Embedding Anatomical Characteristics in 3D Models of Lower-limb

Sockets through Statistical Shape Modelling

Ana Costa

, Daniel Rodrigues

, Marina Castro

, Soﬁa Assis

and H

elder P. Oliveira

3,4 a

Faculty of Engineering of the University of Porto, Portugal

Adapttech, Porto, Portugal

INESC-TEC, Porto, Portugal

Faculty of Sciences of the University of Porto, Portugal

Keywords:

Principal Component Analysis, Point Cloud Registration, Statistical Shape Models, Lower Limb Sockets, 3D

Scanning.

Abstract:

Lower limb amputation is a condition affecting millions of people worldwide. Patients are often prescribed

with lower limb prostheses to aid their mobility, but these prostheses require frequent adjustments through an

iterative and manual process, which heavily depends on patient feedback and on the prosthetist’s experience.

New computer-aided design and manufacturing technologies have been emerging as a way to improve the

ﬁtting process by creating virtual socket models. Statistical Shape modelling was used to create 3D models

of transtibial (TT) and transfemoral (TF) sockets. Their generalization errors were, respectively, 6.8 ± 1.8

mm and 10.5 ± 1.6 mm, while speciﬁcity errors were 9.7 ± 0.6 mm and 9.8 ± 0.2 mm. In both models, a

visual analysis showed that biomechanically meaningful features were captured: the largest variations found

for both types were in the length of the residual limb and in the perimeter variation along the limb. The results

obtained proved that statistical shape modelling methods can be applied to TF and TT sockets, with several

potential applications in the orthoprosthetic ﬁeld: generation of new plausible shapes and on-demand socket

design adjustments.

1 INTRODUCTION

Lower limb loss has been deﬁned as a complete loss,

in the transverse anatomical plane, of any part of the

lower limb, for any reason. The incidence of limb loss

is expected to increase in the coming years, reach-

ing 3.6M in the United States in 2050 (Varma et al.,

2014). The two most common levels of amputa-

tion are transtibial (below the knee) and transfemoral

(above the knee).

Amputation can have a devastating effect on both

physical and mental health, with mobility being the

main aspect of an amputee’s satisfaction with life

(Wurdeman et al., 2018). To improve mobility, pa-

tients are prescribed with prostheses. Adjusting a

prosthesis to a patient is a difﬁcult process, with the

most prolonged, iterative part being the ﬁtting of the

socket to the residual limb. This process heavily de-

pends, not only on the skill and experience of the

prosthetist, but also on patient feedback, which can

https://orcid.org/0000-0002-6193-8540

sometimes be unreliable, with no quantitative infor-

mation involved (Patern

o et al., 2018).

There have been several attempts, both in indus-

try and academia, to improve this process by apply-

ing digital technologies to both the manufacturing and

the design of the socket. Computer-Aided Design and

Manufacturing (CAD/CAM) systems allow for digi-

talization of sockets, creating a virtual representation

that can be corrected with more accuracy and preci-

sion through digital tools (Mehmood et al., 2019). In

comparison with the traditional process, this is a faster

and cheaper method of socket adjustment. Most im-

portantly, this method provides a signiﬁcant improve-

ment in the quality of life of amputees compared to

traditional ﬁtting techniques (Karakoc¸ et al., 2017).

TF and TT sockets are usually based on a small

set of base designs. This implies that shape vari-

ability is limited and there are restrictions on what

can be considered a valid shape. However, 3D scan-

ners used in the prosthetic ﬁeld nowadays are blind to

this prior knowledge, which could improve the qual-

ity of their 3D representations. From this observation,

528

Costa, A., Rodrigues, D., Castro, M., Assis, S. and Oliveira, H.

Embedding Anatomical Characteristics in 3D Models of Lower-limb Sockets through Statistical Shape Modelling.

DOI: 10.5220/0010339805280535

In Proceedings of the 16th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2021) - Volume 4: VISAPP, pages

528-535

ISBN: 978-989-758-488-6

our work hypothesizes that, with a diverse enough

dataset, a generic model capturing the variability be-

tween shapes can be built. From this model, it would

then be possible to ﬁt new examples to the universe of

possible shapes, thus detecting and eliminating unre-

alistic scanning artifacts. Such a model would also al-

low generation of new plausible shapes, which could

bypass the need for an initial plaster mold of the resid-

ual limb, as well as open up new pathways for the ap-

plication of machine and deep learning algorithms to

this ﬁeld by using data augmentation. In this work, we

validate these hypotheses by using statistical shape

modelling techniques with a dataset of 3D scans of

lower limb sockets and exploring its applications in

the generation of new shapes.

