We are given an image I and a library of templates L, such that L is an ove
rcomplete basis for I. The templates can represent objects, faces, features
, analytical functions, or be single pixel templates (canonical templates).
There are infinitely many ways to decompose I as a linear combination of t
he library templates. Each decomposition defines a representation for the i
mage I, given L.
What is an optimal representation for I given L and how to select it? We ar
e motivated to select a sparse/compact representation for I, and to account
for occlusions and noise in the image. We present a concave cost function
criterion on the linear decomposition coefficients that satisfies our requi
rements. More specifically, we study a "weighted L-p norm" with 0 < p < 1.
We prove a result that allows us to generate all local minima for the L-p n
orm, and the global minimum is obtained by searching through the local ones
. Due to the computational complexity, i.e., the large number of local mini
ma, we also study a greedy and iterative "weighted L-p Matching Pursuit" st
rategy.