# GD4Shapes: Geodesic distance with fixed parameterization for 2D Shapes

S. Herold-Garcia, H.L. Varona-Gonzalez, X. Gual-Arnau

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S. Herold-Garcia, H.L. Varona-Gonzalez, X. Gual-Arnau. GD4Shapes: Geodesic distance with fixed parameterization for 2D Shapes. Software Impacts, 2025, 25, pp.100775. 10.1016/j.simpa.2025.100775. hal-05354012

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# GD4Shapes: Geodesic distance with fixed parameterization for 2D Shapes

S. Herold-Garcia<sup>a, *, f</sup>, H.L. Varona-Gonzalez<sup>a, b</sup>, X. Gual-Arnau<sup>c</sup>

## ABSTRACT

Keywords:

Shapespacedistance

Template matching

Fixed parameterization

Geodesicdistance

Shape analysis within shape space provides a robust framework for examining geometric properties of objects, enabling comparisons invariant to translation, rotation, and scaling. A key task is computing geodesic distances between shapes, which quantify similarity but are computationally intensive due to the need for exhaustive parameterization searches. Recent advancements propose heuristic methods to simplify these computations, such as fixing parameterizations based on the major axis of shapes, significantly reducing computational costs while maintaining high accuracy (e.g., 96.03% in erythrocyte classification). This article introduces a software tool that leverages this heuristic to efficiently compute shape-spaced distances, aligning shapes considering their major axis, and using templates like circles and ellipses. The tool accelerates morphological analysis, making it ideal for high performance applications in fields like biology and medicine. By streamlining the computation of geodesic distances between shapes and enabling rapid retrieval of information, this software improves research workflows and supports the study of shape-dependent features in diverse fields from cellular morphology to diagnostic hematology.

Code metadata

Current code version

Permanent link to code/repository used for this code version

Permanent link to reproducible capsule

Legal code license

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Software code languages, tools and services used

Compilation requirements, operating environments and dependencies

If available, link to developer documentation/manual

Support email for questions

v2.0

https://github.com/SoftwareImpacts/SIMPAC-2025-46

MIT License

Zenodo

MATLAB

MATLAB 2021b

https://zenodo.org/records/14503529/files/GD4Shapes_v2.0_UserManual_2024.pdf?

download=1

silena.herold@gmail.com

## 1. Introduction

Shape analysis within the shape space provides a robust framework for examining the geometric properties of objects in images, such as contours, dimensions, and spatial relationships. This space refers to a mathematical space where shapes are compared after normalizing for translation, rotation, and scaling, and involves representing shapes in a way that is invariant to these transformations, enabling comparison based on intrinsic geometry. Elastic analysis introduces flexibility by

considering deformations that preserve the overall structure, using metrics like the square root velocity (SRV) representation. The discipline remains highly active in the scientific community, characterized by recent advanced contributions from leading researchers for both 2D and 3D environments [1–9].

One of the core tasks in this field is determining the geodesic distance between two shapes to provide a quantitative measure of their similarity or difference, critical for classification and comparison. This process explores all possible parameterizations, searching for the

<footnote><sup>a</sup> Center for Risk Analysis and Environmental Modeling (CEERMA), Federal University of Pernambuco, Recife, PE, Brazil</footnote>

<footnote><sup>b</sup> Laboratory of Physical, Coastal and Estuarine Oceanography (LOFEC), Oceanography Department (DOCEAN), Federal University of Pernambuco, Recife, PE, Brazil</footnote>

<footnote><sup>c</sup> Department of Mathematics, Institute of New Imaging Technologies (INIT), University Jaume I, Castelló, Spain</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: silena.herold@gmail.com (S. Herold-Garcia), humberto.varona@ufpe.br (H.L. Varona-Gonzalez), gual@uji.es (X. Gual-Arnau).</footnote>

<footnote>https://doi.org/10.1016/j.simpa.2025.100775</footnote>

<footnote>Received 5 March 2025; Received in revised form 11 May 2025; Accepted 27 May 2025</footnote>
optimal rotation and reparameterization of a curve to best match another [1], selecting the smallest distance as the geodesic distance. While this method is accurate, it is computationally intensive and time-consuming, limiting its practicality for large-scale or time-sensitive applications.

Shape is a key predictor of functional and structural criteria in anatomical and biochemical systems, particularly within complex bio-systems[10]. Shapeanalysis, particularly via the shape space approach, is extensively used in image analysis to detect patterns, segment structures, and quantify deformations, making it essential in medicine, biology, and computational geometry. Recent applications highlight its versatility across disciplines, including studies on Glioblastoma Multiforme tumors [1,2,11], leaf outlines [1], carotid arteries [1], Diffusion Tensor MRI of fiber and white matter tracts [1,11], vertebrae in mice [1,2], electrocardiogram signals [11], endometrial tissue surfaces [11,12], subcortical brain surfaces [11], and plant roots [13]. Additionally, its use extends to anthropology and other image analysis domains, demonstrating broad applicability [14].

