Pierre-Emmanuel Leni

Case-Based Reasoning adaptation of numerical representations of human organs by interpolation

Case-Based Reasoning (CBR) and interpolation tools can provide solutions to unknown problems by adapting solutions from other problems already solved. We propose a generic approach using an interpolation tool during the CBR-adaptation phase. The application EquiVox, which attempts to design three dimensional representations of human organs according to external measurements, was modelled. It follows the CBR-cycle with its adaptation tool based on Artificial Neural Networks and its performances are evaluated and discussed. The results show that this adaptation tool meets the requirements of radiation protection experts who use such prototypes and also what the limits are of such tools in CBR-adaptation. When adaptations are guided by experience grained through trial and error by experts, interpolation tools become well-suited methods for automatically and quickly providing adaptation strategies and knowledge through training phases.

Kolmogorov Superposition Theorem and Its Application to Multivariate Function Decompositions and Image Representation

In this paper, we present the problem of multivariate function decompositions into sums and compositions of monovariate functions. We recall that such a decomposition exists in the Kolmogorovs superposition theorem, and we present two of the most recent constructive algorithms of these monovariate functions. We first present the algorithm proposed by Sprecher, then the algorithm proposed by Igelnik, and we present several results of decomposition for gray level images. Our goal is to adapt and apply the superposition theorem to image processing, i.e. to decompose an image into simpler functions using Kolmogorov superpositions. We synthetise our observations, before presenting several research perspectives.

Kolmogorov Superposition Theorem and its application to wavelet image decompositions

This paper deals with the decomposition of multivariate functions into sums and compositions of monovariate functions. The global purpose of this work is to find a suitable strategy to express complex multivariate functions using simpler functions that can be analyzed using well know techniques, instead of developing complex Ndimensional tools. More precisely, most of signal processing techniques are applied in 1D or 2D and cannot easily be extended to higher dimensions. We recall that such a decomposition exists in the Kolmogorovs superposition theorem. According to this theorem, any multivariate function can be decomposed into two types of univariate functions, that are called inner and external functions. Inner functions are associated to each dimension and linearly combined to construct a hash-function that associates every point of a multidimensional space to a value of the real interval [0, 1]. Every inner function is the argument for one external function. The external functions associate real values in [0, 1] to the image by the multivariate function of the corresponding point of the multidimensional space. Sprecher, in Ref. 1, has proved that internal functions can be used to construct space filling curves, i.e. there exists a curve that sweeps the multidimensional space and uniquely matches corresponding values into [0, 1]. Our goal is to obtain both a new decomposition algorithm for multivariate functions (at least bi-dimensional) and adaptive space filling curves. Two strategies can be applied. Either we construct fixed internal functions to obtain space filling curves, which allows us to construct an external function such that their sums and compositions exactly correspond to the multivariate function; or the internal function is constructed by the algorithm and is adapted to the multivariate function, providing different space filling curves for different multivariate functions. We present two of the most recent constructive algorithms of monovariate functions. The first method is due to Sprecher (Ref. 2 and Ref. 3). We provide additional explanations to the existing algorithm and present several decomposition results for gray level images. We point out the main drawback of this method: all the function parameters are fixed, so the univariate functions cannot be modified; precisely, the inner function cannot be modified and so the space filling curve. The number of layers depends on the dimension of the decomposed function. The second algorithm, proposed by Igelnik in Ref. 4, increases the parameters flexibility, but only approximates the monovariate functions: the number of layers is variable, a neural networks optimizes the monovariate functions and the weights associated to each layer to ensure convergence to the decomposed multivariate function. We have implemented both Sprechers and Igelniks algorithms and present the results of the decompositions of gray level images. There are artifacts in the reconstructed images, which leads us to apply the algorithm on wavelet decomposition images. We detail the reconstruction quality and the quantity of information contained in Igelniks network.

Adapting numerical representations of lung contours using Case-Based Reasoning and Artificial Neural Networks

In case of a radiological emergency situation involving accidental human exposure, a dosimetry evaluation must be established as soon as possible. In most cases, this evaluation is based on numerical representations and models of subjects. Unfortunately, personalised and realistic human representations are often unavailable for the exposed subjects. However, accuracy of treatment depends on the similarity of the phantom to the subject. The EquiVox platform (Research of Equivalent Voxel phantom) developed in this study uses Case-Based Reasoning principles to retrieve and adapt, from among a set of existing phantoms, the one to represent the subject. This paper introduces the EquiVox platform and Artificial Neural Networks developed to interpolate the subject’s 3D lung contours. The results obtained for the choice and construction of the contours are presented and discussed.

Kolmogorov Superposition Theorem and Wavelet Decomposition for Image Compression

Kolmogorov Superposition Theorem stands that any multivariate function can be decomposed into two types of monovariate functions that are called inner and external functions: each inner function is associated to one dimension and linearly combined to construct a hash-function that associates every point of a multidimensional space to a value of the real interval [0,1]. These intermediate values are then associated by external functions to the corresponding value of the multidimensional function. Thanks to the decomposition into monovariate functions, our goal is to apply this decomposition to images and obtain image compression.

