124 research outputs found
Analysis of the human corneal shape with machine learning
Cette thèse cherche à examiner les conditions optimales dans lesquelles les surfaces cornéennes antérieures peuvent être efficacement pré-traitées, classifiées et prédites en utilisant des techniques de modélisation géométriques (MG) et d’apprentissage automatiques (AU).
La première étude (Chapitre 2) examine les conditions dans lesquelles la modélisation géométrique peut être utilisée pour réduire la dimensionnalité des données utilisées dans un projet d’apprentissage automatique. Quatre modèles géométriques ont été testés pour leur précision et leur rapidité de traitement : deux modèles polynomiaux (P) – polynômes de Zernike (PZ) et harmoniques sphériques (PHS) – et deux modèles de fonctions rationnelles (R) : fonctions rationnelles de Zernike (RZ) et fonctions rationnelles d’harmoniques sphériques (RSH). Il est connu que les modèles PHS et RZ sont plus précis que les modèles PZ pour un même nombre de coefficients (J), mais on ignore si les modèles PHS performent mieux que les modèles RZ, et si, de manière plus générale, les modèles SH sont plus précis que les modèles R, ou l’inverse. Et prenant en compte leur temps de traitement, est-ce que les modèles les plus précis demeurent les plus avantageux? Considérant des valeurs de J (nombre de coefficients du modèle) relativement basses pour respecter les contraintes de dimensionnalité propres aux taches d’apprentissage automatique, nous avons établi que les modèles HS (PHS et RHS) étaient tous deux plus précis que les modèles Z correspondants (PZ et RR), et que l’avantage de précision conféré par les modèles HS était plus important que celui octroyé par les modèles R. Par ailleurs, les courbes de temps de traitement en fonction de J démontrent qu’alors que les modèles P sont traités en temps quasi-linéaires, les modèles R le sont en temps polynomiaux. Ainsi, le modèle SHR est le plus précis, mais aussi le plus lent (un problème qui peut en partie être remédié en appliquant une procédure de pré-optimisation). Le modèle ZP était de loin le plus rapide, et il demeure une option intéressante pour le développement de projets. SHP constitue le meilleur compromis entre la précision et la rapidité.
La classification des cornées selon des paramètres cliniques a une longue tradition, mais la visualisation des effets moyens de ces paramètres sur la forme de la cornée par des cartes topographiques est plus récente. Dans la seconde étude (Chapitre 3), nous avons construit un atlas de cartes d’élévations moyennes pour différentes variables cliniques qui pourrait s’avérer utile pour l’évaluation et l’interprétation des données d’entrée (bases de données) et de sortie (prédictions, clusters, etc.) dans des tâches d’apprentissage automatique, entre autres. Une base de données constituée de plusieurs milliers de surfaces cornéennes antérieures normales enregistrées sous forme de matrices d’élévation de 101 by 101 points a d’abord été traitée par modélisation géométrique pour réduire sa dimensionnalité à un nombre de coefficients optimal dans une optique d’apprentissage automatique. Les surfaces ainsi modélisées ont été regroupées en fonction de variables cliniques de forme, de réfraction et de démographie. Puis, pour chaque groupe de chaque variable clinique, une surface moyenne a été calculée et représentée sous forme de carte d’élévations faisant référence à sa SMA (sphère la mieux ajustée). Après avoir validé la conformité de la base de donnée avec la littérature par des tests statistiques (ANOVA), l’atlas a été vérifié cliniquement en examinant si les transformations de formes cornéennes présentées dans les cartes pour chaque variable étaient conformes à la littérature. C’était le cas. Les applications possibles d’un tel atlas sont discutées.
