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Handbook of Mathematical Models in Computer Vision

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David Marr's theory was a pioneering step towards understanding visual perception.
In his view human vision was based on a complete surface reconstruction
of the environment that was then used to address visual subtasks. This approach
was proven to be insufficient by neuro-biologists and complementary ideas from
statistical pattern recognition and artificial intelligence were introduced to better
address the visual perception problem. In this framework visual perception is
represented by a set of actions and rules connecting these actions. The emerging
concept of active vision consists of a selective visual perception paradigm
that is basically equivalent to recovering from the environment the minimal piece
information required to address a particular task of interest.

Mathematical methods are an alternative to tackle visual perception. The central
idea behind these methods is to reformulate the visual perception components
as optimization problems where the minima of a specifically designed objective
function "solve" the task under consideration. The definition of such functions is
often an ill-posed problem since the number of variables to be recovered is much
larger than the number of constraints. Furthermore, often the optimization process
itself is ill-posed due the non-convexity of the designed function inducing the
presence of local minima. Variational, statistical and combinatorial methods are
three distinct and important categories of mathematical methods in computational
vision.

Variational techniques are either based on the optimization of cost functions
through the calculus of variations or on the design of partial differential equations
v^hose steady state corresponds to the solution of the visual perception task. Such
techniques have gained significant attention over the past decade and have been
used to address image restoration and enhancement, image segmentation, tracking
and stereo reconstruction among other problems. The possibility to use the
calculus of variations in the optimization process is the most important strength
of these methods combined with the fact that one can integrate many terms and
build quite complicated objective functions at the expense of converging toward
local minima.
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