An iterative precision vector to optimise the CBR adaptation of EquiVox

The case-based reasoning (CBR) approach consists in retrieving solutions from similar past problems and adapting them to new ones. Interpolation tools can easily be used as adaptation tools in CBR systems. The accuracies of interpolated results depend on the set of known solved problems with which the interpolation tools have previously been trained. To be sufficiently accurate, an interpolation tool must be trained with a large number of known cases. However, CBR systems are also relevant if the number of known cases is restricted. In addition, the training of interpolation tools is generally seen by users as a black box. This paper presents a generic method to optimise CBR adaptations driven by trained interpolation tools and also takes into account remarks made by users about known solution accuracy. This method was applied to the CBR system called EquiVox which retrieves, reuses (interpolates), revises and retains three-dimensional numerical representations of organ contours and thus enhances its own performance.

New adaptive and progressive image transmission approach using function superpositions

We present a novel approach to adaptive and progressive image transmission, based on the decomposition of an image into compositions and superpositions of monovariate functions. The monovariate functions are iteratively constructed and transmitted, one after the other, to progressively reconstruct the original image: the progressive transmission is performed directly in the 1D space of the monovariate functions and independently of any statistical properties of the image. Each monovariate function contains only a fraction of the pixels of the image. Each new transmitted monovariate function adds data to the previously transmitted monovariate functions. After each transmission step, by using the updated monovariate functions the image is reconstructed with an increased resolution. Finally, once all the monovariate functions have been transmitted, the original image is reconstructed exactly. This approach is characterized by its flexibility and robustness to packet loss: any numbers of intermediate transmissions and reconstructions are possible, and in case of packet loss, the global appearance of the transmitted image is preserved. Moreover, the intermediate images can be reconstructed at any resolution, and for any number of intermediate reconstructions, the original image will be exactly reconstructed. Finally, the quantity of data to be transmitted only depends on the image size and is independent of the number of intermediate reconstructions. Our main contributions are the modification of the decomposition scheme defined by the Kolmogorov superposition theorem to enable multiresolution image reconstructions and its application for progressive image transmission, using successively increasing resolutions. We illustrate this approach on several images and evaluate the reconstruction quality, decomposition flexibility, and error resilience during transmission.

Development of a 4D numerical chest phantom with customizable breathing

Respiratory movement information is useful for radiation therapy, and is generally obtained using 4D scanners (4DCT). In the interest of patient safety, reducing the use of 4DCT could be a significant step in reducing radiation exposure, the effects of which are not well documented. The authors propose a customized 4D numerical phantom representing the organ contours. Firstly, breathing movement can be simulated and customized according to the patients anthroporadiametric data. Using learning sets constituted by 4D scanners, artificial neural networks can be trained to interpolate the lung contours corresponding to an unknown patient, and then to simulate its respiration. Lung movement during the breathing cycle is modeled by predicting the lung contours at any respiratory phases. The interpolation is validated comparing the obtained lung contours with 4DCT via Dice coefficient. Secondly, a preliminary study of cardiac and œsophageal motion is also presented to demonstrate the flexibility of this approach. The application may simulate the position and volume of the lungs, the œsophagus and the heart at every phase of the respiratory cycle with a good accuracy: the validation of the lung modeling gives a Dice index greater than 0.93 with 4DCT over a breath cycle.

Progressive transmission of secured images with authentication using decompositions into monovariate functions

Abstract. We propose a progressive transmission approach of an image authenticated using an overlapping subimage that can be removed to restore the original image. Our approach is different from most visible watermarking approaches that allow one to later remove the watermark, because the mark is not directly introduced in the two-dimensional image space. Instead, it is rather applied to an equivalent monovariate representation of the image. Precisely, the approach is based on our progressive transmission approach that relies on a modified Kolmogorov spline network, and therefore inherits its advantages: resilience to packet losses during transmission and support of heterogeneous display environments. The marked image can be accessed at any intermediate resolution, and a key is needed to remove the mark to fully recover the original image without loss. Moreover, the key can be different for every resolution, and the images can be globally restored in case of packet losses during the transmission. Our contributions lie in the proposition of decomposing a mark (an overlapping image) and an image into monovariate functions following the Kolmogorov superposition theorem; and in the combination of these monovariate functions to provide a removable visible “watermarking” of images with the ability to restore the original image using a key.

New representations for multidimensional functions based on Kolmogorov superposition theorem. Applications on image processing

Mastering the sorting of the data in signal (nD) can lead to multiple applications like new compression, transmission, watermarking, encryption methods and even new processing methods for image. Some authors in the past decades have proposed to use these approaches for image compression, indexing, median filtering, mathematical morphology, encryption. A mathematical rigorous way for doing such a study has been introduced by Andrei Nikolaievitch Kolmogorov (1903-1987) in 1957 and recent results have provided constructive ways and practical algorithms for implementing the Kolmogorov theorem. We propose in this paper to present those algorithms and some preliminary results obtained by our team by applying them to image processing problems such as compression, progressive transmission and watermarking.