La troisième étude (Chapitre 4) traite de la classification non-supervisée (clustering) de surfaces cornéennes antérieures normales. Le clustering cornéen un domaine récent en ophtalmologie. La plupart des études font appel aux techniques d’extraction des caractéristiques pour réduire la dimensionnalité de la base de données cornéennes. Le but est généralement d’automatiser le processus de diagnostique cornéen, en particulier en ce qui a trait à la distinction entre les cornées normales et les cornées irrégulières (kératocones, Fuch, etc.), et dans certains cas, de distinguer différentes sous-classes de cornées irrégulières. L’étude de clustering proposée ici se concentre plutôt sur les cornées normales afin de mettre en relief leurs regroupements naturels. Elle a recours à la modélisation géométrique pour réduire la dimensionnalité de la base de données, utilisant des polynômes de Zernike, connus pour leur interprétativité transparente (chaque terme polynomial est associé à une caractéristique cornéenne particulière) et leur bonne précision pour les cornées normales. Des méthodes de différents types ont été testées lors de prétests (méthodes de clustering dur (hard) ou souple (soft), linéaires or non-linéaires. Ces méthodes ont été testées sur des surfaces modélisées naturelles (non-normalisées) ou normalisées avec ou sans traitement d’extraction de traits, à l’aide de différents outils d’évaluation (scores de séparabilité et d’homogénéité, représentations par cluster des coefficients de modélisation et des surfaces modélisées, comparaisons statistiques des clusters sur différents paramètres cliniques). Les résultats obtenus par la meilleure méthode identifiée, k-means sans extraction de traits, montrent que les clusters produits à partir de surfaces cornéennes naturelles se distinguent essentiellement en fonction de la courbure de la cornée, alors que ceux produits à partir de surfaces normalisées se distinguent en fonction de l’axe cornéen.
La dernière étude présentée dans cette thèse (Chapitre 5) explore différentes techniques d’apprentissage automatique pour prédire la forme de la cornée à partir de données cliniques. La base de données cornéennes a d’abord été traitée par modélisation géométrique (polynômes de Zernike) pour réduire sa dimensionnalité à de courts vecteurs de 12 à 20 coefficients, une fourchette de valeurs potentiellement optimales pour effectuer de bonnes prédictions selon des prétests. Différentes méthodes de régression non-linéaires, tirées de la bibliothèque scikit-learn, ont été testées, incluant gradient boosting, Gaussian process, kernel ridge, random forest, k-nearest neighbors, bagging, et multi-layer perceptron. Les prédicteurs proviennent des variables cliniques disponibles dans la base de données, incluant des variables géométriques (diamètre horizontal de la cornée, profondeur de la chambre cornéenne, côté de l’œil), des variables de réfraction (cylindre, sphère et axe) et des variables démographiques (âge, genre). Un test de régression a été effectué pour chaque modèle de régression, défini comme la sélection d’une des 256 combinaisons possibles de variables cliniques (les prédicteurs), d’une méthode de régression, et d’un vecteur de coefficients de Zernike d’une certaine taille (entre 12 et 20 coefficients, les cibles). Tous les modèles de régression testés ont été évalués à l’aide de score de RMSE établissant la distance entre les surfaces cornéennes prédites (les prédictions) et vraies (les topographies corn¬éennes brutes). Les meilleurs d’entre eux ont été validés sur l’ensemble de données randomisé 20 fois pour déterminer avec plus de précision lequel d’entre eux est le plus performant. Il s’agit de gradient boosting utilisant toutes les variables cliniques comme prédicteurs et 16 coefficients de Zernike comme cibles. Les prédictions de ce modèle ont été évaluées qualitativement à l’aide d’un atlas de cartes d’élévations moyennes élaborées à partir des variables cliniques ayant servi de prédicteurs, qui permet de visualiser les transformations moyennes d’en groupe à l’autre pour chaque variables. Cet atlas a permis d’établir que les cornées prédites moyennes sont remarquablement similaires aux vraies cornées moyennes pour toutes les variables cliniques à l’étude.This thesis aims to investigate the best conditions in which the anterior corneal surface of normal
corneas can be preprocessed, classified and predicted using geometric modeling (GM) and machine
learning (ML) techniques. The focus is on the anterior corneal surface, which is the main
responsible of the refractive power of the cornea.
Dealing with preprocessing, the first study (Chapter 2) examines the conditions in which GM
can best be applied to reduce the dimensionality of a dataset of corneal surfaces to be used in ML
projects. Four types of geometric models of corneal shape were tested regarding their accuracy and
processing time: two polynomial (P) models – Zernike polynomial (ZP) and spherical harmonic
polynomial (SHP) models – and two corresponding rational function (R) models – Zernike rational
function (ZR) and spherical harmonic rational function (SHR) models. SHP and ZR are both known
to be more accurate than ZP as corneal shape models for the same number of coefficients, but which
type of model is the most accurate between SHP and ZR? And is an SHR model, which is both an
SH model and an R model, even more accurate? Also, does modeling accuracy comes at the cost
of the processing time, an important issue for testing large datasets as required in ML projects?
