Symmetry Completion Test: A Novel Approach for Visual Distortion

Mapping and Correction Using Symmetry Constraints

Ye Ling

1 a

, David M. Frohlich

2 b

, Tom H. Williamson

3 c

and Jean-Yves Guillemaut

1 d

Centre for Vision, Speech and Signal Processing, University of Surrey, Guildford, U.K.

Digital World Research Centre, University of Surrey, Guildford, U.K.

Guy’s and St Thomas’ NHS Foundation Trust, London, U.K.

Keywords:

Metamorphopsia, Visual Distortion, Image Warping, Assistive Technology.

Abstract:

Metamorphopsia, commonly referred to as distorted vision, is a serious visual impairment which remains

uncorrectable by optical glasses or contact lenses. This paper presents a novel approach to digitally map visual

distortion based on patient feedback. The approach is based on the use of low-level geometrical constraints

(central symmetry) which provide a simple and intuitive mechanism for a patient to provide feedback on their

perceived visual distortion. We derive a set of fundamental constraints and show how visual distortion mapping

can be framed as an optimisation problem. Critically, a parametric distortion model based on MLS is used

to reduce the dimensionality of the problem and enable detailed visual distortion estimation. An extensive

evaluation using simulated data demonstrates the accuracy and robustness of the approach. This approach

opens up the possibility of correcting for visual distortion by applying the inverse mapping on the input stream

to for instance VR see-through devices or screen-based devices.

1 INTRODUCTION

Vision constitutes a vital sensory function facilitat-

ing the human body’s perception of the external en-

vironment. The World Health Organization (WHO)

reports that the global population afﬂicted with visual

impairment approaches approximately 2.2 billion in-

dividuals (WHO-Newsroom, 2023). Common con-

ditions such as myopia and presbyopia can be cor-

rected through the utilization of optical glasses or

contact lenses. However, there are numerous other

visual impairments that are not amenable to cor-

rection, thereby necessitating the use of low-vision

aids, which are only partially effective for a subset

of these impairments. Examples of such aids in-

clude large print books, handheld magniﬁers, CCTV

screen magniﬁers, microscope and telescope devices,

tinted sunglasses and ﬁlters, and assistive settings on

screens that enable modiﬁcations to font size, con-

trast and colour. The possibility of incorporating

some of these functions into head-mounted displays

https://orcid.org/0000-0002-9973-2302

https://orcid.org/0000-0003-3483-9915

https://orcid.org/0000-0002-1879-449X

https://orcid.org/0000-0001-8223-5505

has been explored experimentally, on and off, for

over 20 years (Massof, 1998). In recent years, there

has been a growing commercial viability of such sys-

tems, leading to the emergence of various low vision

“see-through” headsets in the market, such as eSight

Eyewear, NuEyes Pro Smart Glasses and IrisVision

(Deemer et al., 2018). Most of these devices typically

incorporate basic manual adjustments for magniﬁca-

tion, contrast, and colour settings, and in some cases,

they may also feature text recognition capabilities for

speech output. These show some beneﬁt to low vi-

sion sufferers, but a challenge now exists to extend

the functionality of these headsets to more complex

conditions and customize them to the unique capabil-

ities of individuals (Zhao et al., 2019), such as meta-

morphopsia. This paper introduces a new approach

for the mapping and correction of metamorphopsia or

distorted vision, which is a common symptom of age-

related macular degeneration and other retinal disor-

ders.

Metamorphopsia, a condition characterized by vi-

sual distortion, results in the perception of straight

lines as curved, as depicted in Figure 1. Figure 1a

shows the well-known painting “Mona Lisa”. How-

ever, from the perspective of an individual suffering

from metamorphopsia, some regions are perceived as

414

Ling, Y., Frohlich, D., Williamson, T. and Guillemaut, J.

Symmetry Completion Test: A Novel Approach for Visual Distortion Mapping and Correction Using Symmetry Constraints.

DOI: 10.5220/0012378000003660

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

In Proceedings of the 19th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2024) - Volume 4: VISAPP, pages

414-425

ISBN: 978-989-758-679-8; ISSN: 2184-4321

(a) Normal vision. (b) Distorted vision.

Figure 1: Illustration of the effects of metamorphopsia. An

image seen by a healthy person (a) will appear distorted (b)

to a person suffering from metamorphopsia.

distorted. Figure 1b shows a simulated representation

of this perceived distorted vision. According to pre-

vious studies (Bouwens and van Meurs, 2003; Bex,

2010; Burke, 1999; Cohen et al., 2003; Jensen and

Larsen, 1998; Mansouri et al., 2009; Zur and Ullman,

2003; Ugarte et al., 2013; Simunovic, 2015), this is-

sue is considered to be caused by the displacement of

photoreceptors or the re-organization of visual cortex

and perception adjustment after sensory input inter-

ruption from the retina in macular disease. Traditional

optical correction methods involving glasses and con-

tact lenses fail to adequately address these distortions

since they primarily stem from retinal damage rather

than a mere obstruction of light passage through the

eye. It also moves with the eye, making ﬁxed optical

correction useless. Regrettably, no effective clinical

treatment method for this problem has been identi-

ﬁed.

