Shape-to-Scale: Linking Geometry and Scale

• Shape-to-Scale is a family of constructions that interrelate geometric shape and scale through analysis, editing, and multiscale modeling.
• It underpins diverse methodologies such as editable 3D implicit geometry, contour normalization, and scale-induced segmentation, offering robust solutions for variable applications.
• The literature distinguishes approaches that decouple scale as a nuisance parameter from those that couple it to preserve intrinsic geometric structures.

The literature suggests that Shape-to-Scale is not a single formalism but a family of constructions in which shape and scale are either separated, coupled, or transformed into one another for analysis, editing, inference, or comparison. In recent work, the phrase can denote turning rigid completed implicit 3D objects into editable, part-aware volumetric assets; encoding contours so that intrinsic geometry is preserved while absolute size is discarded and optionally restored; treating shape-and-scale spaces as quotients distinct from pure shape spaces; or defining scale spaces whose smoothing behavior is itself induced by the geometry of the object or region under study (Vermandere et al., 13 Feb 2026, Romero et al., 1 Jul 2025, Anderson, 2018).

1. Principal meanings in the literature

The expression is used across several technical settings. In some papers, scale is a deformation variable to be applied to a shape while preserving structure; in others, scale is a nuisance variable to be factored out; in still others, scale is a degree of freedom retained in a relational quotient or a multiscale parameter induced by geometry. This suggests that the common denominator is not a single representation, but a recurring question: what part of a geometric description should remain invariant, and what part should remain editable or inferable as scale.

Domain	Operative meaning	Representative paper
3D implicit editing	Convert static completed SDFs into editable, part-aware scaleable objects	"Adaptive Scaling with Geometric and Visual Continuity of completed 3D objects" (Vermandere et al., 13 Feb 2026)
2D contour representation	Remove global size mathematically, preserve intrinsic contour geometry	"ShapeEmbed: a self-supervised learning framework for 2D contour quantification" (Romero et al., 1 Jul 2025)
Relational geometry	Distinguish pure shape from shape-and-scale by quotient choice	"Shape Derivatives" (Anderson, 2018)
Region-based scale spaces	Let shape determine the scale space itself	"Coarse-to-Fine Segmentation With Shape-Tailored Scale Spaces" (Sundaramoorthi et al., 2016)
Detection loss design	Couple box shape and box scale into anisotropic penalties	"Shape-IoU: More Accurate Metric considering Bounding Box Shape and Scale" (Zhang et al., 2023)
DS-InSAR	Separate angular scattering structure (“shape”) from magnitude behavior (“scale”)	"Shape-to-Scale InSAR Adaptive Filtering and Phase Linking under Complex Elliptical Models" (Yao et al., 16 Sep 2025)

A useful high-level distinction follows from these usages. Some formulations are shape-preserving under scale removal, as in contour embeddings and similarity-shape analysis. Others are scale-preserving under shape-aware control, as in editable 3D objects and anisotropic regression losses. A third class is shape-induced multiscale analysis, in which the definition of coarse and fine structure depends on the geometry of the object or region itself.

2. Editable implicit geometry: completed objects made scale-aware

A direct, literal use of the term appears in the 3D implicit-editing pipeline of "Adaptive Scaling with Geometric and Visual Continuity of completed 3D objects" (Vermandere et al., 13 Feb 2026). The paper addresses what it identifies as a practical gap between 3D object completion and 3D object editing. Completion networks such as AutoSDF and texture completion methods such as Texture Fields or IF-Net texture can infer missing geometry and appearance from partial observations, but the outputs are essentially static implicit volumes. A completed Signed Distance Field faithfully reconstructs geometry at one size, yet does not expose editable parts, articulation, or protected regions. Global rescaling or naive local deformation therefore stretches repeated elements, breaks proportions, and can create discontinuities between geometry and texture.

