Interpolation of perimetric test grids using artificial neural networks

INTERPOLATION OF PERIMETRIC TEST GRIDS USING ARTIFICIAL NEURAL NETWORKS

CLEMENS JÜRGENS,1 ULRICH SCHIEFER,2 ROLAND BURTH2 and

ANDREAS ZELL1

1Wilhelm-Schickard-Institute for Computer Science, Department of Computer Architecture, University of Tübingen; 2Department of Pathophysiology of Vision and Neuro-Ophthalmology, University Eye Hospital Tübingen; Tübingen, Germany

Abstract

Objective: To interpolate the test grid of the Tübingen automated perimeter (TAP) using neural networks. Methods: TAP uses a threshold-oriented slightly supraliminal strategy, resulting in a high test point density with centripetal condensation (191 test locations within the central 30° visual field). Defect depth is scored by six luminance classes. In this study, the authors used a dataset of 702 perimetric records. They designed a feed-forward neural network with the test point coordinates as input data and the corresponding luminance classes as output data. For training, they used a subset of 191-n test points and, in a second step, they tested the interpolation capability with regard to local luminance class for the omitted n test points, with n = 19; 38; 47. Results: Correctly predicted local luminance classes ranged from 83% for 19 to 73% for 47 omitted test points. Conclusions: Artificial neural networks are able to interpolate perimetric test grids. This is an essential prerequisite for individually generated grid-independent examination procedures.

Introduction

In perimetry, there are technical and practical reasons for increasing the spatial distribution of test locations. Several studies have evaluated different perimetric test grids for optimizing the examination strategy and/or for saving time.1-7 Other authors have generated individually-tailored stimulus arrangements instead of predefined test grids.8,9 However, this spatial variability may reduce comparability of visual field examinations, which would be a great disadvantage, especially in follow-up studies. In order to make visual fields with different stimulus arrangements comparable, a powerful and feasible method of interpolating perimetric test grids is necessary. Artificial neural networks have been successfully applied for classifying difficult spatial patterns:10 the two-spiral-problem,11 i.e., a famous benchmark problem; in our approach, this tech-

Address for correspondence: Dr. med. Clemens Jürgens, Fachbereich Informationsund Kommunikationswesen, Fachhochschule Hannover, Ricklinger Stadtweg 120, D-30459 Hannover, Germany.

Email: clemens.juergens@ik.fh-hannover.de

Perimetry Update 2002/2003, pp. 13–19

Proceedings of the XVth International Perimetric Society Meeting, Stratford-upon-Avon, England, June 26–29, 2002

edited by David B. Henson and Michael Wall

© 2004 Kugler Publications, The Hague, The Netherlands

Perimetric records

This study is based on 702 perimetric records, which were classified by perimetric specialists and assigned to eight characteristic scotoma subgroups. (Table 1 represents an overview with a diagrammatic figure for each subgroup.)

Table 1. Seven hundred and two perimetric records of the 30° central visual field (TAP) assigned to eight characteristic scotoma subgroups

aRetinal nerve fiber layer defect; bvisual field defect; cdifferential luminance sensitivity

Artificial neural network

Artificial neural networks (ANNs) are typically used in the fields of pattern recognition, classification, clustering, simulation, and time series analysis. We used a feedforward network with two input units, to present the X- and Y-coordinates as input data, and seven output units, representing normal sensitivity, and six luminance classes. The corresponding output unit of each measured luminance class was set to 1, whereas all other output units were set to 0. ANN had two hidden layers with nine hidden units in each layer and shortcut connections. The network topology is shown in Figure 2.

We used standard back-propagation12 as a learning algorithm, which is the most common algorithm for ANNs. It updates the weights after each training pattern, using a gradient descent method to find the minimum of the error function. Standard back-

Fig. 2. Feed-forward neural network with shortcut connections for test grid interpolation (not all connections are shown): two input units represent the test point coordinates; two hidden layers each with nine hidden units; seven output units represent normal sensitivity; plus six luminance classes.

propagation has two parameters; in our analysis we used the following values: (a) gradient step width η = 0.8; (b) maximum propagated error dmax = 0.2.

For training, we omitted an increasing number of test points: 19; 38; 47 (Fig. 3 A- C). These test points were used in a second step to test ANN interpolation capability.

For each visual field, a neural network was trained using batchman, a batch version of the Stuttgart neural network simulator (SNNS). After training, the unknown test points were presented to the trained network: the output unit with the highest output value was considered to be the predicted luminance class. A pattern was rated as correct if the predicted and measured luminance classes were exactly the same.

Results

Using 172 test points for training and 19 for testing (Fig. 3A), we were able to correctly assign 83% of the luminance classes. With 38 test points being omitted (Fig.

A B

C

Fig. 3. Number of test points omitted for training: A – 19; B – 38 ; C – 47. Gray points indicate the training set; the omitted testing points are shaded in black.

3B), we found 73% correct results, with 47 test points being omitted (Fig. 3C), we achieved 73%.

Having trained a neural network with all the 191 test points, it can generate the luminance class for any test location within the examination area. Figures 4 A–D show the learning progress for quadrantanopia.

Discussion

With our approach, we show that artificial neural networks can be used to interpolate perimetric test grids of the TAP. With an increasing number of test locations, which

A B

C D

Fig. 4. Interpolation of quadrantanopia using all 191 test locations for training. The learning progress at different training cycles is shown: A – 1; B – 10; C – 20; D – 2000.

are used for training, we achieve better interpolation results. This demonstrates the importance of a high spatial test point resolution for determining the size and shape of a scotoma.

Interpolation of visual fields is a helpful tool in clinical decision-making. It can help in the comparison of perimetric tests that use different test grids. As a next step, we would like to apply this procedure to follow-up studies with different stimulus arrangements for threshold testing in order to make point-wise trend analysis possible. This may lead to optimal individually generated test grids.

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