Digital approaches open up novel avenues to over-

come those limitations by manipulating the image

presented to the user with a view to restoring healthy

vision. The advancement of Augmented Reality (AR)

headsets and 3D displays serves as a solid hardware

foundation to support these efforts. The fundamen-

tal concept underlying the utilization of digital tech-

nologies involves the application of inverted deforma-

tions within the video stream transmitted through see-

through devices, thereby enabling the compensation

of visual distortions experienced by the affected eye.

Thus, the estimation of the visual distortion mapping

becomes the major task.

This paper proposes a monocular approach for ac-

curate mapping of visual distortion termed the “Sym-

metry Completion Test”. Being monocular, the ap-

proach can be applied separately to each eye and it

does not rely on having a distortion-free reference

eye to guide the distortion mapping process. The

key idea behind the approach is to leverage simple

low-level geometric constraints (central symmetry) in

order to interactively discover the distortion experi-

enced by a patient. A distortion model based on Mov-

ing Least Squares (MLS) is also introduced in order

to parametrise distortion using a small number of vari-

ables. Patient feedback on each symmetry constraint

is used to formulate an energy function which is op-

timised to retrieve the model parameters representing

the visual distortion experienced by each participant.

The paper makes the following key contributions.

First, it introduces the central symmetry constraints

that form the foundation for the distortion mapping

test. These are rigorously developed with full math-

ematical detail and derivation provided. Second, we

demonstrate how these can be incorporated into an ef-

fective optimisation framework to retrieve visual dis-

tortion. In particular, we show how a low-dimensional

parametric model can be leveraged to overcome the

high-dimensionality issues pertaining to estimating

visual distortion across the visual ﬁeld. Finally, we

carry out an extensive evaluation to validate the ap-

proach.

The remainder of this paper is organized as fol-

lows. Section 2 gives an overview of the literature

relevant to this study, highlighting the existing body

of knowledge in the ﬁeld. Section 3 introduces the

methodology and implementation details. Section 4

presents the results obtained from the simulation ex-

periments. Section 5 concludes by summarising the

ﬁndings and discussing avenues for future work.

2 RELATED WORK

The idea of utilizing computer vision technology to

improve impaired human vision can be traced back

to the 80s. (Peli and Peli, 1984) introduced an ap-

plication that employed adaptive image enhancement

techniques to enhance visual perception in individu-

als with low vision. Additionally, enlargement tech-

niques have gained signiﬁcant popularity in the realm

of low vision improvement, as evidenced by studies

(Vargas-Mart

ın et al., 2005; Szpiro et al., 2016; Zhao

et al., 2019). Colour inversion is a useful method for

low-vision people. (Szpiro et al., 2016) introduces

a method for low vision improvement by inverting

colours while (Peli, 1994) uses white and black to

present text. Both of them achieve the purpose of

improving low vision by using high contrast. In addi-

tion, edge detection emerges as another viable method

for enhancing visual acuity in individuals with low vi-

sion, as exempliﬁed by studies (Vargas-Mart

ın et al.,

2005; Szpiro et al., 2016; Zhao et al., 2019). While

these above methods have proven effective in address-

ing various visual impairments, they fall short in their

ability to correct metamorphopsia. As described in

Symmetry Completion Test: A Novel Approach for Visual Distortion Mapping and Correction Using Symmetry Constraints

415

Section 1, metamorphopsia presents unique and un-

predictable visual distortions. Consequently, correct-

ing metamorphopsia proves to be a formidable chal-

lenge in the absence of an accurate mapping of the

speciﬁc visual distortions associated with this condi-

tion.

The methodology known as square completion

task, as introduced in (Wiecek et al., 2015), presents

an approach to map the visual distortion. This method

involves presenting four corner points and a central

ﬁxation point to the participant. Among the four cor-

ners, one serves as a reference point, while the partic-

ipant’s task is to adjust the positions of the remaining

three corners until they form a regular square within

their visual ﬁeld. Subsequently, the participant is

required to conﬁrm a midpoint on each side of the

square. Consequently, data for eight points can be ob-

tained based on a single corner of the square. This

process is repeated with different reference points un-

til all four corners have been tested. Ultimately, the

mean value of the eight points obtained from each

corner is employed to describe the participant’s visual

distortion. Although it attempts to quantify the distor-

tion, this approach suffers from two key limitations.

Firstly, it only provides a sparse characterisation of

the distortion pattern, being limited to 8 points within

the ﬁeld of view. This restricted sampling may fail

to capture the full extent and complexity of the visual

distortion. Secondly, the use of averaging to fuse the

results from the different trials may introduce errors

when confronted with asymmetric distortions.