This framework has been applied to cellular contour analysis, enabling the mathematical modeling of cell boundaries as parameterized 2D closed curves. Researchers use it to separate cell shape dynamics from kinematic factors, aiding parameter estimation, diagnostics, synthetic shape generation, and predictive modeling of morphology evolution [15]. It also quantifies erythrocyte anomalies in blood smears, enhancing the diagnosis and prognosis of hematological diseases such as thrombotic thrombocytopenic purpura [16], hemolytic uremic syndrome [16], and sickle cell disease [17,18]. An innovative approach considers the specific geometry of healthy red blood cells (circular) and sickle cells (sickle-shaped) to establish a unique parameterization based on each cell's major axis [19]. Distances are then computed using this fixed parameterization, simplifying computational methods by avoiding the need to consider all possible parameterizations. Furthermore, each cell is aligned to two templates (a circle and an ellipse), and classification is performed by measuring the shape distances using the fixed parameterization. This approach significantly reduces computational costs for shape distance calculations and has been successfully applied to erythrocyte morphological classification, achieving a 96.03% accuracy rate in both supervised classification and unsupervised clustering tasks.

This article highlights the importance of a software that applies this approach to compute shape-spaced distances by fixing a parameterization based on each shape's major axis. This approach accelerates results while maintaining the performance of traditional methods. By integrating this tool into research workflows, it enhances efficiency in morphological analysis, particularly for objects that can be aligned with reference shapes (e.g., circles and ellipses). Its ability to generate geometrically meaningful distances streamlines shape classification and improves result interpretability, making it invaluable for high-throughput studies in shape-dependent biological and structural research.

## 2. Software details

GD4Shapes is a MATLAB 2021a package designed to offers a streamlined approach to computing geodesic distances between two closed plane shapes in shape space. It provides an integrated set of functions, such as: configure the type of shape to obtain the distance, and the type of alignment for the shape to be analyzed; to load the shape to be analyzed and to compute the geodesic distance between the shapes. Furthermore, it allows to visualize the original shapes, the aligned shapes, the computed distance, and the geodesic paths obtained, as well as to export the graphs of interest as needed. In Fig. 1 it is shown the steps to quickly load and analyze two shapes with GD4Shapes.

The operation mode of this MATLAB package is very simple. It starts by defining the basic shape to obtain the distance between the two shapesto be analyzed: (1) contour, for obtaining the distance between

two shapes;(2) circle, for obtaining the distance between a shape and a circular template; and (3) ellipse, for obtaining the distance between a shape and an elliptical template. The alignment selector button allows defining the type of alignment to be applied to the shapes before computing the distance. From here, in every case, it is possible to load the shape to be analyzed. In the first case, two shapes can be loaded. In the second and third cases, only the second shape can be loaded, as the first is already defined as a template. Both shapes are displayed in the same graph. After loading the shapes, the geodesic distance can be computed using the corresponding button. The number of points to parameterize each shape must also be specified. The software visualizes the centered shapes, with their centroids at the origin of the axes, as well as the aligned shapes, considering the type of alignment defined with the selector button to fix parameterization. Additionally, the geodesic path between the curves is displayed. Finally, the geodesic distance is shown. At this point, all graphs are customized for subsequent export as an image with a resolution of 500 DPI (Fig. 1).

## 3. Software algorithms

Basically, this package computes geodesic distances between two closed plane shapes in shape space, using the SRV representation of closed curves in R-squared and the standard elastic metric. This metric quantifies the “cost” of deforming one shape into another by differently penalizing stretching (γ/γ₀t) and bending (γ/γ₀θ) [20]. To compute the distance between two shapes, the optimal reparameterization function is obtained, which corresponds to an optimal alignment of the shapes. This alignment minimizes deviations in stretching and bending terms [20], and yields the smallest distance value. For shapes with dominant orientation features, such as major and minor axis structures, optimal alignment typically coincides with matching their principal axes (e.g., major axes), as this reduces deviations in the stretching term of the metric.

The proposed method leverages these principles and exploits the specific geometry of sickle-shaped cells, which have sickle or elongated shapes, to perform a previous alignment along their major axis. Subsequently, a parameterization is established that optimally incorporates this alignment, enabling the computation of a distance value close to the minimum geodesic distance. This eliminates the need for exhaustive iterations, significantly reducing computation time while still providing a result that closely approximates the true geodesic distance [19].

