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**11 - 14**of**14**### Part II Learning and interpolation 115 Chapter 7 Learning

"... Learning and interpolation are two approaches to solve the problem of how to build a reasonable estimate of an unknown function on the basis of a finite number of samples. Such problems arise in various frameworks ranging from partial differential equations through geometric modeling in image synthe ..."

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Learning and interpolation are two approaches to solve the problem of how to build a reasonable estimate of an unknown function on the basis of a finite number of samples. Such problems arise in various frameworks ranging from partial differential equations through geometric modeling in image synthesis to learning and adaptive control. In this chapter, we present an overview of various existing methods the purpose of which is to estimate functions from samples. 7.1 The learning problem The classical problem that each method has to solve can be stated in the following way: given a number of measures (xn, yn) ∈ R d × R, for n = 1... N, we want to find a function f mapping R d to R such that f(xn) = yn for n = 1... N 7.1.1 What is the best solution? Such a problem can easily be solved in the framework of parametric estimation, where the unknown function is determined by a small number of parameters (as in linear regression). In such a case, the underlying linear system is overdetermined and has usually no solution. A model of measures with a Gaussian noise can be used: yn = f(xn) + ɛn where the ɛn are i.i.d. Gaussian variables of zero mean and of standard deviation σ. Regression consists in finding the parameter combination that maximizes the likelihood (conditional density w.r.t the function f). In the Gaussian case, we maximize

### Merger of Ocean Color Data from Multiple Satellite Missions within the SIMBIOS Project

"... The purpose of data merger activities undertaken by the National Aeronautic and Space Administration’s (NASA) Sensor Intercomparison and Merger for Biological and Interdisciplinary Studies (SIMBIOS) Project is to create scientific quality ocean color data encompassing measurements from multiple sate ..."

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The purpose of data merger activities undertaken by the National Aeronautic and Space Administration’s (NASA) Sensor Intercomparison and Merger for Biological and Interdisciplinary Studies (SIMBIOS) Project is to create scientific quality ocean color data encompassing measurements from multiple satellite missions. The fusion of data from multiple satellites will improve the quality of ocean color products over single-mission data sets by expanding spatial and temporal coverage of the world’s oceans and increasing statistical confidence in generated parameters. The merger will also support a variety of new applications by taking advantage of sensor-varying calibration, spectral, spatial, temporal, and ground coverage characteristics. Leading to the data merger goals, the SIMBIOS Project has established a thorough ocean color validation program and has been cross-comparing and cross-calibrating sensor data with in situ measurements and data among the missions. The SIMBIOS Science Team has been studying data merger algorithms based on spectral data assimilation and spatial interpolation. The SIMBIOS Project Office has implemented statistical objective analysis and regression techniques based on artificial neural networks and support vector machines. The accuracy of the merger methods will be evaluated using in situ data, statistical analyses, and simple chlorophyll means – the method already implemented within the SIMBIOS Project. This paper defines challenges and suggests solutions for data merger based on the example of daily chlorophyll concentration products from Moderate Resolution Imaging

### A General Framework for Image Kernel Engineering

"... Understanding image content is a long standing problem of computer science. Despite decades of research in computer vision, an effective solution to this problem does not appear to be in sight. Recent advances in the theory of learning by examples indicate that devising systems which can be trained ..."

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Understanding image content is a long standing problem of computer science. Despite decades of research in computer vision, an effective solution to this problem does not appear to be in sight. Recent advances in the theory of learning by examples indicate that devising systems which can be trained instead of programmed to solve this problem is an interesting alternative to solutions constructed from higher level image analysis and description. In this thesis we consider a number of image understanding problems viewed as classification problems for which a certain number of input/output pairs is given. Within the statistical learning schemes we adopt (binary support vector machines and one-class support vector machines), the solution to each problem is written as a linear combination of certain functions, named kernel functions. These functions, which satisfy some specific mathematical properties, are evaluated on input pairs and encode the prior knowledge on the problem domain. Roughly speaking, kernel functions can be thought of as measuring the similarity between input pairs by extracting certain features from the raw data.