The proposed solution is a volumetric representation called a colored signed distance function (CSDF). The geometric SDF is discretized onto a regular voxel grid of resolution $128 \times 128 \times 128$, corresponding color is mapped onto the same grid, and the result is stored as a 3D texture whose RGB channels encode color and whose alpha channel stores the signed distance value. This gives a single representation in which geometry and appearance can be transformed jointly. Editability is then introduced by automatic part segmentation: a watertight mesh is extracted from the completed SDF with Marching Cubes, decomposed into nearly convex subcomponents using Approximate Convex Decomposition (ACD) via CoACD, and the resulting part labels are remapped back into the voxelized CSDF. After that step, every voxel carries an SDF value $d$, a color value $(r,g,b)$, and a part index $i$.

Scaling is controlled by two parallel planes along a chosen axis, a starting plane and an ending plane. The user moves the ending plane to a new destination, thereby specifying expansion or contraction of the selected region. The region before the starting plane is unchanged, parts beyond the ending plane are translated, and parts inside the interval become the active deformation zone. Within that zone, a new voxel array is created and the CSDF attributes are resampled according to type. Geometry uses linear interpolation of SDF values,

$$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$

appearance uses the same channel-wise interpolation for RGB,

$$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$

and part indices are handled by SDF-aware transition detection rather than numerical interpolation, with boundary location

$$x_b = \text{argmin}_x \left( |\text{SDF}(x)| \; \text{where } p(x) \neq p(x+\delta x) \right).$$

The paper also adds a reevaluation step for SDF correctness near the interfaces between unchanged, translated, and stretched regions. Distances are recomputed only for voxels whose nearest-surface relationship may have changed, which narrows correction to the affected zone. For large deformations, the central mechanism is repetition-based tiling. Rather than stretching a modular region such as a shelf bay, sofa cushion, or slatted part, the method replicates the selected section multiple times and absorbs only the residual mismatch by minor interpolation-based resizing. Interface layers between repetitions are smoothed by matching the last layers of one repeated part to the first layers of the next, except at the last repetition where the terminal boundary reconnects to the original neighboring part.

The evaluation is qualitative rather than metric-heavy. The paper uses objects extracted from Matterport3D indoor scans, completed geometrically with AutoSDF and texturally with IF-Net or Texture Fields, and also uses higher-quality complete 3D objects from Sketchfab. Completed CSDFs are displayed in Unity with a raymarching shader and an interactive plane-based interface. Three deformation modes are compared: Global scaling, Selective scaling, and Selective tiling/repetition. No Chamfer distance, IoU, perceptual texture score, or user study statistics are reported. The reported result is that global scaling performs worst for localized functional structure, selective scaling is better for structurally simple regions, and selective tiling/repetition is best for repeated modular elements such as bookshelves, cabinets, and cushioned furniture. The paper’s own summary is that completed implicit 3D objects can become editable implicit assets rather than fixed end products.

3. Explicit separation of intrinsic shape from absolute size

A second major meaning of shape-to-scale is the explicit removal of size while preserving intrinsic geometry. "ShapeEmbed: a self-supervised learning framework for 2D contour quantification" (Romero et al., 1 Jul 2025) makes this separation mathematically explicit for 2D contours. Starting from a binary mask, the method extracts a simply connected outline with marching squares, interpolates it with a parametric linear spline, and uniformly samples $N$ contour points, $N=64$ by default. The contour is then encoded by its Euclidean distance matrix

$D_{ij} = \|x_i-x_j\|_2.$

Translation and rotation invariance are built in because pairwise distances do not change under those transformations. Scale invariance is enforced by normalizing with the Frobenius norm,

$d$0

using the fact that if $d$1, then $d$2. The learned descriptor therefore discards absolute size by construction.

The remaining nuisance factor is point indexing. For a closed curve sampled from a different starting point or traversal direction, the EDM changes by a cyclic reindexing and possible reversal. ShapeEmbed addresses this with a VAE architecture whose encoder uses circular padding, mirrored-input processing, and an indexation-invariant reconstruction loss,

$d$3

The resulting latent code is invariant to translation, scaling, rotation, reflection, and contour indexing. Crucially, the paper is explicit that discarding size is a design choice rather than a limitation: it recommends storing the pre-normalization matrix norm and concatenating it as an extra scalar feature when downstream tasks require both shape and size, a variant called ShapeEmbed+Sz. This is one of the clearest formal realizations of a shape-to-scale decomposition: intrinsic geometry is encoded in the latent descriptor, while size is restored as a separate channel only when needed.

