Brief problem description
I'm using a Basel Morphable Face Model, which was labeled with MPEG4 and Farkas landmarks. I can generate different faces, use different lighting conditions, rotations, perspective etc.
I want to use it to train an algorithm in order to find these landmarks on real pictures further.
More strictly
Given a labeled set of images (with fixed size, if it's critical), I want to get a function $q\left( x \right)$ (where $x$ is an image), which will minimize loss expectation (Bayes' task)
$$E = \sum\limits_{x \in X, k \in K} P\left( k \mid x \right) \cdot W\left( k, q\left( x \right) \right)$$
$k$ is a tuple of feature points' coordinates like $<(x_1, y_1), (x_2, y_2), \dots>$.
Loss function is either
$$W\left(k, k'\right) = \sum\limits_{i=1}^n ||k_i - k_i'||^2$$
or
$$W\left(k, k'\right) = \sum\limits_{i=1}^n(1\; if\; ||k_i - k_i'|| > \Delta\; otherwise\; 0)$$
Maybe there is a better problem description, and loss expectation minimization is not suitable for this task?
Details
I guess, there are more specific algorithms for this issue than Brute Force (estimate probabilities by training set and use them to find landmarks on input images) or Artificial Neural Networks.
Why not ANN: the problem "as is" is incredibly hard. I have $44$ points, most of which have symmetric ones (so, it's something between $60$ and $80$). Given image $100×100$, I have at least $10'000^{60}$ different solutions and should find the best one (or weighted average in case of $l^2$ loss function). $10^{240}$ is a huge number. Add probabilities estimate to this Math carnival, and you'll get a problem of estimating probability function for $10'000$ dependent random variables via each image of the training set. So, if some general approach will work fast, I'm not sure that it will be accurate enough. The new problem will occur — what exactly will we lose when ANN will be applied?
Maybe two-dimensional context-free grammars can solve this, and I just need a proper representation to take scale and rotations into account.
Maybe the problem can be stated in terms of constraint satisfaction problem and solved efficiently.
Question
Which algorithms you can recommend for my purpose?