Morphing Between Monotonic Spinner Planar Curves Through

Radial-Sign Descriptors

Emna Ghorbel

1,2 a

and Faouzi Ghorbel

CRISTAL Laboratory, GRIFT Research Group ENSI, La Manouba University 2010, La Manouba, Tunisia

Medtech, South Mediterranean University, Tunis, Tunisia

Keywords:

Radial-Sign Descriptor, Shape Morphing, Monotonic Spinner Shape, Shape Generation.

Abstract:

This paper introduces an innovative morphing method leveraging Radial-Sign descriptors for monotonic spin-

ner shapes, offering a robust, efﬁcient, and computationally reﬁned solution to shape blending challenges.

The method encodes two shapes using radial distances and angular sign variations relative to the centroids,

respectively, producing complete, stable, and invertible descriptions. By applying weighted interpolation di-

rectly to these two descriptors and reconstructing in-between shapes through an inverse formula, the approach

ensures smooth, morphologically coherent transitions while preserving essential geometric properties. Unlike

conventional curvature or registration-based techniques, which often require intensive post-processing or face

limitations with signiﬁcantly different shapes, the proposed method adeptly blends both similar and dissimilar

shapes, including those with differing turning number, by introducing additional turns in simpler shapes to

ensure continuity and coherence.

1 INTRODUCTION

Morphing, or blending, is a powerful technique

widely utilized in computer vision and graphics, and

it has gained substantial importance in machine learn-

ing for its ability to generate new data and, then,

augment existing datasets. This process involves the

smooth transformation of one shape or image into an-

other. However, its effective implementation demands

addressing several challenges. The correspondences

between points on source and target shapes is criti-

cal yet challenging, especially in the presence of non-

rigid deformations. Misaligned correspondences can

lead to undesirable artifacts such as unnatural defor-

mations.

In this work, we focus on planar shape Morph-

ing. The process of planar shape morphing is com-

monly divided into two fundamental steps: the ver-

tex correspondence problem and the vertex path prob-

lem (Saba et al., 2014). The vertex correspondence

problem involves establishing a mapping between

points, vertices, or features on the source and target

shapes, ensuring that meaningful and consistent re-

lationships are preserved throughout the transforma-

tion process. This step is crucial for handling vari-

ations in topology, geometry, or complexity between

https://orcid.org/0000-0002-6179-1358

the shapes. Once correspondences are established, the

vertex path problem focuses on determining the in-

termediate shapes that interpolate between the source

and target. This phase ensures a smooth and visually

coherent transition along the morphing trajectory.

Traditional shape blending techniques, such as

polygon interpolation (Alexa et al., 2000; Shapira and

Rappoport, 1995), often struggle with computational

challenges, particularly regarding triangle compatibil-

ity in non-smooth shapes. These methods can result

in discontinuities or require human-interaction.

Methods utilizing curvature descriptors (Surazh-

sky and Elber, 2002; Hirano et al., 2017; Saba et al.,

2014; Sederberg et al., 1993a) are widely adopted

and demonstrate strong performance. Thanks to arc-

length parameterization, the correspondence step is

simpliﬁed to aligning the starting points of the source

and target shapes. However, these approaches often

result in intermediate shapes that are not closed, re-

quiring additional post-processing steps, such as B-

splines, to achieve curve closure, which can be rela-

tively computationally intensive (Hirano et al., 2017;

Saba et al., 2014; Th

evenaz et al., 2000).

As another common approach, Registration-based

methods align shapes through non-linear optimiza-

tion for correspondence purposes (Sebastian et al.,

2003; Srivastava et al., 2010; Klassen et al., 2004;

410

Ghorbel, E. and Ghorbel, F.

Morphing Between Monotonic Spinner Planar Curves Through Radial-Sign Descriptors.

DOI: 10.5220/0013335400003905

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 14th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2025), pages 410-417

ISBN: 978-989-758-730-6; ISSN: 2184-4313

Jin et al., 2021). These methods exhibit highly log-

ical and smoothly continuous deformations. How-

ever, it often incur signiﬁcant computational overhead

and may fail to guarantee smooth transitions when the

source and target shapes are signiﬁcantly different.