The same conceptual distinction appears in Bayesian alignment under similarity transformations. "Bayesian alignment of similarity shapes" (Mardia et al., 2013) extends the unlabeled Bayesian point-configuration model of Green and Mardia to include rotation, translation, and scaling. The paper emphasizes the classical distinction between form analysis, which preserves size, and shape analysis, which removes scale together with rotation and translation. Its central technical point is that adding a scale parameter is not a trivial appendage to the rigid model: the likelihood must be reformulated in a symmetric latent-space way so that one configuration is shrunk by $d$4, the other expanded by $d$5, and both align to latent $d$6-space. This yields a posterior kernel containing both a Procrustes-like quadratic term and a nontrivial Jacobian factor,

$d$7

The paper further develops a two-scale model in which different point groups may have different scale factors $d$8 and $d$9, motivated by protein secondary structure elements that share an overall fold but differ in local length ratios. In this setting, scale is neither ignored nor fused into a single nuisance variable; it is inferred as a structured relational parameter.

4. Shape versus shape-and-scale as quotient spaces

A third, more formal meaning is provided by relational shape theory. "Topological Shape Theory" (Anderson, 2018) treats shape alone and shape-and-scale as distinct quotient constructions. For constellations of $(r,g,b)$0 points in a carrier space $(r,g,b)$1, the unreduced configuration space is

$(r,g,b)$2

If the continuous automorphism group includes scaling, the quotient yields pure shape space; if scaling is not quotiented out, the result is shape-and-scale space. In ordinary Euclidean geometry this becomes the distinction between quotient by $(r,g,b)$3 and quotient by $(r,g,b)$4. The paper states the cone relation explicitly: $(r,g,b)$5 Thus, in the Euclidean setting, shape-and-scale space is the cone over shape space, with the radial variable playing the role of overall scale. In the topological coarse-grained version, the same relation becomes graph-theoretic: adjoining the maximal coincidence-or-collision $(r,g,b)$6 turns the graph $(r,g,b)$7 of topological shape classes into the cone graph $(r,g,b)$8. The paper formulates this as Lemma 1: if $(r,g,b)$9 is a topological shape theory’s shape space, then the cone graph $i$0 is the corresponding topological shape-and-scale theory’s shape-and-scale space.

"Shape Theories. II. Compactness Selection Principles" (Anderson, 2018) sharpens the same distinction by asking when these quotient spaces remain analytically tractable. In Kendall’s similarity-shape setting, the shape space can be written as

$i$1

while the Euclidean shape-and-scale partner satisfies

$i$2

The paper argues that the good behavior of Kendall-type theories depends on compactness and properness: quotients of manifolds by compact Lie groups are Hausdorff, locally compact, and second-countable stratified spaces. Outside that protected class, especially in affine and projective shape theories, the quotient may be merely Kolmogorov-separated. The consequence for shape-to-scale is that the passage from shape to shape-and-scale is mathematically clean in some theories, but not universal across generalized quotient geometries.

"Shape Derivatives" (Anderson, 2018) moves from quotient spaces to differential invariants. The paper places Shape Theory and Shape-and-Scale Theory under one relational framework and derives derivatives by Taylor-expanding finite invariants on minimal nontrivially relational units. In 1D, quotient choices lead to a sequence of derivatives: $i$3 The last expression is the Schwarzian derivative. This hierarchy is a direct formalization of the distinction between retaining scale and quotienting it out: ordinary derivatives arise when only translations are removed, logarithmic and ratio-type derivatives arise when dilations are also removed, and projective shape gives the Schwarzian.