2 RELATED WORK

Statistical Shape Models (SSMs) are based on the as-

sumption that each shape is a deformed version of a

reference shape. Therefore, they can be used to an-

alyze differences in a dataset and also to synthesize

new, similar shapes (Lindner, 2017).

One of the best known SSMs is the Active Shape

Model, which uses landmarks to examine and mea-

sure shape change (Cootes et al., 1992). This and sim-

ilar models, where shapes are represented by a set of

corresponding points, are referred to as Point Distri-

bution Models (Huang et al., 2013). The main steps

in building a statistical shape model, once a dataset

is acquired, are: shape representation, shape registra-

tion, model training and evaluation.

Registration can be deﬁned as the process of

bringing two or more shapes of the same object or

of similar objects into the same reference system

(Castellani and Bartoli, 2014). The problem of regis-

tering a set of shapes can be tackled in a single-stage

fashion or, alternatively, as a two-stage process, where

a point-to-point correspondence is established and,

independently, an optimal alignment is found. There

are several ways to establish correspondence between

3D shapes by applying rigid or non-rigid transfor-

mations. In SSMs, automatic registration typically

falls into one of the following types: parametrization,

distance-based, feature-based or image-based. These

methods often solve the alignment and correspon-

dence issue together. When dealing, in particular,

with point clouds, a number of local feature descrip-

tors have been proposed, such as point signatures,

point feature histograms, signature of histograms of

orientations and rotational projection statistics (Yang

et al., 2016).

Most often, model building uses Principal Com-

ponent Analysis (PCA), an algorithm which ﬁnds the

directions with greatest variance in the training data.

SSMs can have many different applications. Mod-

elling human expression and pose is one of the main

areas of interest due to their large potential for human-

machine interactions (Yang et al., 2011). The other

large ﬁeld of application is medical image segmenta-

tion, for detecting abnormalities in anatomical shapes

(Cha et al., 2018).

To the authors’ best knowledge, relatively little

work done has been done with statistical shape anal-

ysis in prostheses. Even though the SSM is not the

paper’s main focus, (Steer et al., 2020) work on meth-

ods for socket design based on SSM and ﬁnite ele-

ment analysis stands out the most. A statistical shape

model is used to introduce representative morpholog-

ical variation into a ﬁnite element model (which is the

paper’s main focus). This model is constructed from

aligned surface scans of TT plaster casts to which

PCA was applied. The principal modes of variation

found were the residuum length, related to the ampu-

tation height, and the proﬁle, related to how conical

or bulbous the limb is.

3 METHODS

This work entailed the collection of a dataset of point

clouds of lower limb sockets, described in this sec-

tion. The point clouds were registered using heuris-

tic techniques for both alignment and point-to-point

correspondence. The statistical shape models were

built using PCA. All implementations were written in

Python 3.7, using NumPy 1.18 (Van Der Walt et al.,

2011), scikit-learn 0.23 (Pedregosa et al., 2011) and

Open3D 0.9.0 (Zhou et al., 2018).

3.1 Dataset Collection and

Characterization

Table 1: Summary of dataset characteristics. Dimensions

are in millimeters.

Height Perimeter

MinH MaxH BP MP HP

Min 113 166 121 210 230

TT Median 175 250 180 300 325

Max 256 349 256 427 462

Min 138 264 121 252 391

TF Median 225 325 220 400 480

Max 292 426 308 537 650

The point clouds of lower limb sockets were obtained

using a 3D stereoscopy and laser-based scanner which

Embedding Anatomical Characteristics in 3D Models of Lower-limb Sockets through Statistical Shape Modelling

529

Figure 1: Transtibial and transfemoral sockets from the

dataset, in posterior views, with medial leaning and lateral

support highlighted.

digitizes the interior surface of sockets

. These scans

were acquired in clinical settings with the use of the

data for research purposes authorized. Scans have no

associated patient information other than the type of

amputation and the leg. For each scan, multiple mea-

sures were taken to characterize the socket, namely its

full height (MaxH) and the maximum height at which

the perimeter still corresponds to a full circle (MinH),

and the socket perimeters at MinH, at mid-height and

at the bottom (HP, MP and BP, respectively).