In the other hand, Fourier-based methods offer

a relatively fast alternative, inherently producing in-

termediate closed curves due to their periodic nature

while being invariant to starting point(Ghorbel et al.,

2022; Ghorbel et al., 2021). However, these methods

are constrained to contours limiting their applicability

to open curves.

As an efﬁcient and straightforward solution, we

propose a novel blending approach leveraging the

Radial-Sign descriptor (Ghorbel and Burdin, 1994;

Burdin et al., 1992), a complete, stable, and invertible

shape descriptor, designed for the morphing of Mono-

tonic Spinner planar curves. A Monotonic Spinner

planar curve is deﬁned as a curve where the radial dis-

tance from the center of rotation varies monotonically

with the angle. Similarly to curvature-based meth-

ods, the proposed technique requires only a starting

point alignment to establish correspondence between

shapes and generates deformations that are invari-

ant under rigid transformations (translations, scales

and rotations). However, unlike curvature-based ap-

proaches, the proposed method maintain a morpho-

logically coherent transition, preserving the natural

structure and ﬂow of the shapes throughout the pro-

cess. Moreover, the method effectively blend both

similar and dissimilar shapes, including those with

differing turning number. Indeed, it sufﬁces to in-

troduce additional turns in simpler shapes, ensuring

continuity and coherence in the resulting transition.

Therefore, it allows the interpolation between shapes

with distinct topologies.

In the following, Section 2 recalls the Radial-

Sign Descriptor and introduce the Blending approach.

Section 3 concerns the experiments where Morphing

quality, Morphing invariance, comparison with cur-

vature methods, and application on the MPEG-7 CE

Benchmark are presented. Finally, Section 4 con-

cludes this work.

2 PROPOSED MORPHING

APPROACH

In this section, we begin by recalling the Radial-Sign

descriptors introduced in (Ghorbel and Burdin, 1994;

Burdin et al., 1992), emphasizing their properties that

are essential for shape morphing. Therefore, we pro-

pose a novel morphing method that uses Radial-Sign

descriptors to achieve smooth and coherent transitions

between monotonic spinner contours.

2.1 Radial-Sign Descriptor

Let R = (O,i, j) be a Cartesian coordinate system in

the Euclidean plane, and let s denote a normalized arc

length. Consider γ(s), a monotonic spinner curve with

respect to its centroid O, parameterized by its normal-

ized arc length. In polar coordinates, this parameteri-

zation can be expressed as:

γ(s) =



ρ(s)cos(θ(s)),ρ(s)sin(θ(s))



where ρ(s) is the radial function and θ(s) is the polar

angle. Differentiating this representation with respect

to s and taking the norm of both sides of the resulting

equation, we use the fact that ∥γ

′

(s)∥

= 1 which leads

to the following relation,

′

(s)

1 − ρ

′

(s)

ρ(s)

The solution to this differential equation is given by,

θ(s) = θ(0) +

σ(l)

1 − ρ

′

(l)

ρ(l)

dl, (1)

where σ(l) ∈ {−1,1} is the sign of θ

′

(l). Note that to

obtain invariance to scale, it sufﬁces to normalize the

radius function by dividing ρ(s) by the scale factor

scale

, deﬁned as the maximum radius:

scale

= max

ρ(s).

This normalization leads to the scale-invariant radial

function,

ρ(s) =

ρ(s)

scale

Figures 1 and 2 show respectively a 1-turning and 4-

turning monotonic spinner curves with respect to their

centroids and their corresponding polar coordinates.

Figure 3 illustrates the radial-sign computation frame-

work. Starting from Cartesian coordinates, the trans-

formation to polar coordinates is achieved by calculat-

ing the radial function ρ(s) and the polar angle θ(s).

Subsequently, the sign σ(s) of the derivative of the

polar angle θ(s) is computed.