A useful method for mapping and compensating

for distorted vision is presented in (Bozzelli et al.,

2020; Cimmino et al., 2021). In this approach, par-

ticipants are tasked with adjusting a generated Am-

sler grid by manipulating the positions of its vertices

until the grid assumes the appearance of a regular,

straight-line grid. Subsequently, the developed appli-

cation utilizes this mapping of geometrical deforma-

tion to correct visual distortion in real-time through

the manipulation of the video stream in an AR head-

set. However, this method is not without its limita-

tions. Many participants reported discomfort while

wearing the AR headset during the operation, which

adversely affected their overall experience. Addition-

ally, the calibration procedure required for accurate

mapping was found to be time-consuming, further im-

peding the efﬁciency of the process. Consequently, a

mere 28% of participants successfully completed the

entire testing procedure, indicating the need for fur-

ther improvement in terms of comfort and usability to

enhance user participation and compliance.

The interactive line manipulation method repre-

sents an innovative approach aimed at visualizing

the distorted view experienced by individuals with

metamorphopsia (Ichige et al., 2019; Moritake et al.,

2021; Zhu et al., 2022). This technique detects dis-

tortion through the analysis of horizontal and verti-

cal straight lines and subsequently corrects the distor-

tion by adjusting the parameters associated with the

anchor points within the affected area. By applying

the derived deformation to an input image, a compen-

satory effect on the visual distortion can be achieved.

However, one notable challenge encountered in the

implementation of this method is the uncontrollable

duration of the testing process. Particularly in cases

where the distortion exhibits complexities, the exper-

iment duration tends to be signiﬁcantly prolonged.

This issue poses practical limitations, as it hampers

the efﬁciency and feasibility of the technique, neces-

sitating further exploration and reﬁnement to expedite

the testing procedure without compromising the accu-

racy of the distortion analysis and correction.

Another interesting approach is the one introduced

by (Zaman et al., 2020; Ong et al., 2022). Different

from the methods introduced before, the mapping of

visual distortion is obtained ﬁrst and then the purpose

of correcting the visual distortion is achieved by ap-

plying the mapping. In this study, a novel approach is

introduced, wherein the distorted areas are substituted

with black holes of equivalent size. Thus, the defor-

mation is suppressed by integration with the normal

vision of the healthy eye. The overall deformation

is effectively mitigated. While this particular method

may not directly map or correct the visual deforma-

tion, it offers an interesting means of alleviating dis-

torted vision. The incorporation of black holes to re-

place the affected areas holds promise in reducing the

visual impact of the distortion by leveraging the in-

tegration of the remaining intact visual information.

Although further research is necessary to fully eval-

uate and optimize the effectiveness of this approach,

it represents a noteworthy avenue for mitigating the

effects of distorted vision.

None of these previous tests have yet demon-

strated suitability for extensive clinical implementa-

tion. This paper aims to address some of the previous

shortcomings by introducing a novel and practical ap-

proach that enables accurate dense visual distortion

mapping. A key insight is the combination of the use

of low-level geometric constraints (central symmetry

constraints), that provide a simple way for the patient

to provide constraints on distortion, with the use of

a low-dimensional parametric model of visual distor-

tion, to achieve dense and scalable mapping. An ini-

tial demonstration of the test was presented in (Ling

et al., 2023).

VISAPP 2024 - 19th International Conference on Computer Vision Theory and Applications

416

3 METHODOLOGY

This section describes the proposed approach for

monocular mapping of visual distortion from low-

level geometric constraints provided by the user. The

section starts by formulating the problem in mathe-

matical terms and stating the assumptions. It then

introduces the proposed geometric constraints under-

pinning the approach. Next, it describes how visual

distortion is inferred from these by framing the prob-

lem as an energy minimisation problem and introduc-

ing some regularisation constraints. Finally, the im-

plementation details are provided.

3.1 Problem Statement and

Assumptions

Consider the problem of recovering a 2D deforma-

tion ﬁeld d deﬁned over the visual ﬁeld I of a pa-

tient’s eye affected by metamorphopsia. d represents

the perceived distortion as a 2D displacement vector

at each point in I . Note that d models visual distor-

tion only in a single eye. However, the approach is

easily extended to model binocular metamorphopsia

by recovering a separate displacement ﬁeld for each

eye. Without loss of generality, we describe the re-

covery of the deformation ﬁeld for a single eye in the

rest of the paper.

Given that the visual distortion manifests itself as

a result of retinal issues (e.g. detached retina), it

tracks the gaze direction of the patient. Visual dis-

tortion is therefore mapped in the visual ﬁeld centred

around the gaze direction of the patient. To eliminate

the dependency on gaze direction, the patient is re-

quested to rest their head on a chin-rest and maintain

their focus on a ﬁxation point located at the centre

of the screen throughout the test. This assumption

may be relaxed in the future through the use of eye-

tracking technology.

3.2 Central Symmetry Constraints

The patient’s distortion is not directly observable. The

main idea behind the proposed approach is to derive

a constraint on the visual distortion through the com-

pletion of a simple interactive test involving low-level

geometric constraints, more speciﬁcally central sym-

metry. While other types of low-level constraints may

be considered (e.g. axial symmetry, collinearity or or-

thogonality), central symmetry was selected for the

intuitiveness of the resulting test.