Focusing on low J values (number of model coefficients) to address these issues in consideration
of dimensionality constraints that apply in ML tasks, it was found, based on a number of evaluation
tools, that SH models were both more accurate than their Z counterparts, that R models were both
more accurate than their P counterparts and that the SH advantage was more important than the R
advantage. Processing time curves as a function of J showed that P models were processed in quasilinear time, R models in polynomial time, and that Z models were fastest than SH models.
Therefore, while SHR was the most accurate geometric model, it was the slowest (a problem that
can partly be remedied by applying a preoptimization procedure). ZP was the fastest model, and
with normal corneas, it remains an interesting option for testing and development, especially for
clustering tasks due to its transparent interpretability. The best compromise between accuracy and
speed for ML preprocessing is SHP.
The classification of corneal shapes with clinical parameters has a long tradition, but the
visualization of their effects on the corneal shape with group maps (average elevation maps,
standard deviation maps, average difference maps, etc.) is relatively recent. In the second study
(Chapter 3), we constructed an atlas of average elevation maps for different clinical variables
(including geometric, refraction and demographic variables) that can be instrumental in the
evaluation of ML task inputs (datasets) and outputs (predictions, clusters, etc.). A large dataset of
normal adult anterior corneal surface topographies recorded in the form of 101Ă—101 elevation
matrices was first preprocessed by geometric modeling to reduce the dimensionality of the dataset
to a small number of Zernike coefficients found to be optimal for ML tasks. The modeled corneal
surfaces of the dataset were then grouped in accordance with the clinical variables available in the
dataset transformed into categorical variables. An average elevation map was constructed for each
group of corneal surfaces of each clinical variable in their natural (non-normalized) state and in
their normalized state by averaging their modeling coefficients to get an average surface and by
representing this average surface in reference to the best-fit sphere in a topographic elevation map.
To validate the atlas thus constructed in both its natural and normalized modalities, ANOVA tests
were conducted for each clinical variable of the dataset to verify their statistical consistency with
the literature before verifying whether the corneal shape transformations displayed in the maps
were themselves visually consistent. This was the case. The possible uses of such an atlas are
discussed.
The third study (Chapter 4) is concerned with the use of a dataset of geometrically modeled
corneal surfaces in an ML task of clustering. The unsupervised classification of corneal surfaces is
recent in ophthalmology. Most of the few existing studies on corneal clustering resort to feature
extraction (as opposed to geometric modeling) to achieve the dimensionality reduction of the dataset. The goal is usually to automate the process of corneal diagnosis, for instance by
distinguishing irregular corneal surfaces (keratoconus, Fuch, etc.) from normal surfaces and, in
some cases, by classifying irregular surfaces into subtypes. Complementary to these corneal
clustering studies, the proposed study resorts mainly to geometric modeling to achieve
dimensionality reduction and focuses on normal adult corneas in an attempt to identify their natural
groupings, possibly in combination with feature extraction methods. Geometric modeling was
based on Zernike polynomials, known for their interpretative transparency and sufficiently accurate
for normal corneas. Different types of clustering methods were evaluated in pretests to identify the
most effective at producing neatly delimitated clusters that are clearly interpretable. Their
evaluation was based on clustering scores (to identify the best number of clusters), polar charts and
scatter plots (to visualize the modeling coefficients involved in each cluster), average elevation
maps and average profile cuts (to visualize the average corneal surface of each cluster), and
statistical cluster comparisons on different clinical parameters (to validate the findings in reference
to the clinical literature). K-means, applied to geometrically modeled surfaces without feature
extraction, produced the best clusters, both for natural and normalized surfaces. While the clusters
produced with natural corneal surfaces were based on the corneal curvature, those produced with
normalized surfaces were based on the corneal axis. In each case, the best number of clusters was
four. The importance of curvature and axis as grouping criteria in corneal data distribution is
discussed.