5. Shape-induced scale spaces and multiscale geometric analysis

Another large cluster of work uses shape to define the relevant notion of scale rather than treating scale as an external parameter. "A Product Shape Congruity Measure via Entropy in Shape Scale Space" (Genctav et al., 2017) constructs a volumetric shape scale space by solving

$i$4

inside a 3D volume $i$5 with boundary $i$6. The parameter $i$7 controls smoothness, and the field is sampled on loci at fixed inward distances from the boundary,

$i$8

Entropy of the sampled values,

$i$9

serves as a congruity measure. The sphere is the entropy-minimizing prototype because for a sphere the equidistant loci $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$0 coincide with level sets of $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$1, making the sample distribution degenerate. Deformation away from the sphere raises entropy, but repeated and congruent parts may lower it by producing repeated field values. Scale is therefore built into the field family $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$2 and the offset parameter $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$3, not appended afterward.

"Coarse-to-Fine Segmentation With Shape-Tailored Scale Spaces" (Sundaramoorthi et al., 2016) makes the same principle operational in segmentation. For each region $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$4, the method defines a scale space by the Heat Equation with no-flux boundary conditions,

$$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$5

The segmentation energy integrates region-fitting error over a continuum of diffusion times,

$$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$6

Because diffusion is confined to the region, coarse and fine structure are determined by the region’s own geometry rather than by global image smoothing. The paper’s central claim is that this creates preference for coarse structure without smoothing across boundaries. A practical contribution is that the integrated scale-space effect can be approximated by solving time-independent elliptic equations at the native image scale rather than the full time-dependent PDE family.

"Disconnected Skeleton: Shape at its Absolute Scale" (Aslan et al., 2011) uses scale in yet another sense: a 2D shape should be described at its own absolute scale, meaning the maximum allowable regularization that still permits morphological analysis. Instead of retaining a multiscale skeleton hierarchy, the paper diffuses a shape to a single, shape-dependent coarse scale and extracts only the stable local symmetry branches that remain. Under strong regularization, the skeleton becomes disconnected because branches terminate as soon as the evolving curve locally becomes circular. Disconnectedness is thus the signature of heavy regularization and stable part extraction, not a defect.

Two-scale coupling also appears in physically based matching. "A Hyperelastic Two-Scale Optimization Model for Shape Matching" (Simon et al., 2015) matches 3D surface meshes by combining a fine surface scale, where ICP-like correspondence acts, with a coarse volumetric scale, where deformation cost is measured through nonlinear elasticity on a tetrahedralized body. The target surface geometry may contain $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$7 to $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$8 triangles, while the volumetric mesh typically uses $$\text{SDF}(x) = \text{SDF}_1 + \frac{x - x_1}{x_2 - x_1} (\text{SDF}_2 - \text{SDF}_1),$$9–$$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$0 tetrahedra. The coarse-scale objective is an $$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$1-type boundary-force cost, the fine-scale term is a spring penalty toward current correspondences, and the local optimization is reduced to a second-order cone program after Schur-complement condensation. Here scale separation is not merely computational; it encodes the claim that geometric matching should be regularized by coarse physically meaningful deformation mechanics.

6. Operational uses in detection, texture, SAR, and clustering

In object detection, "Shape-IoU: More Accurate Metric considering Bounding Box Shape and Scale" (Zhang et al., 2023) uses shape to mean the width–height relation of the ground-truth box and scale to mean box size. The paper argues that equal center offsets or width/height deviations can have different IoU effects depending on whether the object is elongated or near-square and whether it is small-scale or large-scale. Its solution is a loss with shape-dependent directional weights,

$$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$2

which modulate both center-distance and size-mismatch terms. The method is therefore a joint shape–scale-aware localization loss, not a pure shape-to-scale predictor.

In single-image geometry, "Shape from Texture using Locally Scaled Point Processes" (Didden et al., 2013) treats texture elements as a latent point process on a plane and models perspective projection as a local scaling field over the image domain. The image intensity of the induced point process is written as

$$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$3

and for a planar surface under perspective projection the final intensity model becomes

$$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$4

Here shape is the plane orientation, scale is the local density/spacing distortion, and inference proceeds by estimating the scaling parameters $$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$5 from the latent point pattern. The method relies more on local density scaling and spacing scaling than on explicit texel-size measurements.