In a nutshell, the patient is presented with three

points on a screen such that one of them is the mid-

point of the segment deﬁned by the other two. These

points would be perceived as satisfying a central sym-

metry constraint by a person with healthy vision, but

will usually not satisfy this constraint for a patient suf-

fering from metamorphopsia if the points fall within

the area of the visual ﬁeld affected by visual distor-

tion. The patient is therefore asked to displace one

of the points to satisfy central symmetry. Such a test

provides a constraint, which can be used to infer the

deformation ﬁeld. Next, we provide a derivation of

the two types of symmetry constraints considered de-

pending on which point is manipulated.

3.2.1 Type 1 Constraint: Side Point Correction

Let us consider three points P, O and Q such that O

is the midpoint of the segment PQ. We also assume

O is located at the centre of the screen and used as a

ﬁxation point. Since P, O and Q satisfy the central

symmetry constraint in screen coordinates, as shown

in Figure 2a, we have:

2O = P + Q. (1)

However, as a result of visual distortion displayed

in Figure 2b, the patient will perceive the points as

distorted with the following locations in the patient’s

view:

= P +d(P), O

= O +d(O), Q

= Q +d(Q). (2)

These will not normally satisfy the symmetry con-

straint.

The patient is required to displace the side point Q

to satisfy the central symmetry constraint, as shown

in Figure 2c. Let us denote by v

(Q) the 2D displace-

ment that needs to be applied to Q in order to satisfy

that constraint and by Q

′

= Q + v

(Q) the resulting

displaced point in screen coordinates. Due to visual

distortion, Q

′

will be perceived at the following loca-

tion in the patient’s view:

′

= Q

′

+ d(Q

′

) = Q + v

(Q) + d(Q

′

). (3)

The perceived points P

, O

and Q

′

now satisfy the

central symmetry constraint in the patient’s view:

= P

+ Q

′

. (4)

Substituting (2) and (3) into (4), we obtain:

2O + 2d(O) = P + d(P) + Q + v

(Q) + d(Q

′

), (5)

which, after simpliﬁcation using (1) and rearranging,

gives:

d(P) + d(Q

′

) −2d(O) + v

(Q) = 0. (6)

(6) deﬁnes a constraint relating the correction made

to the side point Q by the user to enforce central sym-

metry and the visual distortion at P, O and Q

′

Symmetry Completion Test: A Novel Approach for Visual Distortion Mapping and Correction Using Symmetry Constraints

417

(a) Normal vision. (b) Distorted vision.

Figure 2: Illustration of the Type 1 symmetry constraint.

3.2.2 Type 2 Constraint: Central Point

Correction

Let us consider again three points P, O and Q, but this

time such that the user-controlled point Q is the mid-

point of the segment OP. The diagram used to explain

the relationship can be found in Figure 3. As in the

previous case, we assume O is located at the centre of

the screen and used as a ﬁxation point. The deriva-

tion of the resulting constraint is similar to that of the

previous constraint and we therefore only describe the

main steps for brevity, reusing the same notation. It

follows from the symmetry constraint that:

2Q = P + O. (7)

After the displacement of Q by the patient, the per-

ceived points P

, O

and Q

′

satisfy the central sym-

metry constraint in the patient’s view:

′

= P

+ O

. (8)

Substituting (2) and (3) into (8), we have:

2Q + 2v

(Q) + 2d(Q

′

) = P + d(P) + O + d(O), (9)

which can be simpliﬁed using (7) and rearranged to:

(Q) + 2d(Q

′

) −d(P) −d(O) = 0. (10)

(10) deﬁnes a constraint relating the correction made

to the central point Q by the user to enforce symmetry

and the visual distortion at P, O and Q

′

3.3 Cost Function Deﬁnition and

Optimisation

Recovery of the visual distortion d from the previ-

ous constraints is framed as an energy minimisation

(a) Normal vision. (b) Distorted vision.

Figure 3: Illustration of the Type 2 symmetry constraint.

problem. We consider multiple symmetry constraints

of both types introduced earlier and obtained by con-

sidering different point locations to sample the visual

ﬁeld. In our implementation, we consider 24 refer-

ence points arranged in a 5 ×5 regular grid covering

◦

as described in Figure 4. The reference point de-

ﬁnes the location of P for each of the two types of

constraint.

(6) and (10) each deﬁne two constraints on the vi-

sual distortion d (one for each axis). However, each

symmetry constraint generates four unknowns relat-

ing to the displacements at P and Q

′

(two unknowns

per point). As such, this deﬁnes an under-constrained

system of equations and direct optimisation is not

possible. To overcome this, we introduce some reg-

ularisation by using a parametric model to represent

the visual distortion d. More speciﬁcally, the Moving

Least Squares (MLS) introduced in (Schaefer et al.,

2006) is used to parametrise distortion using a small

number of control points. In our implementation,

eight control points are used. These include four mov-

able handles used to control the image deformation

and four ﬁxed corner points used to anchor the image.

This deﬁnes a total of eight degrees of freedom (two

per movable handle). The corner point constraints

embedded in our MLS model are useful to resolve the

rigid image transformation ambiguity (rotation, trans-

lation, scaling) present in the cost function.