Some examples of the dataset can be seen in Fig-

ure 1. Anatomical variations such as length of the

residual limb or musculature (reﬂected in a more con-

ical shape) can be seen, along with more distinctive

features such as the medial leaning of TT sockets. A

total of 30 TT and 21 TF examples were collected.

The analysis of their characteristics is summarized in

Table 1.

3.2 Point Cloud Registration

3.2.1 Alignment

Due to the inherent differences between shapes of TT

and TF sockets, two separate but analogous proce-

dures were followed for registration. For both types

of sockets, the ﬁnal registration result was determined

to be a point cloud with as many points as the smallest

point cloud in the dataset (4,731 for TT and 11,487 for

TF). The acquisition system guarantees that the ac-

quired shapes are all under the same metric referential

and that the vertical axes of the sockets are aligned.

Therefore, the alignment problem is simpliﬁed to a

two-dimensional rotation and a translation.

The translation problem was solved by overlap-

ping the centroids of each socket with the origin of

the coordinate system.

INSIGHT™ Scanner: https://www.adapttech.eu/

insight#knowinsight

The rotation required an additional pre-processing

step: the mirroring of shapes pertaining to the left

leg along a radial plane, to harmonize differences be-

tween left and right leg. Then, to ﬁnd the optimal 2D

rotation matrix to align the shapes, a landmark present

across all shapes (based on domain knowledge) was

chosen. The optimal rotation was ﬁnally deﬁned as

that which brings these landmarks into overlapping

positions. For TT sockets, the landmark chosen was

the center of the posterior proximal support, as seen in

Figure 2a. For TF sockets, the landmark was the cen-

ter of the medial proximal border, as seen on Figure

2b.

A semi-automatic method was employed to de-

tect these landmarks using the measurements taken

for each instance of the dataset.

To overlap the landmarks, a socket from the

dataset was randomly chosen as the target and all

other point clouds were aligned relative to it.

3.2.2 Correspondence

Two different methods were tested to establish point-

to-point correspondence: local feature similarity us-

ing Fast Point Feature Histograms and a custom

heuristic henceforth referred to as Selective Sam-

pling.

Fast Point Feature Histograms. (FPFH) are local

descriptors which have been used in state-of-the-art

applications for point cloud registration. This descrip-

tor relies on the angular relationships between the nor-

mals of a given point and its neighbours to compute

descriptive histograms. From these descriptors, cor-

respondence can be established between points which

have the most similar features by computing the his-

tograms’ distances (Rusu et al., 2009).

Selective Sampling: is based on a registration tech-

(a) Posterior view of TT

socket.

(b) Anterior-Medial

view of TF socket.

Figure 2: Landmarks chosen for TT and TF sockets.

VISAPP 2021 - 16th International Conference on Computer Vision Theory and Applications

530

Figure 3: Two sockets registered through Selective Sam-

pling. For a given height percentage h, a socket’s circular

proﬁle has N points separated by an angle α.

nique often used in statistical shape models, which

is parametrization by sampling evenly spaced points

across a contour. Using domain knowledge and a

more intuitive notion of correspondence, a similar ap-

proach which performs a selective sampling on the

point clouds was designed. Two corresponding points

can be thought of as points which are in the same

relative position on two different sockets. Since the

sockets are approximately paraboloids, this position

can be deﬁned by two values: the angle ω relative to

a known vector and a percentage h of the height in

two regions: above the landmark, and below the land-

mark. These two regions assure that the posterior and

anterior support will always correspond across sock-

ets, even if the lateral support’s height differs between

them.

The reference vector used to deﬁne the angle was

the the origin-landmark vector OL used for alignment

(where origin corresponds to the center of the circum-

ference at the same z as the landmark). This vector is

translated to a given height percentage h, OL

and ro-

tated around the z-axis in N intervals of α degrees.

