Approximation de fonctions et de données discrètes au sens de la norme L1 par splines polynomiales

Data and function approximation is fundamental in application domains like path planning or signal processing (sensor data). In such domains, it is important to obtain curves that preserve the shape of the data. Considering the results obtained for the problem of data interpolation, L1 splines appear to be a good solution. Contrary to classical L2 splines, these splines enable to preserve linearities in the data and to not introduce extraneous oscillations when applied on data sets with abrupt changes. We propose in this dissertation a study of the problem of best L1 approximation. This study includes developments on best L1 approximation of functions with a jump discontinuity in general spaces called Chebyshev and weak-Chebyshev spaces. Polynomial splines fit in this framework. Approximation algorithms by smoothing splines and spline fits based on a sliding window process are introduced. The methods previously proposed in the littérature can be relatively time consuming when applied on large datasets. Sliding window algorithm enables to obtain algorithms with linear complexity. Moreover, these algorithms can be parallelized. Finally, a new approximation approach with prescribed error is introduced. A pure algebraic algorithm with linear complexity is introduced. This algorithm is then applicable to real-time application.L'approximation de fonctions et de données discrètes est fondamentale dans des domaines tels que la planification de trajectoire ou le traitement du signal (données issues de capteurs). Dans ces domaines, il est important d'obtenir des courbes conservant la forme initiale des données. L'utilisation des splines L1 semble être une bonne solution au regard des résultats obtenus pour le problème d'interpolation de données discrètes par de telles splines. Ces splines permettent notamment de conserver les alignements dans les données et de ne pas introduire d'oscillations résiduelles comme c'est le cas pour les splines d'interpolation L2. Nous proposons dans cette thèse une étude du problème de meilleure approximation au sens de la norme L1. Cette étude comprend des développements théoriques sur la meilleure approximation L1 de fonctions présentant une discontinuité de type saut dans des espaces fonctionnels généraux appelés espace de Chebyshev et faiblement Chebyshev. Les splines polynomiales entrent dans ce cadre. Des algorithmes d'approximation de données discrètes au sens de la norme L1 par procédé de fenêtre glissante sont développés en se basant sur les travaux existants sur les splines de lissage et d'ajustement. Les méthodes présentées dans la littérature pour ces types de splines peuvent être relativement couteuse en temps de calcul. Les algorithmes par fenêtre glissante permettent d'obtenir une complexité linéaire en le nombre de données. De plus, une parallélisation est possible. Enfin, une approche originale d'approximation, appelée interpolation à delta près, est développée. Nous proposons un algorithme algébrique avec une complexité linéaire et qui peut être utilisé pour des applications temps réel

Approximation spline L1C1 par fenêtres glissantes pour le signal et l'image

National audienceDans cet article, nous traitons le problème d'approximation de nuages de points par une courbe spline ou surface au sens de la norme L1. L'utilisation de cette norme permet de préserver la forme des données même en cas de changement brutal de celle-ci. Dans nos précédents travaux, nous avons introduit une méthode par fenêtre glissante de cinq points pour l'approximation courbe spline L1 et une méthode de croix glissante de neuf points pour l'approximation surface spline L1 de données type grille. Malgré leur complexité linéaire, ces méthodes peuvent demeurer lentes lorsqu'elles sont appliquées sur un large flot de données. Par conséquent, sur la base de nouveaux résultats algébriques sur l'approximation L1 sur un nombre restreint de données, nous proposons ici des méthodes reposant sur des fenêtres de taille inférieure et nous comparons les différentes méthodes. In this article, we adress the problem of approximating scattered data points by C1-smooth polynomial spline curves and surfaces using L1-norm optimization. The use of this norm helps us to preserve the shape of the data even near to abrupt changes. In our previous work, we introduced a five-point sliding window process for L1 spline curve approximation and a nine-point cross sliding window process for L1 spline surface approximation of grid datasets. Nethertheless, these methods can be still time consuming despite their linear complexity. Consequently, based on new algebraic results obtained for L1 approximation on restricted sets of points in both planar and spatial cases, we define in this article methods with smaller windows and we lead a comparison between the methods

Path planning with PH G2 splines in R2

International audienceIn this article, we justify the use of parametric planar Pythagorean Hodograph spline curves in path planning. The elegant properties of such splines enable us to design an efficient interpolator algorithm, more precise than the classical Taylor interpolators and faster than an interpolator based on arc length computations