The solution is found by ﬁnding the visual dis-

tortion d which minimises the following energy func-

VISAPP 2024 - 19th International Conference on Computer Vision Theory and Applications

418

Figure 4: Reference points distribution within the visual

ﬁeld. The red points denote the reference points, while the

green point denotes the ﬁxation point.

tion:

E(d) =

∑

i=0

(d, i)) +

∑

i=0

(d, i)), (11)

where N

and N

denote the number of Type 1 and

Type 2 symmetry constraints respectively. The indi-

vidual errors for each type of constraint are derived

from (6) and (10) and deﬁned as follows:

(d, i) = ∥d(P

) + d(Q

′

) −2d(O) + v

)∥, (12)

(d, i) = ∥2v

) + 2d(Q

′

) −d(P

) −d(O)∥.

(13)

For robustness to errors in the satisfaction of the con-

straints, the Huber loss L

is used:

(e) =

(

, if

≤ δ,

δ(

−

δ), otherwise.

(14)

The parameters δ

and δ

are set to the median of

the individual errors {e

(d, i)}

i=1

and {e

(d, i)}

i=1

respectively. Optimization is performed using the pat-

ternsearch algorithm implemented in Matlab (Audet

and Dennis Jr, 2002; Kolda et al., 2006; Lewis et al.,

2007).

3.4 Implementation and Practical

Considerations

To ensure standardised testing conditions, partici-

pants are required to position themselves in front of

the screen and rest their heads on the chin-rest to

maintain a ﬁxed distance from the screen. Being a

monocular test, the participant needs to cover the fel-

low eye. Throughout the test, participants are re-

quired to maintain their focus on the central ﬁxation

point. As described in Section 3, there are two dif-

ferent central symmetrical constraints. Thus, the par-

ticipant needs to displace the movable point to satisfy

the constraints one by one. The test procedure is as

follows:

(a) Type 1 constraint: cen-

tral ﬁxation point (green)

as centre of symmetry.

(b) Type 2 constraint: blue

point as centre of symme-

try.

Figure 5: Screenshot showing the two different central sym-

metric constraints presented to the patient.

Step 1: The participant is required to manipulate the

blue point to fulﬁl a type 1 constraint, as shown

in Figure 5a. Once the participant has conﬁrmed

that the three points satisfy the constraint, they can

proceed to the next randomly selected reference

point.

Step 2: Repeat Step 1 until all reference points have

been tested.

Step 3: The participant is required to manipulate the

blue point to fulﬁl a type 2 constraint, as shown

in Figure 5b. Once the participant has conﬁrmed

that the three points satisfy the constraint, they can

proceed to the next randomly selected reference

point.

Step 4: Repeat Step 3 until all reference points have

been tested.

Step 5: Compute the distortion map by optimising

the cost function and save the data.

This process bears some similarities with the

square completion task proposed by (Wiecek et al.,

2015), as both methods rely on point-based operations

and employ geometric constraints. However, there are

notable distinctions between the two approaches. In

contrast to the square completion task, this method

presents only three dots to the participant, reducing

the complexity of the task. Additionally, the geo-

metric constraints utilised by the two methods dif-

fer. While this method adopts central symmetric con-

straints, the square completion task primarily utilizes

four corner points and four mid-points on the side to

establish a regular square. In summary, when com-

pared to the square completion task, this method of-

fers enhanced ease of operation. Moreover, it success-

fully overcomes the square completion task’s inherent

limitation in handling asymmetrical patterns. By em-

ploying a simpliﬁed dot arrangement and a more prin-

cipled set of geometric constraints, we achieve a more

straightforward and accurate approach to mapping the

visual distortion.

Symmetry Completion Test: A Novel Approach for Visual Distortion Mapping and Correction Using Symmetry Constraints

419

Sym.

Macrop.

Sym.

Microp.

Asym.

Macrop.

Asym.

Microp.

Complex

Distortion 1.

Complex

Distortion 2.

Central

“kink”.

Figure 6: The seven distortion patterns used in the simulation experiments.

4 EXPERIMENTAL EVALUATION

To evaluate the approach, we conducted a series of

experiments using simulated visual distortions. The

evaluation considers seven different distortion pat-

terns whose choice was informed by a discussion with

an experienced ophthalmologist. The selected distor-

tion patterns comprise symmetrical macropsia, sym-

metrical micropsia, asymmetrical macropsia, asym-

metrical micropsia, two complex distortions, and a

central “Kink” distortion, as depicted in Figure 6.

Each distortion pattern provides Ground Truth (GT)

information on the distortion at each pixel in the vi-

sual ﬁeld and is used to calculate the correction re-

quired to enforce the central symmetry constraint for

each reference point considered, thereby simulating

the response of a patient.

