is that for every closed dilation surface, the set of directions of closed geodesics is dense in the circle; equivalently, the set of directions covered by a cylinder is dense in $\rho = (k,l,B_{kl})$5 (Boulanger et al., 2021).

The proof uses degenerations of Delaunay polygonations under the Teichmüller flow

$\rho = (k,l,B_{kl})$6

together with an induction on surface complexity. The argument classifies limiting Delaunay polygons into three degeneration types, organizes them into Delaunay pieces, and shows that either lower-complexity limit components inherit the density statement inductively or cyclic domains force cylinders of large modulus, which in turn force periodic trajectories (Boulanger et al., 2021).

This geometric usage is not an image-processing or machine-learning operation, but it preserves the underlying intuition of dilation: a structure is built from scaling transformations, and the global dynamics are governed by how straight trajectories behave under those discrete or continuous scalings. A plausible implication is that “digital dilation” has become a cross-disciplinary term for scale-modulated locality, even when the underlying mathematics ranges from morphological set growth to affine holonomy.

7. Scope, distinctions, and recurring design principles

Across these literatures, digital dilation has several non-equivalent meanings. In morphology it is a supremum-based set expansion; in HDR lighting it is connected-component growth followed by energy-preserving averaging; in CNNs it is sparse sampling over a larger receptive field; in C-FE DIC it is a kernel support radius; in wavelet coding it is neighborhood expansion over significance maps; in classical filtering it is zero-inserted kernel enlargement; in dilation surfaces it is part of the affine structure group (Riva et al., 2021, Debevec et al., 2022, Salehi et al., 2021, Lu et al., 2023, Wu et al., 2010, Orhei et al., 2021, Boulanger et al., 2021).

Despite these differences, several design principles recur. First, dilation is repeatedly used to enlarge support without proportionally enlarging parameter count or local stencil complexity. Second, many successful variants combine small and large supports rather than using a single large dilation rate. Third, uncontrolled dilation can create artifacts: gridding in CNNs, sparse or hole-like coverage in large-rate filters, excessive diffusion in HDR lighting, or degeneration to standard FE behavior when the dilation window becomes too small (Yu et al., 2017, Salehi et al., 2021, Debevec et al., 2022, Lu et al., 2023).

The term therefore names not one operation but a family of scale-expansion mechanisms on discrete structures. What remains invariant is the technical role of dilation as a controlled enlargement of influence: the operator acts over a wider domain than its compact parametrization would ordinarily permit, and the success of that enlargement depends on how well the expanded support is matched to the task, the data, and the failure modes of the underlying method.