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A viscosity solution method for Shape-From-Shading without image boundary data

Emmanuel PradosFabio CamilliOlivier Faugeras — 2006

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we propose a solution of the Lambertian shape-from-shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solution. The power of our approach is twofolds: (1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Moreover, it allows to characterize the viscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, (1992) 867–884],...

Inference for copula modeling of discrete data: a cautionary tale and some facts

Olivier P. Faugeras — 2017

Dependence Modeling

In this note, we elucidate some of the mathematical, statistical and epistemological issues involved in using copulas to model discrete data. We contrast the possible use of (nonparametric) copula methods versus the problematic use of parametric copula models. For the latter, we stress, among other issues, the possibility of obtaining impossible models, arising from model misspecification or unidentifiability of the copula parameter.

Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles

Olivier Paul FaugerasLudger Rüschendorf — 2017

Mathematica Applicanda

Our purpose is both conceptual and practical. On the one hand, we discuss the question which properties are basic ingredients of a general conceptual notion of a multivariate quantile. We propose and argue that the object “quantile” should be defined as a Markov morphism which carries over similar algebraic, ordering and topological properties as known for quantile functions on the real line. On the other hand, we also propose a practical quantile Markov morphism which combines a copula standardization...

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