4.1 Ablation Study

An ablation study is conducted to validate the bene-

ﬁt of the two types of constraints introduced. To this

end, the proposed approach is evaluated using type 1

only, type 2 only and both types of constraints. Sev-

eral metrics, including the structural similarity index

measure (SSIM), peak signal-to-noise ratio (PSNR),

and root mean square error (RMSE), are used to quan-

tify the similarity between the mapped distortion and

the GT. Since metamorphopsia primarily affects the

central area of the visual ﬁeld, measurements are fo-

cused on a square region centred on the central ﬁxa-

tion point. This square has a side length equivalent to

a visual angle of 10

◦

, extending 5

◦

outward from the

central ﬁxation point. Moreover, the number of itera-

tions and cost function executions are also counted to

measure the computational complexity.

As can be seen in Table 1, the proposed approach

using both types of constraints uses far less comput-

ing power than the other approaches considering only

one type of constraint while achieving comparable

or better performance. Figure 7 also shows that the

approach combining both types of constraints over-

comes the limitation of the type 1 constraint which

Table 1: Ablation study results analysing the effect of both

types of constraints on performance.

Metric

Type 1 Type 2 Types 1 & 2

Sym.

Macrop.

PSNR (dB) ↑ 10.12 11.61 11.86

SSIM ↑ 0.60 0.66 0.68

RMSE (

◦

) ↓ 0.24 0.12 0.13

Iteration ↓ 90 158 66

Func-Count ↓ 1149 2028 869

Sym.

Microp.

PSNR (dB) ↑ 9.61 11.49 10.26

SSIM ↑ 0.57 0.66 0.60

RMSE (

◦

) ↓ 0.25 0.13 0.17

Iteration ↓ 264 148 108

Func-Count ↓ 3183 1892 1394

Asym.

Macrop.

PSNR (dB) ↑ 8.89 11.87 9.66

SSIM ↑ 0.52 0.72 0.57

RMSE (

◦

) ↓ 0.22 0.14 0.17

Iteration ↓ 174 162 118

Func-Count ↓ 2167 2039 1493

Asym.

Microp.

PSNR (dB) ↑ 8.25 10.19 8.55

SSIM ↑ 0.46 0.58 0.49

RMSE (

◦

) ↓ 0.27 0.15 0.22

Iteration ↓ 170 174 154

Func-Count ↓ 2121 2166 1867

Complex

PSNR (dB) ↑ 8.93 10.39 10.90

SSIM ↑ 0.51 0.60 0.64

RMSE (

◦

) ↓ 0.26 0.20 0.19

Iteration ↓ 106 126 110

Func-Count ↓ 1311 1648 1364

Complex

PSNR (dB) ↑ 10.114 10.329 10.326

SSIM ↑ 0.596 0.604 0.598

RMSE (

◦

) ↓ 0.22 0.22 0.21

Iteration ↓ 164 152 68

Func-Count ↓ 2048 1924 889

Central

“kink”

PSNR (dB) ↑ 15.82 12.59 15.37

SSIM ↑ 0.82 0.69 0.81

RMSE (

◦

) ↓ 0.05 0.09 0.06

Iteration ↓ 184 126 90

Func-Count ↓ 2242 1589 1141

cannot retrieve symmetric distortion patterns when

used on its own. The green square denotes the spe-

ciﬁc region of interest used to calculate the metrics.

4.2 Robustness to Noise Analysis

Displacements estimated directly from the GT in or-

der to satisfy the central symmetry constraint provide

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420

Table 2: Experimental results for different noise levels.

Metric No Noise

Gassian Noise

5 10 15 20 25

Mean Std Mean Std Mean Std Mean Std Mean Std

Sym.

Macrop.

PSNR (dB) ↑ 15.81 10.97 0.92 10.09 1.30 9.50 1.10 9.07 0.96 8.79 0.94

SSIM ↑ 0.82 0.64 0.05 0.59 0.07 0.55 0.07 0.52 0.07 0.49 0.07

RMSE (

◦

) ↓ 0.07 0.15 0.03 0.19 0.05 0.24 0.08 0.29 0.10 0.34 0.11

Sym.

Microp.

PSNR (dB) ↑ 14.93 10.98 1.29 9.67 1.06 9.24 0.84 8.86 0.99 8.63 0.86

SSIM ↑ 0.79 0.64 0.07 0.55 0.07 0.52 0.06 0.49 0.08 0.47 0.06

RMSE (

◦

) ↓ 0.07 0.16 0.04 0.22 0.06 0.26 0.09 0.31 0.11 0.35 0.12

Asym.

Macrop.

PSNR (dB) ↑ 14.44 10.10 1.11 9.85 1.19 9.36 1.10 9.04 0.88 8.78 0.89

SSIM ↑ 0.78 0.59 0.06 0.57 0.07 0.53 0.07 0.51 0.07 0.49 0.07

RMSE (

◦

) ↓ 0.08 0.17 0.04 0.20 0.06 0.25 0.09 0.30 0.11 0.36 0.13

Asym.

Microp.