A convolutional neural network to detect scoliosis treatment in radiographs

Purpose The aim of this work is to propose a classification algorithm to automatically detect treatment for scoliosis (brace, implant or no treatment) in postero-anterior radiographs. Such automatic labelling of radiographs could represent a step towards global automatic radiological analysis. Methods Seven hundred and ninety-six frontal radiographies of adolescents were collected (84 patients wearing a brace, 325 with a spinal implant and 387 reference images with no treatment). The dataset was augmented to a total of 2096 images. A classification model was built, composed by a forward convolutional neural network (CNN) followed by a discriminant analysis; the output was a probability for a given image to contain a brace, a spinal implant or none. The model was validated with a stratified tenfold cross-validation procedure. Performance was estimated by calculating the average accuracy. Results 98.3% of the radiographs were correctly classified as either reference, brace or implant, excluding 2.0% unclassified images. 99.7% of brace radiographs were correctly detected, while most of the errors occurred in the reference group (i.e. 2.1% of reference images were wrongly classified). Conclusion The proposed classification model, the originality of which is the coupling of a CNN with discriminant analysis, can be used to automatically label radiographs for the presence of scoliosis treatment. This information is usually missing from DICOM metadata, so such method could facilitate the use of large databases. Furthermore, the same model architecture could potentially be applied for other radiograph classifications, such as sex and presence of scoliotic deformity.Acknowledgements The authors are grateful to the ParisTech BiomecAM chair program on subject-specific musculoskeletal modelling (with the support of ParisTech and Yves Cotrel Foundations, Société Générale, Proteor and Covea)

Vertebral rotation estimation from frontal X-rays using a quasi-automated pedicle detection method

Purpose Measurement of vertebral axial rotation (VAR) is relevant for the assessment of scoliosis. Stokes method allows estimating VAR in frontal X-rays from the relative position of the pedicles and the vertebral body. This method requires identifying these landmarks for each vertebral level, which is time-consuming. In this work, a quasi-automated method for pedicle detection and VAR estimation was proposed. Method A total of 149 healthy and adolescent idiopathic scoliotic (AIS) subjects were included in this retrospective study. Their frontal X-rays were collected from multiple sites and manually annotated to identify the spinal midline and pedicle positions. Then, an automated pedicle detector was developed based on image analysis, machine learning and fast manual identification of a few landmarks. VARs were calculated using the Stokes method in a validation dataset of 11 healthy (age 6–33 years) and 46 AIS subjects (age 6–16 years, Cobb 10°–46°), both from detected pedicles and those manually annotated to compare them. Sensitivity of pedicle location to the manual inputs was quantified on 20 scoliotic subjects, using 10 perturbed versions of the manual inputs. Results Pedicles centers were localized with a precision of 84% and mean difference of 1.2 ± 1.2 mm, when comparing with manual identification. Comparison of VAR values between automated and manual pedicle localization yielded a signed difference of − 0.2 ± 3.4°. The uncertainty on pedicle location was smaller than 2 mm along each image axis. Conclusion The proposed method allowed calculating VAR values in frontal radiographs with minimal user intervention and robust quasi-automated pedicle localization.The authors are grateful to the ParisTech BiomecAM chair program on subject-specific musculoskeletal modeling for funding (with the support of ParisTech and Yves Cotrel Foundations, Société Générale, Proteor and Covea)

Effect of Face Blurring on Human Pose Estimation: Ensuring Subject Privacy for Medical and Occupational Health Applications

The face blurring of images plays a key role in protecting privacy. However, in computer vision, especially for the human pose estimation task, machine-learning models are currently trained, validated, and tested on original datasets without face blurring. Additionally, the accuracy of human pose estimation is of great importance for kinematic analysis. This analysis is relevant in areas such as occupational safety and clinical gait analysis where privacy is crucial. Therefore, in this study, we explore the impact of face blurring on human pose estimation and the subsequent kinematic analysis. Firstly, we blurred the subjects’ heads in the image dataset. Then we trained our neural networks using the face-blurred and the original unblurred dataset. Subsequently, the performances of the different models, in terms of landmark localization and joint angles, were estimated on blurred and unblurred testing data. Finally, we examined the statistical significance of the effect of face blurring on the kinematic analysis along with the strength of the effect. Our results reveal that the strength of the effect of face blurring was low and within acceptable limits (<1°). We have thus shown that for human pose estimation, face blurring guarantees subject privacy while not degrading the prediction performance of a deep learning model

Automatic Segmentation and Identification of Spinous Processes on Sagittal X-Rays Based on Random Forest Classification and Dedicated Contextual Features

X-ray based quantitative analysis of spine parameters is required in routine diagnosis or treatment planning. Existing tools commonly require manual intervention. Attempts towards automation of the whole procedure have mainly focused on vertebral bodies, whereas other regions such as the posterior arch also bear considerable amount of useful information. In this study, we combine a specific design of contextual visual features with a multi-class Random Forest classifier to perform pixel-wise segmentation and identification of all cervical spine spinous processes, on sagittal radiographs. Segmentations were evaluated on 62 radiographs, comparing to manual tracing. Correct identification was obtained for all subjects, and segmentation returned mean SD values of: Dice coefficient =88 8%; Hausdorff distance =2.1 1.4 mm and; mean surface distance =0.6 0.4 mm. The derived geometric parameters can be used to reduce the amount of manual intervention needed for spine modeling or to measure clinical indices

Dissecting the brains of central bankers: the case of the ECB's Governing Council members on reforms

Since 2009, European central bankers have supported some reforms, in order to draw roadmaps to get out of the euro debt crisis. This paper tests whether the educational and professional background of European central bankers matter for the type of reforms each of them advocated. Through a textual analysis of public speeches delivered by the European central bankers, we draw a cognitive map for each of them and, thus, of the reforms they propose as ways out of the euro debt crisis. Our results show that their occupational background is an important determinant of their respective economic reform proposals