Ill-posed problems in early vision

By Mario Bertero, Tomaso A. Poggio, V. Torre. 1988

Mathematical results on ill-posed and ill-conditioned problems are reviewed and the formal aspects of regularization theory in the linear case are introduced. Specific topics in early vision and their regularization are then analyzed rigorously, characterizing existence, uniqueness, and stability of solutions. A fundamental difficulty that arises in almost every vision problem is scale, that is, the resolution at which to operate. Methods that have been proposed to deal with the problem include scale-space techniques that consider the behavior of the result across a continuum of scales. From the point of view of regulation theory, the concept of scale is related quite directly to the regularization parameter lambda . It suggested that methods used to obtained the optimal value of lambda may provide, either directly or after suitable modification, the optimal scale associated with the specific instance of certain problems. >

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Citations in the corpus

Generalization in anti-causal learning (2018, background): “The connections between inverse problems and statistical learning theory (in the weak generalization regime) have recently been discussed by (Kurková, 2004; Rosasco et al., 2005) and shown to have practical consequences (Bertero et al., 1988).”
Automatic differentiation in machine learning: a survey (2015, background): “On the other hand, in computer vision, many problems are formulated as the minimization of an appropriate energy functional (Bertero et al, 1988; Chambolle, 2000).”
Markov Random Field Modeling in Image Analysis (2001, methodology): “It has been developed into a general framework, called regularization (Poggio et al. 1985; Bertero et al. 1988), for a variety of low level vision problems.”
View independent human body pose estimation from a single perspective image (2004, background): “To make the problemmoretractable,researchershave resortedto assuminga scaledorthographic camerain the uncalibratedcaseor a calibratedcamerain theperspectivecase,bothof whicharemorerestrictivethan one would like in practice.”
1 On what it means to see , and what we can do about it (2008, unknown): “From the computational standpoint, this development amounted to shifting the focus of research from " inverse optics " approaches (Bertero et al., 1988), which aim to recover the solid geometry of the viewed scene, to managing feature-based evidence for task-specific hypotheses about the input…”
Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization (2006, background): “By the deﬁnition of almost minimizer (see equation (2.1)), we have We can rewrite the ﬁrst inequality as The term in the bracket is less than or equal to ((n − 1 )/n)ε E n − 1 (because of the second inequality) and thus”
Multiresolution Markov models for signal and image processing (2002, background): “…second closely related problem is one that has been widely studied in the field of computer vision, namely that of reconstructing surfaces from regular or irregularly sampled measurements of surface height and/or of the normal to the surface (as in the shape-from-shading problem [34], [53], [152]).”
CoMoSpeech: One-Step Speech and Singing Voice Synthesis via Consistency Model (2023, background): “The quality of the acoustic feature produced by the acoustic model, typically mel-spectrogram, crucially affects the quality of the synthesized speeches.”
A Theoretical Framework for Convex Regularizers in PDE-Based Computation of Image Motion (2001, background): “(48) From the perspective of regularization, Table 1 reveals another useful classification in this context: while image-driven models correspond to the class of quadratic regularizers (Bertero et al., 1988), flowdriven models belong to the more general class of nonquadratic convex regularizers.”
A Developmental Roadmap for Learning by Imitation in Robots (2007, background): “One approach to solve ill-posed problems, [29], [30], consists in using additional constraints that restrain the set of admissible solutions, to guarantee a unique solution.”
An information-theoretic approach to traffic matrix estimation (2003, background): “There is extensive experience with ill-posed linear invers e problems from fields as diverse as seismology, astronomy, and med ical imaging [1, 2, 17, 18, 26], all leading to the conclusion that some sort of side information must be brought in, producing a reco nstruction which may be good or bad de”
Regeneration Learning: A Learning Paradigm for Data Generation (2023, background): “• First, for the generation tasks where Y contains much more complex information than X , the generation models face severe one-to-many mapping problems (ill-posed or ill-condition problem) (Bertero, Poggio, and Torre 1988), which increases the learning difficulty.”