The fourth study presented in this thesis (Chapter 5) explores the ML paradigm to verify whether
accurate predictions of normal corneal shapes can be made from clinical data, and how. The
database of normal adult corneal surfaces was first preprocessed by geometric modeling to reduce
its dimensionality into short vectors of 12 to 20 Zernike coefficients, found to be in the range of
appropriate numbers to achieve optimal predictions. The nonlinear regression methods examined
from the scikit-learn library were gradient boosting, Gaussian process, kernel ridge, random forest,
k-nearest neighbors, bagging, and multilayer perceptron. The predictors were based on the clinical
variables available in the database, including geometric variables (best-fit sphere radius, white-towhite diameter, anterior chamber depth, corneal side), refraction variables (sphere, cylinder, axis)
and demographic variables (age, gender). Each possible combination of regression method, set of
clinical variables (used as predictors) and number of Zernike coefficients (used as targets) defined
a regression model in a prediction test. All the regression models were evaluated based on their
mean RMSE score (establishing the distance between the predicted corneal surfaces and the raw
topographic true surfaces). The best model identified was further qualitatively assessed based on
an atlas of predicted and true average elevation maps by which the predicted surfaces could be
visually compared to the true surfaces on each of the clinical variables used as predictors. It was
found that the best regression model was gradient boosting using all available clinical variables as
predictors and 16 Zernike coefficients as targets. The most explicative predictor was the best-fit
sphere radius, followed by the side and refractive variables. The average elevation maps of the true
anterior corneal surfaces and the predicted surfaces based on this model were remarkably similar
for each clinical variable
Comparative analysis of some modeal reconstruction methods of the cornea from corneal elevation data
Purpose. A comparative study of the ability of some modal schemes to reproduce corneal shapes of varying complexity is performed, using both standard radial polynomials and the radial basis functions (RBF). Our claim is that the correct approach in the case of highly irregular corneas should combine several bases. Methods. Standard approaches of reconstruction by Zernike and other types of radial polynomials are compared with the discrete least squares fit (LSF) by the RBF in three theoretical surfaces, synthetically generated by computer algorithms in the lack of measurement noise. For the reconstruction by polynomials the maximal radial order 6 was chosen, which corresponds to the first 28 Zernike polynomials or the first 49 Bhatia-Wolf polynomials. The fit with the RBF has been carried out using a regular grid of centers. Results. The quality of fit was assessed by computing for each surface the mean square errors (MSE) of the reconstruction by LSF, measured at the same nodes where the heights were collected. Another criterion of the fitting quality used was the accuracy in recovery of the Zernike coefficients, especially in the case of incomplete data. Conclusions. The Zernike (and especially, the Bhatia-Wolf) polynomials constitute a reliable reconstruction method of a non-severely aberrated surface with a small surface regularity index (SRI). However, they fail to capture small deformations of the anterior surface of a synthetic cornea. The most promising is a combined approach that balances the robustness of the Zernike fit with the localization of the RBF
Comparison of parametric methods for modeling corneal surfaces
Corneal topography is a medical imaging technique to get the 3D shape of the cornea as a set of 3D points of its anterior
and posterior surfaces. From these data, topographic maps can be derived to assist the ophthalmologist in the diagnosis of
disorders. In this paper, we compare three different mathematical parametric representations of the corneal surfaces leastsquares fitted to the data provided by corneal topography. The parameters obtained from these models reduce the
dimensionality of the data from several thousand 3D points to only a few parameters and could eventually be useful for
diagnosis, biometry, implant design etc. The first representation is based on Zernike polynomials that are commonly used
in optics. A variant of these polynomials, named Bhatia-Wolf will also be investigated. These two sets of polynomials are
defined over a circular domain which is convenient to model the elevation (height) of the corneal surface. The third
representation uses Spherical Harmonics that are particularly well suited for nearly-spherical object modeling, which is
the case for cornea. We compared the three methods using the following three criteria: the root-mean-square error (RMSE),
the number of parameters and the visual accuracy of the reconstructed topographic maps. A large dataset of more than
2000 corneal topographies was used. Our results showed that Spherical Harmonics were superior with a RMSE mean
lower than 2.5 microns with 36 coefficients (order 5) for normal corneas and lower than 5 microns for two diseases
affecting the corneal shapes: keratoconus and Fuchs’ dystrophy
An adaptive algorithm for the cornea modeling from keratometric data
In this paper we describe an adaptive and multi-scale algorithm for the parsimonious t of the corneal surface data that allows to adapt the number of functions used in the reconstruction to the conditions of each cornea. The method implements also a dynamical selection of the parameters and the management of noise. It can be used for the real-time reconstruction of both altimetric data and corneal power maps from the data collected by keratoscopes, such as the Placido rings based topographers, decisive for an early detection of corneal diseases such as keratoconus. Numerical experiments show that the algorithm exhibits a steady exponential error decay, independently of the level of aberration of the cornea. The complexity of each anisotropic gaussian basis functions in the functional representation is the same, but their parameters vary to fit the current scale. This scale is determined only by the residual errors and not by the number of the iteration. Finally, the position and clustering of their centers, as well as the size of the shape parameters, provides an additional spatial information about the regions of higher irregularity. These results are compared with the standard approximation procedures based on the Zernike polynomials expansions
Comparative Analysis of Some Modal Reconstruction Methods of the Shape of the Cornea from Corneal Elevation Data
Purpose: A comparative study of the ability of some modal schemes to reproduce corneal shapes of varying complexity was performed, by using both standard radial polynomials and radial basis functions (RBFs). The hypothesis was that the correct approach in the case of highly irregular corneas should combine several bases.