Rotating OL

around the vertical z-axis by this angle

creates a set P

of evenly spaced target coordinates in

the shape of a circular proﬁle:

(h, ω) = OL

× R

(i × α) (1)

where i ∈ [0, ... , N] and R

is the rotation matrix

around z. Each point P

(h, ω) in P

is then matched

to its closest neighbour in the point cloud A which

has not yet been matched (to ensure a one-to-one cor-

respondence):

P(h, ω) = arg min

p∈A

(||p − P

(h, ω)||) (2)

The ﬁnal registration result is then controlled by

two parameters: N, the number of points in each

circular proﬁle, and K the number of proﬁles which

should be sampled across the socket height. Figure

3 shows a schematic representation of the selective

sampling process.

3.3 Model Building

With properly aligned and registered socket shapes,

a statistical shape model F can be built from the

dataset. PCA was performed for each separate type

of socket, since the base designs of TT and TF vary

considerably. To perform PCA on a registered train-

ing set, Singular Value Decomposition is applied to

the shapes F − F, where F is the mean of the aligned

shapes. The resulting matrix is a matrix P of eigen-

vectors, as well as their corresponding eigenvalues λ.

Any valid shape set can then be deﬁned as:

F = F +

∑

m=1

(3)

where M is the number of eigenvectors in the model

(subspace dimension), b

are scalar weights and PC

are the principal components. To assure plausible

variation, b

has to be limited. Typical values to con-

strain this variation are [−3λ

, 3λ

]. For both TT

and TF sockets, two different models were built, us-

ing the shapes registered through Selective Sampling

and FPFH.

3.4 Evaluation

Generalization reﬂects the ability of a model to de-

scribe instances of the object that have not been

seen during model building. If a model is overﬁt-

ted, then its ability to generalize when faced with a

new example will be very low. Generalization can be

measured with a leave-one-out methodology (Davies

et al., 2010).

Speciﬁcity is a measure of the similarity between

generated shapes and the ones present in the training

set. It is quantitatively deﬁned by generating a new

population of instances and averaging the distance be-

tween a new generated shape and the closest shape to

it in the training set. The new population of instances

should be created with weights that assure plausible

shapes. This, combined with generalization, evalu-

ates both the generative and reconstructive abilities of

the model.

A visual expert evaluation was also performed by

determining the individual meaning of each principal

component (PC) captured by the model and and at-

tributing it an anatomical interpretation (when such

interpretation existed in the prosthetics domain).

Generalization and speciﬁcity are two com-

mon approach-independent metrics used in the ﬁeld

(Davies et al., 2010). Due to the reduced size of the

dataset, no explicit training/testing set division was

done. This omission is counterweighted by the evalu-

ation of generalization - a proxy metric of overﬁtting.

Embedding Anatomical Characteristics in 3D Models of Lower-limb Sockets through Statistical Shape Modelling

531

Table 2: Weight factors of each visually interpretable principal component, as validated by ortoprosthetists.

PC1 PC2 PC3 PC4 PC5 PC6

Min Max Min Max Min Max Min Max Min Max Min Max

TT -30 30 - - -20 30 -5 15 - - -15 15

TF -30 30 -30 30 -15 10 -15 30 -12 12 -10 15

Figure 4: Generalization comparison between transtibial

models registered through Selective Sampling and FPFH.

4 RESULTS

Results on generalization and speciﬁcity are pre-

sented for models based on point clouds registered

through Selective Sampling and FPFH. This allows

for a comparison between the accuracy of both meth-

ods, since a better registration will lead to a better

model. Speciﬁcity and generalization were computed

using as distance measure a normalized (by the num-

ber of points in question) Mean Absolute Distance

(mean of the absolute distance between correspond-

ing points, MAD).

Regarding expert evaluation, orthoprosthetists

contributed to results by deﬁning weight limits that

assure the generation of plausible shapes (Table 2).

These weights were applied for the generation of

new shapes, both for speciﬁcity calculation and for

the visualization of PC effects on the average shape

(using Equation (3)). For components with no visual

interpretation, the weights used were [−3λ

, 3λ

as common in the literature.

Results subject to comparison were not found in

the authors’ literature review.