For the two shapes to be analyzed, three options are available for the first shape: a curve, a circular template, or an elliptical template. The second shape can be any closed curve. The fixed parameterization methodology ensures high accuracy, making the software an effective tool for shape classification tasks. However, its use for computing distances using the provided geometric templates requires consideration of the similarity between the studied shapes and these templates, as efficiency may decline for highly dissimilar cases (e.g., contours with significant non-convexity).

## 4. Related works

In [19], this software was used to compute distances between erythrocytes of three classes: normal cells, sickle cells, and cells with other deformations. Fig. 2 demonstrates distance calculations between: (a) two sickle cells; (b) a circle template and a normal cell; (c) an ellipse template and a sickle cell. These distances were applied in the supervised classification of erythrocytes, which is crucial for diagnosing and treating sickle cell anemia crises, helping assess their severity. This method reduced computation time from big-O(Nk²) to big-O(k), where N is the number of contour points and k is the number of cells analyzed. The results show that the method is highly effective for classification. The F1 score, which combines precision and recall into a single metric, was used to validate the results. A higher F1 score, closer to 1, indicates better classification performance. In this study,
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Steps to quickly load and analyzes two shapes with GD4Shapes.</figcaption></figure>

the Normal class achieved an F1 score of 99.75%, the Sickle Cell class scored 93.86%, and the Other Deformations class reached 94.01%.

To visualize the computational cost reduction, **Table 1** shows the runtime in seconds for both approaches in an example of calculating distances between erythrocytes from the three classes analyzed in [19]. The hardware used was an Intel(R) Core(TM) i7-8750H CPU @ 2.20 GHz 2.21 GHz, 16 GB RAM, Windows 11. The contour resolution was 295 points, and all possible parameterizations were considered, with $N = 295$. Code parallelization was not employed to obtain the results. It can be observed that the original approach required a runtime of ~37 s to compute the distance between two cells, as the iterative process inherently depends on the number of parameterizations analyzed. Reducing the computations to 100 and 50 parameterizations decreased the runtime to ~12.5 s and ~7.5 s, respectively. However, with the proposed method, the runtime is reduced to ~0.15 s, as $N = 1$ eliminates this iterative process. Furthermore, in supervised shape classification using the original approach, the distance between every pair of analyzed cells must be computed, causing the runtime to scale quadratically ($k^2$, where $k$ is the number of cells). This scaling becomes linear when employing the distance to geometric templates, as only two distances need to be computed for each cell. In [19], an analysis of variance verified that the distances obtained using this method for the supervised classification of erythrocytes are statistically consistent, confirming the method's validity in this case. These results make the approach suitable for clinical applications in sickle cell disease treatment and feasible for use in resource-limited settings or real-time analysis.

## 5. Impact overview

This MATLAB package calculates the geodesic distance between two plane curves using a novel methodology that reduces computational time. It advances shape analysis, particularly in fields where shapes are

<table><caption>**Table 1** Runtime (in seconds) for computing the distances between cells using both approaches.</caption><thead><tr><th></th><th>Normal</th><th>Sickle</th><th>Other deform</th><th>Circle template</th><th>Ellipse template</th></tr></thead><tbody><tr><th colspan="6">Original approach</th></tr><tr><th>Normal</th><td>36.88</td><td>36.74</td><td>36.94</td><td>37.01</td><td>36.73</td></tr><tr><th>Sickle</th><td></td><td>36.88</td><td>36.83</td><td>37.64</td><td>37.04</td></tr><tr><th>Other Deform</th><td></td><td></td><td>37.10</td><td>36.73</td><td>36.78</td></tr><tr><th colspan="6">Proposed approach</th></tr><tr><th>Normal</th><td>0.15</td><td>0.15</td><td>0.14</td><td>0.15</td><td>0.15</td></tr><tr><th>Sickle</th><td></td><td>0.14</td><td>0.15</td><td>0.15</td><td>0.15</td></tr><tr><th>Other Deform</th><td></td><td></td><td>0.15</td><td>0.14</td><td>0.15</td></tr></tbody></table>

studied with geometric templates like circles and ellipses. For instance, in [19], the software was applied to classify erythrocytes in blood samples from sickle cell anemia patients, achieving over 96% accuracy in distinguishing normal, sickle-shaped, and cells with other morphological abnormalities. Such precision underscores its potential to enhance the modeling and morphological classification of red blood cells, crucial for studying pathologies like sickle cell disease [17–19]. Sickle cell disease is a growing global health issue, with over 500,000 affected babies born annually, mainly in sub-Saharan Africa. It is a leading cause of death in children and young adults, highlighting the urgent need for better screening, registries, and treatments [21,22]. By integrating these computational models into automated diagnostic tools, medical professionals can accurately evaluate morphological changes associated with such pathologies, improving patient care and quality of life. The reduced computational cost also makes the method feasible for use in resource-limited settings or real-time analysis systems.