In distributed-scatterer InSAR, "Shape-to-Scale InSAR Adaptive Filtering and Phase Linking under Complex Elliptical Models" (Yao et al., 16 Sep 2025) gives the term a specialized statistical meaning. The paper separates shape, defined as the scale-invariant angular structure of a multitemporal complex scattering vector, from scale, defined as its radial or magnitude behavior. The angular part follows the complex angular central Gaussian model,

$$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$6

with shape matrix identifiable only up to a positive scale factor. Homogeneous pixels are selected by an angular-consistency adaptive filter using Tyler’s $$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$7-estimator, and phase linking is then performed under a complex generalized Gaussian model whose shape parameter $$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$8 captures non-Rayleigh amplitude fluctuations. The framework is explicitly described as centered around scale-invariant second-order statistics.

In cluster analysis, "Shape complexity in cluster analysis" (Aguilar et al., 2022) turns shape into feature scaling factors before $$\mathbf{C}(x) = \mathbf{C}_1 + \frac{x - x_1}{x_2 - x_1} (\mathbf{C}_2 - \mathbf{C}_1),$$9-means. After ordinary standard-deviation normalization, the paper introduces additional positive multipliers $$x_b = \text{argmin}_x \left( |\text{SDF}(x)| \; \text{where } p(x) \neq p(x+\delta x) \right).$$0 and defines pairwise distances

$$x_b = \text{argmin}_x \left( |\text{SDF}(x)| \; \text{where } p(x) \neq p(x+\delta x) \right).$$1

together with the shape complexity

$$x_b = \text{argmin}_x \left( |\text{SDF}(x)| \; \text{where } p(x) \neq p(x+\delta x) \right).$$2

Because one factor emphasizes large distances and the other small distances, the authors interpret stationary points under

$$x_b = \text{argmin}_x \left( |\text{SDF}(x)| \; \text{where } p(x) \neq p(x+\delta x) \right).$$3

as candidate scalings that sharpen what they call midrange distances. The method is explicitly presented as a route from the multidimensional shape of the data cloud to per-feature scale factors.

7. Limitations, misconceptions, and open problems

A recurring misconception is that shape-to-scale always means “infer size from shape.” The literature does not support such a restriction. Some methods remove size entirely and only reintroduce it as an auxiliary scalar; some preserve size as a relational variable; some use shape to define a scale space; some use scale to edit shape; and some couple both into a loss or a statistical model. The shared structure is the controlled handling of invariance versus retained degrees of freedom, not a single predictive map.

Several limitations recur. In editable 3D implicit geometry, the $$x_b = \text{argmin}_x \left( |\text{SDF}(x)| \; \text{where } p(x) \neq p(x+\delta x) \right).$$4 voxelized CSDF trades fidelity against memory, convex decomposition is geometric rather than semantic, and repetition works best for modular rather than organic forms (Vermandere et al., 13 Feb 2026). In contour representation learning, the method is restricted to simply connected 2D contours, scale is intentionally normalized out unless the saved norm is appended, and the EDM grows quadratically with the number of sampled contour points (Romero et al., 1 Jul 2025). In quotient-space theory, the clean cone relation between shape and shape-and-scale holds in the Euclidean/similarity setting but not universally; many generalized theories, especially affine and projective ones, fall outside compactness protections and may be merely Kolmogorov-separated (Anderson, 2018). In entropy-based shape scale spaces, evidence is qualitative and preliminary rather than statistically validated against human judgments or alternative descriptors (Genctav et al., 2017). In DS-InSAR, the angular-consistency filter requires iterative estimation and Monte Carlo bootstrap thresholding, so the statistical separation of shape and scale carries real computational cost (Yao et al., 16 Sep 2025). In clustering, the optimization is nonconvex, computationally expensive, and best interpreted as heuristic candidate generation rather than a blind automatic procedure (Aguilar et al., 2022).

A broader open problem follows from these limitations. The literature suggests no single universal passage from shape to scale across domains. In some settings, such as Kendall-type relational geometry, the distinction is formal and exact; in others, such as implicit 3D editing or DS-InSAR, it is algorithmic and application-specific; in still others, such as segmentation or multiscale congruity, scale is induced by PDE evolution on the object or region itself. What unifies these lines of work is the insistence that shape should not be treated independently of the way scale is removed, retained, edited, or operationalized.