4.1 Transtibial Statistical Shape Model

4.1.1 Model Performance

By analysing the cumulative variance of the models,

it is possible to determine how many PCs are required

to obtain a descriptive model. To describe 95% of

the variance in the training dataset, the Selective Sam-

Figure 5: Speciﬁcity comparison between transtibial mod-

els registered through Selective Sampling and FPFH.

pling model requires 6 PCs, while the FPFH one re-

quires 22.

As shown in Figure 4, the model built from Selec-

tive Sampling registration has a lower reconstruction

error, meaning better generalization. As expected,

generalization improves with the number of principal

components used for reconstruction. With 6 PCs, Se-

lective Sampling produces a reconstruction error of

6.8 ± 1.8 mm, while FPFH at 22 PCs surpasses this

error, with 21.6 ± 2.4 mm.

Speciﬁcity, which represents the similarity be-

tween the generated shapes and the ones present in

the training set, is again superior using Selective Sam-

pling (Figure 5), with an error of 9.7 ± 0.6 mm.

Speciﬁcity is a crucial parameter in this work, given

that one of the proposed applications is the creation

of plausible shapes.

Since the models registered through Selective

Sampling outperformed FPFH in all metrics, this

model was chosen for the subsequent analyses.

4.1.2 Principal Component Analysis

In partnership with orthoprosthetists, it was possible

to derive anatomical interpretations of the variance

components captured by the model. PCs in which no

relevant anatomical or design information was identi-

ﬁed were omitted.

The ﬁrst PC, which represents around 70% of the

total variance, is related to the length of the residual

limb, as can be seen in Figure 6. This is a natural

variation, since the level of amputation is highly vari-

able, depending on the patient’s anatomy and injury

degree.

VISAPP 2021 - 16th International Conference on Computer Vision Theory and Applications

532

Figure 6: Representation of the principal components PC1

(residual limb length), PC3 (residual limb volume), PC4

(patellar coverage) and PC6 (medial leaning) with eigen-

values (λ), over the average transtibial shape (µ) and weight

factors (WF) from Table 2 (posterior view).

The third PC, accounting for 5% of variance, cor-

responds to the circular proﬁle variation along the lon-

gitudinal axis, distinguishing between more conical

or cylindrical sockets. This is related to the muscula-

ture of the residual limb, which varies between sub-

jects and also in time, since residual limbs are prone

to atrophy and volume variations. Figure 6 shows

this effect: the leftmost sockets exhibits perimeter re-

duction along its longitudinal axis, creating a conical

shape, unlike the rightmost one, where the perimeter

varies a lot less.

The fourth PC, responsible for 2.5% of the vari-

ance, is related to the level of coverage of the kneecap.

Different designs can cover more or less of the knee

structure, which is reﬂected in Figure 6. For instance,

the leftmost socket, with higher lateral and medial

walls and more patellar coverage, has a design evok-

ing a suprapatellar patellar-tendon bearing socket (Pa-

tern

o et al., 2018).

The sixth PC, representing 1% of the variance,

represents a lateral-medial leaning in the socket (Fig-

ure 6), along with a narrowing of the distal end. This

is frequently observed and depends on several factors:

the level of amputation and the natural muscular pro-

ﬁle and bone structure of the lower limb.

Figure 7: Generalization comparison between transfemoral

models registered through Selective Sampling and FPFH.

Figure 8: Speciﬁcity comparison between transfemoral

models registered through Selective Sampling and FPFH.

4.2 Transfemoral Statistical Shape

Model

4.2.1 Model Performance

A similar analysis to the one performed for the TT

model can be done for the TF model. To describe 95%

of the variance in the training dataset, the Selective

Sampling model requires 11 PCs, while the FPFH one

requires 16.

For a model with 11 PC, representing 95% of

the variation in the training set, the generalization

through Selective Sampling is 10.5 ± 1.6 mm, mean-

ing an unseen shape would be, on average, recon-

structed with this error. As shown in Figure 7, the Se-

lective Sampling model again outperformed the FPFH

registration. The generalization ability for this model

is inferior to the one found for TT sockets.

Finally, the speciﬁcity of the TF model was also

inferior to the TT one, with an error of 9.8 ± 0.2 mm

for the Selective Sampling model (Figure 8).