Under the assumption of a normalized arc-length

parameterization, the angular function θ(s) can be re-

placed by the sign function of its derivative, σ(s),

without any loss of information. As a result, the set

of descriptors {(ρ(s),σ(s))} becomes a compact and

robust representation that is invariant to rigid transfor-

mations while maintaining the property of complete-

ness (Crimmins, 1982).

A complete descriptor ensures that two shapes are

identical if and only if their descriptors are equal. Be-

yond invariance and completeness, additional prop-

erties such as invertibility and stability are essential

Morphing Between Monotonic Spinner Planar Curves Through Radial-Sign Descriptors

411

for tasks such as morphing and shape reconstruction.

the invertibility guarantees that shapes can be recon-

structed from their descriptors through explicit for-

mulas, enabling the generation of intermediate curves

by interpolating between descriptors (Ghorbel et al.,

2022). The stability on the other hand, ensures that

small variations in the descriptor space result in cor-

respondingly small variations in the shape space, and

vice versa, making the descriptors robust to perturba-

tions (Ghorbel, 1992). Figures 4, 5 and 6 illustrate

these properties with practical examples.

For shape morphing, these properties are particu-

larly critical. Morphing requires descriptors that al-

low the reconstruction of shapes as continuous and

progressive deformations, without introducing abrupt

changes in shape. To achieve this, the focus is placed

on monotonic spinner curves, which exhibit a con-

stant sign for σ(s). This constancy ensures both the

invertibility and stability of the descriptors, facilitat-

ing smooth transitions between shapes during morph-

ing.

Figure 1: An example of a monotonic spinner with 1-

turning number (case. star-shaped) curve with polar coordi-

nates.

2.2 Radial-Sign Morphing

Since the Radial-Sign descriptor satisﬁes the four key

properties of Completeness, Invariance, Invertibility,

and Stability (CIIS) for monotonic spinner contours,

it is well-suited for performing morphing, as intro-

duced in (Ghorbel et al., 2022).

Let γ

and γ

be two monotonic spinner con-

tours parameterized by their normalized arc lengths s,

and described by their radial-sign functions f (γ

) =

Figure 2: An example of a monotonic spinner with 4-

turning number curve with polar coordinates. the red line

corresponds to the Radial measure.

Figure 3: The radial-sign descriptor framework. Starting

from Cartesian coordinates, the transformation to polar co-

ordinates is achieved by calculating the radial function ρ(s)

and the polar angle θ(s). Then, the sign σ(s) of the deriva-

tive of θ(s) is computed.

(ρ

(s),σ

(s)) and f (γ

) = (ρ

(s),σ

(s)), respec-

tively. The interpolation between the two contours

is performed in the space of the Radal-sign invariant

descriptors. For t ∈ [0,1], the interpolated radial func-

tion is:

= (1 −t)ρ

+tρ

While the sign function should be equal,

= σ

(s) = σ

(s)

Finally, the intermediate contours γ

are reconstructed

by computing the θ

angle from the σ

as follows,

(l)

1 − ρ

′

(l)

dl,

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

412

(a) (b)

Figure 4: Invertibility of the descriptors for star-shaped

curve (1-turning number). (a) Original curve (b) Recon-

structed curve.

(a) (b)

Figure 5: Invertibility of the descriptors for monotonic spin-

ner shape (4,5-turning number). (a) Original curve (b) Re-

constructed curve.

This method ensures a smooth and consistent mor-

phing process, leveraging the invariance and stability

of the Radial-Sign descriptors to generate meaningful

intermediate contours. Figure 7 presents two Radial-

Sign morphing examples between a pair of curves.

3 EXPERIMENTS

In this section, we report and analyze the experi-

mental results. First, we present some morphing

sequences in order to evaluate the quality visually.

Therefore, the robusteness of Radial-sign based de-

formations to rigid transformation is tested. Then, we

compare the proposed approach based on the Radial-

(a) (b)

Figure 6: Stability of the descriptors. (a) Radial representa-

tion for the original and perturbed curves. (b) Reconstruc-

tion of the original and the pturbed curves.