PSNR (dB) ↑ 12.51 9.18 1.13 9.38 1.31 9.03 1.27 8.83 0.95 8.56 0.91

SSIM ↑ 0.71 0.52 0.06 0.53 0.09 0.50 0.09 0.49 0.07 0.47 0.07

RMSE (

◦

) ↓ 0.09 0.20 0.05 0.21 0.06 0.27 0.10 0.30 0.11 0.34 0.11

Complex

PSNR (dB) ↑ 9.37 10.02 0.46 9.46 0.49 8.71 0.51 8.63 0.52 8.27 0.53

SSIM ↑ 0.55 0.58 0.03 0.55 0.04 0.49 0.04 0.48 0.05 0.45 0.05

RMSE (

◦

) ↓ 0.27 0.22 0.03 0.25 0.04 0.32 0.06 0.34 0.08 0.41 0.12

Complex

PSNR (dB) ↑ 10.46 10.06 0.57 9.48 0.62 8.80 0.59 8.60 0.63 8.38 0.56

SSIM ↑ 0.61 0.59 0.04 0.55 0.05 0.50 0.05 0.48 0.05 0.46 0.05

RMSE (

◦

) ↓ 0.21 0.22 0.02 0.25 0.05 0.31 0.08 0.33 0.09 0.36 0.09

Central

“kink”

PSNR (dB) ↑ 14.87 13.05 1.33 10.91 1.56 9.76 1.31 9.20 0.84 8.72 0.91

SSIM ↑ 0.80 0.73 0.05 0.63 0.08 0.56 0.08 0.52 0.06 0.49 0.07

RMSE (

◦

) ↓ 0.07 0.09 0.03 0.14 0.05 0.22 0.07 0.26 0.09 0.35 0.11

(a) GT. (b) Type 1. (c) Type 2. (d) Types 1 &

Figure 7: Example of mapped visual distortion for the

symmetrical micropsia pattern using different types of con-

straints.

an ideal patient response that is unlikely to be possi-

ble in practice. To simulate human errors and have a

more realistic evaluation scenario, random Gaussian

noise is introduced to corrupt those ideal displace-

ments. The noise is determined based on the radial

visual angle distances and deﬁned as follows for each

coordinate:

f (x) =

√

2πσ

−

2σ

(15)

where σ represents the percentage of the radial dis-

tance to the central ﬁxation point. This ensures that

the magnitude of the added Gaussian noise is depen-

dent on the distance from the central ﬁxation point, to

model the decrease in visual acuity the further away

the point is from the fovea. To assess the robust-

ness of the method, simulation experiments are con-

ducted using 6 different noise levels, corresponding

to σ equals to 0% (no noise), 5%, 10%, 15%, 20% and

25%, with 50 iterations for each noise level. Perfor-

mance is again evaluated by comparing the mapped

distortions to the GT for each distortion pattern using

the SSIM, PSNR and RMSE metrics calculated over

the central region of the ﬁeld of view.

Experimental results with the different noise levels

are shown in Table 2. The mean and standard devi-

ation of the 50 simulation experiments of each noise

level are presented. To facilitate visual analysis of the

experimental results, box and whisker plots are used,

as depicted in Figure 8 in the case of the symmetrical

macropsia pattern. It is evident that all three similar-

ity measurement methods exhibit ﬂuctuations within

a certain range, attributable to the inclusion of ran-

dom Gaussian noise in each experiment. As expected,

it can be observed that the similarity decreases as the

noise level increases.

4.3 Effect of Distortion Pattern

We now evaluate how performance is impacted by the

distortion pattern. The analysis is conducted with a

noise level of σ = 10% which was identiﬁed as most

representative of the noise level seen with real pa-

tients. As can be seen in Figure 9, the symmetry com-

pletion test exhibits some robustness to noise, with

nearly all PSNR values exceeding 8 dB, the majority

of SSIM values surpassing 0.5 and most RMSE val-

ues falling below 0.3

◦

. However, it is important to

note that the performance in the case of complex dis-

tortion patterns is notably poorer compared to other

distortion patterns. This suggests that the method ex-

hibits limitations when confronted with complex dis-

Symmetry Completion Test: A Novel Approach for Visual Distortion Mapping and Correction Using Symmetry Constraints

421

Figure 8: The different noise similarity results for the symmetrical macropsia pattern.

Figure 9: The seven distortion patterns similarity results of the noise level σ = 10%. The letter labels from left to right in

each ﬁgure correspond to seven distortion patterns: (a) Symmetrical macropsia; (b) Symmetrical micropsia; (c) Asymmetrical

macropsia; (d) Asymmetrical micropsia; (e) Complex distortion 1; (f) Complex distortion 2; (g) Central “Kink”.

Result

Heatmap

Sym. Sym. Asym. Asym. Comp. Comp. Central

Macropsia Micropsia Macropsia Micropsia Distortion 1 Distortion 2 “Kink”

Figure 10: Estimated distortions and their error maps for some of the simulated tests in case of the seven different simulated

distortion patterns for the noise level of σ = 10%.

tortions, as the designed distortion model, which only

employs four handles in the central area, struggles to

adequately address such complexities.