Methods: Standard approaches of reconstruction by Zernike and other types of radial polynomials were compared with the discrete least-squares fit (LSF) by the RBF in three theoretical surfaces, synthetically generated by computer algorithms in the absence of measurement noise. For the reconstruction by polynomials, the maximal radial order 6 was chosen, which corresponds to the first 28 Zernike polynomials or the first 49 Bhatia-Wolf polynomials. The fit with the RBF was performed by using a regular grid of centers.
Results: The quality of fit was assessed by computing for each surface the mean square errors (MSEs) of the reconstruction by LSF, measured at the same nodes where the heights were collected. Another criterion of the fit quality used was the accuracy in recovery of the Zernike coefficients, especially in the case of incomplete data.
Conclusions: The Zernike (and especially, the Bhatia-Wolf) polynomials constitute a reliable reconstruction method of a nonseverely aberrated surface with a small surface regularity index (SRI). However, they fail to capture small deformations of the anterior surface of a synthetic cornea. The most promising approach is a combined one that balances the robustness of the Zernike fit with the localization of the RBF
Procedimiento de reconstrucciĂłn de la topografĂa corneal a partir de datos altĂmetros o de curvatura
NĂşmero de publicaciĂłn: ES2392619 A1 (12.12.2012) TambiĂ©n publicado como: ES2392619 B1 (22.10.2013) NĂşmero de Solicitud: Consulta de Expedientes OEPM (C.E.O.) P201000842(08.06.2010)Procedimiento de reconstrucciĂłn de la topografĂa corneal a partir de datos altimĂ©tricos o de curvatura. La invenciĂłn consiste en un mĂ©todo de reconstrucciĂłn de la superficie de la cara anterior de la cĂłrnea, a partir de los datos medidos en un conjunto discreto de puntos por medio de un topĂłgrafo corneal o equipo equivalente. Se trata de un procedimiento que obtiene una expresiĂłn analĂtica de la superficie, combinando un ajuste por polinomios de Zernike o con esfera de mejor ajuste, con una reconstrucciĂłn por funciones de base radial gaussianas. Se logra obtener una descripciĂłn detallada de la superficie corneal, permitiendo un diagnĂłstico más fiable de patologĂas, o la implementaciĂłn de tratamientos customizados. Este procedimiento es fácilmente implementable en cualquier topĂłgrafo corneal, tomĂłgrafo de coherencia Ăłptica, equipos de lámpara de hendidura y equivalentes, de los existentes en el mercado, como sustituto del mĂ©todo estándar basado en polinomios de Zernike.Universidad de AlmerĂ
Optimal sampling patterns for Zernike polynomials
A pattern of interpolation nodes on the disk is studied, for which the inter-
polation problem is theoretically unisolvent, and which renders a minimal
numerical condition for the collocation matrix when the standard basis of
Zernike polynomials is used. It is shown that these nodes have an excellent
performance also from several alternative points of view, providing a numer-
ically stable surface reconstruction, starting from both the elevation and the
slope data. Sampling at these nodes allows for a more precise recovery of the
coefficients in the Zernike expansion of a wavefront or of an optical surface.
Keywords:
Interpolation
Numerical condition
Zernike polynomials
Lebesgue constant
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