Figure 7: Two examples of Radial-sign based Morphing be-

tween a pair of curves.

sign descriptor to curvature-based methods with B-

splines (Surazhsky and Elber, 2002; Sederberg et al.,

1993b; Saba et al., 2014) according to turning number

challenge and some qualitative properties. Finally, we

provide qualitative results on the monotonic spinner

shapes from the MPEG-7 CE dataset (Latecki et al.,

2000).

3.1 Morphing Quality

In this part, we propose to evaluate the morphing

quality visually. Let’s begin by showing various

examples of the Radial-sign blending under various

conditions, including open, closed, and non-simple

curves. Figure 8 demonstrates the deformation be-

tween two open curves, where the intermediate shapes

preserve the continuity and smoothness of the curve

while gradually transitioning from the source curve

to to the target curve. Figure 9 showcases an exam-

ple of deformation between two shapes that exhibit

multiple n-turning points. This scenario highlights

the robustness of the morphing process in handling

shapes with intricate geometries. Finally, Figure 10

illustrates a deformation process between a simple

and a non-simple shape. By introducing additional

turns in the source shape to match those of the tar-

Morphing Between Monotonic Spinner Planar Curves Through Radial-Sign Descriptors

413

get, the method ensures both continuity and coher-

ence throughout the morphing sequence. Despite the

complexities introduced by the differences in topol-

ogy, the approach successfully preserves the underly-

ing structure of the shapes. This demonstrates the ver-

satility and robustness of the proposed method in han-

dling challenging geometries, enabling smooth and

morphologically consistent transitions even between

shapes with signiﬁcant dissimilarities.

Figure 8: Example of Deformation between two open

curves.

Figure 9: Example of Deformation between two n-turning

number shapes.

Figure 10: Example of Deformation between two a simple

shape and a complex one.

Figure 11: Invariance of the Radial-Sign deformation under

rigid transformations.

3.2 Invariance to Rigid

Transformations

To evaluate the robustness of the Radial-Sign morph-

ing approach under rigid transformations, we propose

to blend two monotonic spinner curves after the ap-

plication of different transformations. The aim is to

validate that the intermediate shapes remain invari-

ant when the input shapes are subject to scaling, rota-

tion, and translation. These transformations simulate

practical scenarios where shapes may vary in size, ori-

entation, or spatial location. In this speciﬁc test, the

source curve was manipulated by applying a rotation

of π/4, a scaling factor of a = 2, and a translation

by (1,1) in Cartesian coordinates. Figure 11 show-

cases the results of this experiment. The intermediate

shapes, regardless of the applied transformations, stay

the same. This performance is attributed to the invari-

ance properties of the Radial-Sign descriptors.

ICPRAM 2025 - 14th International Conference on Pattern Recognition Applications and Methods

414

3.3 Comparison with Curvature-Based

Methods

We compare our method to curvature-based ap-

proaches, focusing on three aspects: Turning number

challenge and qualitiative properties.

3.3.1 Turning Number Challenge

Blending two curves with different turning num-

bers presents additional challenges. Speciﬁcally, de-

termining the correct behavior of the interpolation

method in such cases remains complex. Methods

that aim to preserve the natural deformation of the

curves often involve intricate transformations, includ-

ing folding or unfolding of the curves. This behav-

ior complicates the testing and evaluation of our ap-

proach, as it seeks to maintain as much consistency

with the original curve’s geometry as possible, with-

out introducing unnatural deformations.

In our experiments, we blended a circle into an 8-

shaped curve. The results, shown in Figure 11, high-

light the unpredictable behavior observed across all

methods. Nevertheless, our proposed radial-sign ap-

proach mitigates issues associated with turning num-

bers and provides a more stable and intuitive inter-

polation process. The comparison demonstrates the

superior performance of our method, particularly in

terms of the quality of the intermediate shapes. The

shapes produced by our approach are smoother and

preserve their concavities throughout the transforma-

tion. In contrast, curvature-based methods often in-

troduce distortions or lose concavities during interpo-

lation, necessitating additional post-processing steps

such as B-splines to close the curves. In another hand,

our method naturally ensures closed curves without

requiring such steps.