Furthermore, red blood cell morphology has been extensively studied in the shape feature space, with recent methods achieving high
<figure><img src="image_3.png" /><figcaption>Geodesic curves</figcaption></figure>

<figure><img src="image_4.png" /></figure>

d=0.25165

<figure><img src="image_5.png" /><figcaption>Geodesic curves</figcaption></figure>

<figure><img src="image_6.png" /></figure>

d=0.12181

<figure><img src="image_7.png" /><figcaption>Geodesic curves</figcaption></figure>

<figure><img src="image_8.png" /></figure>

d=0.18746

<caption>**Fig. 2.** Examples of geodesic distances: (a) Two sickle cell; (b) Circle template and a normal cell; (c) Ellipse template and a sickle cell.</caption>

performance [23–25]. However, analyzing the cell deformation process in this space is not straightforward. This software provides a foundation for studying morphological changes by treating cell contours as deformable closed planar curves. This approach will enable a more targeted analysis of erythrocyte deformation in patients with conditions such as sickle cell disease.

This method has the potential to be extended to other geometric shape analysis scenarios, especially when objects share similarities with the circle and ellipse templates used for distance calculation. Such as, in [19] the software was applied to analyze objects from the Flavia plant leaf dataset, a pioneering resource for leaf classification. The results demonstrated that the shape-based distances between objects of the same class are consistently smaller than those between objects of different classes. Furthermore, it could be evaluated for use in other biological contexts, such as the morphological analysis of

Human Umbilical Vein Endothelial Cells during angiogenesis [26,27], sperm morphology analysis in fertility studies, where abnormal shapes (elongated, rounded, or irregular) can be classified [28], and the evaluation of breast tumor where there are some defined shape categories (spherical, segmental, discoidal, and irregular) [29–31]. Additionally, its use could be evaluated in non-biological fields, such as the study of granulometric parameters in sediments, which is vital for understanding sediment transport and deposition processes[32,33] or certain characteristics of soils [34,35].

Finally, this software supports education and collaborative research in shape analysis by visualizing deformations and computing geodesics, enhancing the understanding of this non-Euclidean space. Its open-source nature promotes interdisciplinary collaboration, fostering innovation and methodological exchange across fields. As a result, it becomes a versatile resource with applications ranging from academic
research to automated tools that enhance primary healthcare for high-
incidence pathologies. Its integration into public health support systems
can help shape policies and strategies to address the burden of diseases
like sickle cell disease[21,22].

6. Future development of the software

The current version of GD4Shapes includes only circle and ellipse
templates. Future versions will explore adding more templates for an-
alyzing different shapes. Additionally, the ability to compute distances
between a shape and a set of shapes will be incorporated, providing an
overview of the first shape's similarity to the others studied.

7. Final remarks

GD4Shapes is a fast and efficient tool for calculating geodesic dis-
tances between two closed planar shapes. It uses an innovative method
that considers a shape's geometry to fix a parameterization based on its
major axis, handling cases where one shape may be a cell contour or a
template circle or ellipse.

CRediT authorship contribution statement

S. Herold-Garcia: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Formal analysis, Conceptualization. H.L. Varona-Gonzalez: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Conceptualization. X. Gual-Arnau: Conceptualization, Methodology, Supervision, Funding acquisition, Writing – original draft, Writing – review & editing.

Declaration of GenerativeAI and Al-assisted technologies
in the writing process

During the preparation of this work, the author(s) used ChatGPT in
order to check the correctness of the English in some paragraphs. After
using this tool/service, the authors reviewed and edited the content as
needed and take full responsibility for the content of the publication.

Declaration of competing interest

The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Acknowledgments

S.H.G acknowledges Territorial Program for the Development of Health Products and Services, CITMA, Santiago de Cuba (Project PT241SC003-006 Improving Diagnostic Capabilities for Clinical Monitoring of Patients with Sickle Cell Anemia in Santiago De Cuba).

H.L.V.G. thanks the Center for Risk Analysis and Environmental Modeling (CEERMA) and the Laboratory of Physical, Coastal and Estuarine Oceanography (LOFEC). Oceanography Department (DOCEAN), Federal University of Pernambuco.

X.G. A received support from the Spanish Ministry of Science and In-
novation, Spain through projects PID2020-115930GA-100 and PID2022-
141699NB-I00.

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