Additionally, to visualise the disparities between

the generated images and the GT, error heatmaps are

computed. The heatmaps use a uniform scale across

all the results to facilitate comparison, as illustrated

in Figure 10. In line with the results shown in Fig-

ure 9, the error for the complex distortion patterns ex-

hibits more pronounced deviations compared to other

patterns. In contrast, the results for symmetrical and

asymmetrical patterns align closely with the GT, in-

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422

Normal Sym. Sym. Asym. Asym. Comp. Comp. Central

Image Macropsia Micropsia Macropsia Micropsia Distortion 1 Distortion 2 “Kink”

Figure 11: Correction applied on distortion results (the image should look normal if the correction is perfect) on different

contents for some of the simulated tests in the case of the seven different simulated distortion patterns.

dicating the superior performance of the symmetry

completion method for such distortions. As previ-

ously discussed, the method demonstrates limitations

when confronted with complex distortions. More-

over, the error heatmap for the Central “Kink” pattern

reveals some deviations. This can be attributed to the

lack of handles at the central point of the designed

MLS distortion model. This causes some difﬁculties

in accurately mapping the distortion associated with

the distortion at the central ﬁxation point.

4.4 Correction Results

Here we assess the efﬁcacy of the correction by com-

paring images that have been corrected by inverting

the mapped distortion, to the original undistorted im-

ages. Different types of content are considered to il-

lustrate the effect across different possible use cases.

As illustrated in Figure 11, the symmetrical and asym-

metrical distorted patterns can be corrected very ef-

fectively. For complex and central “kink” distortion

patterns, the distortion has only been partially cor-

rected with some clearly visible residual distortion.

4.5 Comparison Against Square

Completion Task

As previously discussed, this method bears some

similarities with the square completion task from

(Wiecek et al., 2015) since both methods use points

and geometric rules to operate. Therefore, a set of

comparison experiments is conducted using the seven

distortion patterns. The comparative evaluation is car-

ried out in the case of the noise-free input measure-

ments. The GT distortion is used to calculate the loca-

tion of the 16 points (two squares) in the square com-

pletion task. The symmetry completion test proceeds

as described previously using 24 reference points.

The two methods are then compared by assessing the

estimated distortion at the 16 points. The 16 points

representing the squares for the seven distortion pat-

terns are displayed in Figure 12. Green represents the

GT, blue shows the results of the symmetry comple-

tion test and red corresponds to the square comple-

tion task results. The symmetry completion test re-

sults are more accurate than the results of the square

completion task as the blue squares are mostly per-

fectly aligned with the GT (the green square). Again,

the symmetry completion test results for the complex

distortion patterns are misaligned at the corner of the

bigger square which means the distortions of these

two patterns cannot be corrected perfectly. This is

consistent with the previous analysis showing that this

method suffers from limitations with complex distor-

tions. Compared to the square completion task, our

method successfully overcomes the square comple-

tion task’s inherent limitation in handling asymmetri-

cal patterns, while also being able to produce a dense

visual distortion map.

5 CONCLUSIONS AND FUTURE

WORK

We presented a novel approach for mapping visual

distortion based on low-level geometric constraints

(central symmetry). We demonstrated how an energy

Symmetry Completion Test: A Novel Approach for Visual Distortion Mapping and Correction Using Symmetry Constraints

423

Sym. Macrop. Sym. Microp. Asym. Macrop. Asym. Microp.

Complex 1. Complex 2. Kink.

Figure 12: Comparative results for the seven distortion patterns. The green colour is the GT, the blue colour is the result of

the symmetry completion test and the red colour is the result of the square completion task.

function can be deﬁned based on these low-level con-

straints and effectively optimised using a parametric

model of distortion based on MLS. Analysis of the

simulation data reveals that the symmetry completion

method is able to accurately map visual distortion and

exhibits some robustness to noise. The predeﬁned

distortion model, constrained by a limited number of

handles and ﬁxed positions, suffers from inaccuracies

when estimating more complex distortions. Further-

more, the absence of a handle at the central point re-

stricts the method’s ability to address distortion oc-

curring at this location.

The demonstrated capability of this approach on

simulated data to estimate visual distortion in indi-

viduals affected by metamorphopsia provides a com-

pelling foundation for the development of an applica-

tion utilizing see-through devices. The next step is to

clinically evaluate the method with real patients living

with metamorphopsia. The approach has received a

favourable ethics opinion and is about to be clinically

evaluated at St Thomas’ Hospital in London with pa-

tients suffering from metamorphopsia, following the

protocol outlined in (Ling et al., 2023). Moreover,

as the method requires the participant to maintain the

focus on the central ﬁxation point, employing eye-

tracking technology may be a good strategy to moni-

tor the satisfaction of this constraint. Leveraging the

integration of an eye tracker also holds promising po-

tential in dynamically compensating for the distorted

vision experienced by individuals with metamorphop-

sia. A particularly interesting avenue for future work

is to investigate how the technology could beneﬁt pa-

tients as a corrective device. In the future, we would

like to explore how these algorithms may be deployed

across other types of devices such as headsets or 3D

tablets and also extend the correction to dynamically

adapt it to gaze direction through eye tracking.

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