3.3.2 Qualitative Properties

To further validate the Radial-Sign descriptor’s per-

formance, we compare qualitatively the morphing ap-

proaches according to several properties.

Table 1: Comparison Between the Curvature-Based and

Radial-Sign-Based Morphing Methods.

Curvature Radial-Sign

Invertibility Numerical Analytical

Postprocessing Yes No

Curve type C

Monotonic − Spinner

Complexity O(n

) O(n)

Invariance SE(2) SE(2)

Table 1 highlights several key differences between

the curvature-based and Radial-Sign morphing meth-

(a)

(b)

(c)

(d)

(e)

Figure 12: Comparison of the methods when interpolating

two curves with different turning numbers. In this case,

γ (left) has turning number 1 and γ

′

has turning number

0. (a) linear interpolation (b) curvature (Sederberg et al.,

1993b) (c) curvature (Surazhsky and Elber, 2002) (d) cur-

vature (Saba et al., 2014) (e) our method.

ods. First, the curvature method relies on numeri-

cal approaches for invertibility, whereas the Radial-

Sign method provides an analytical inversion, lead-

ing to more stable and precise results. Additionally,

curvature-based methods often require postprocessing

to close the curves, while the Radial-Sign approach

operates without this need. In terms of smoothness,

the curvature-based method is efﬁcient on C

curves

while monotonic spinner curves are needed for the

Radial-Sign method. In another hand, for the inter-

polation part, the Radial-Sign method has a linear

time complexity of O(n), compared to the quadratic

O(n

) complexity of curvature-based methods, mak-

ing it signiﬁcantly more efﬁcient. Finally, Radial-

Sign Morphing ensure the invariance under Euclidean

transformations as well as the curvature-based ap-

proaches.

3.4 Morphing on the MPEG-7 CE

Dataset

In this section, we propose extracting monotonic spin-

ner curves, speciﬁcally star-shaped ones, from the

MPEG-7 CE dataset (Latecki et al., 2000) and gen-

erating in-between curves belonging to the same cat-

egory (intra-class) and in-between curves belonging

to different categories (inter-class). The MPEG-7

dataset is a widely used 2D shape dataset consisting

of 70 categories, with 20 images per category. Fig-

ure 13 illustrates some star-shaped samples from the

dataset. After extracting contours from the black and

Morphing Between Monotonic Spinner Planar Curves Through Radial-Sign Descriptors

415

Figure 13: Star-shaped samples from MPEG-7 CE dataset.

Figure 14: Examples of inter-class deformations between

shapes belonging to MPEG-7 CE.

white MPEG-7 CE images, we apply the proposed

morphing method. Figures 15 and 14 illustrate the

results of the morphing process. Intra-class morph-

ing produces smooth and visually coherent transitions

between shapes within the same class. For inter-class

morphing, the Radial-Sign blending approach gener-

ates meaningful intermediate shapes while maintain-

ing a low computational cost, demonstrating the efﬁ-

ciency and practicality of the proposed method.

4 CONCLUSION

This work introduces a morphing approach based

on the Radial-Sign descriptor, tailored for mono-

tonic spinner planar curves. By leveraging the com-

pleteness, stability, and invertibility of these descrip-

tors, the proposed method generates intermediate

shapes that are invariant under rigid transformations

while ensuring progressive and continuous deforma-

tions. Experimental results validate the effectiveness

of this approach in achieving smooth, high-quality in-

between curves, outperforming traditional curvature-

based methods in terms of computational efﬁciency,

Figure 15: Examples of intra-class deformations between

shapes belonging to MPEG-7 CE.

shape closure, and turning number issue. For future

work, the focus will be on advancing 3D classiﬁca-

tion techniques, exploring methods for compression

and reconstruction, and further enhancing morphing

techniques, particularly with sliced monotonic spin-

ner shapes.

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