Similarly to the TT case, the model registered

Embedding Anatomical Characteristics in 3D Models of Lower-limb Sockets through Statistical Shape Modelling

533

Figure 9: Representation of the inﬂuence of principal com-

ponents: PC1 (residual limb length), PC2 (residual limb

volume), PC3 (height of the lateral support), PC4 (adduc-

tion angle), PC5 (posterior proximal contour) and PC6 (is-

chial support) with eigenvalues (λ), over the average trans-

femoral shape (µ), with weight factors (WF) from Table 2.

PC1, PC2, PC4 and PC6 are shown in a posterior view, PC3

in a medial view and PC5 in an anterior view.

through Selective Sampling outperformed FPFH in all

metrics, and was, therefore, the one chosen for subse-

quent analyses.

4.2.2 Principal Component Analysis

The most signiﬁcant variation in the TF shapes is

again relative to the length of the residual limb, rep-

resenting 64% of the variance (Figure 9).

The second PC is related to the circular proﬁles

along the longitudinal axis, responsible for 12% of

variance. A more conical stump can be seen on the

rightmost socket of Figure 9, while the leftmost one

would be appropriate for a more cylindrical stump.

This proﬁle is dependent on the amputee’s muscula-

ture and time after amputation, since the muscles are

subject to atrophy and volume reductions.

The third PC, which accounts for 5% of variance,

represents the height of the lateral support. In Figure

9, the leftmost point cloud has a higher lateral support,

which decreases in the shapes to its right.

PC number four is related with the adduction an-

gle of the femur and the ilio-femoral angle, two im-

portant measurements taken during the ﬁtting process,

and represents 4% of variance. The ﬁrst is deﬁned be-

tween a longitudinal axis and the femur while in max-

imal adduction and is typically larger for women. The

second is deﬁned between the femur line and the lat-

eral support. Figure 9 shows that, from left to right,

these angles decrease.

The ﬁfth PC varies the shape of the posterior prox-

imal contour of the socket. In Figure 9, it is possible

to see that the valley between the lateral and ischial

support is more pronounced in the leftmost shape.

This contour has an important effect in the aesthetic

effect of the socket, as well as in comfort while sit-

ting.

The sixth PC, which accounts for 2.5% of vari-

ance, is related to the prominence of the ischial seat.

In Figure 9 (left of the top contour), it is possible to

see that, from left to right, this support becomes wider

and more pronounced.

5 DISCUSSION & CONCLUSIONS

By observing the features captured by the models, it

is possible to conclude that they represent important

anatomical variations in socket design - such as length

and proﬁle, in coherence with (Steer et al., 2020), but

also subtler design characteristics. Given this, these

models can be useful for generating new shapes with

speciﬁc characteristics by manipulating the inﬂuence

of their respective PCs. Additionally, the model’s

good generalization abilities allow it, for instance, to

be used to reconstruct new 3D scans in socket acqui-

sition systems, which may minimize acquisition arti-

facts.

The analysis performed showed the TT model out-

performed the TF in all metrics. This can be due to

a larger variation across base TF designs, which, in

turn, can lead to a poorer registration process, or sim-

ply due to the lower number of samples used to build

the model.

Two import aspects limit our work. Firstly, the

high dependence on the registration process, which

heavily impacts the input of the PCA model and,

therefore, the accuracy of the results. To improve re-

sults and reduce that dependency, other more robust

local descriptors could be tested. Secondly, the in-

VISAPP 2021 - 16th International Conference on Computer Vision Theory and Applications

534

ability to directly compare with other authors. This

stems from the lack of relevant work in this ﬁeld (as

far as we were able to ascert) and the lack of reference

datasets for the task. Towards mitigating the latter, we

are making some example data available in Section 6.

Nevertheless, this work shows that mathemati-

cally representing socket point cloud data through sta-

tistical shape models encodes biomechanically rele-

vant information, allowing a range of potential ap-

plications with clinical interest like the generation of

new plausible socket shapes (to support data-intensive

learning workﬂows) or the automatic rotation of sock-

ets’ point clouds into relevant anatomical planes (for

improved user experience in CAD/CAM software).

6 CONTRIBUTIONS

Some examples of TT and TF sockets shapes gener-

ated with the SSM are available on a GitHub reposi-

tory: https://github.com/adapttech-ltd/SocketSSM.

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