Optimal Image Alignment with Random Measurements

OPTIMAL IMAGE ALIGNMENT WITH RANDOM MEASUREMENTS

ABSTRACT
We consider the problem of image alignment using random
measurements. More specifically, this paper is concerned
with estimating a transformation that aligns a given reference
image with a query image, assuming that not the images
themselves but only random measurements are available.
According to the theory behind compressed sensing,
random projections of signal manifolds nearly preserve pairwise
Euclidean distances when the reduced space is sufficiently
large. This suggests that image alignment can be performed
effectively based on a sufficient number of random
measurements. We build on our previous work in order to
show that the corresponding objective function can be decomposed
as the difference of two convex functions (DC).
Thus, the optimization problem becomes equivalent to a DC
program that can be solved by an outer-approximation cutting
plane method, which always converges to the globally
optimal solution.
1. INTRODUCTION
The problem of image alignment is of paramount importance
and enjoys numerous applications in various fields including
pattern recognition, computer vision and medical image
analysis, to name just a few [1]. The comparison of two visual
patterns is generally only meaningful if they are aligned
first, so that their distance reflects their structural and geometric
differences. Image alignment consists in estimating
the relative transformation between patterns. The transformed
version of a pattern can be described as a point of a
(possibly nonlinear) manifold in a high dimensional space,
which is usually called the transformation manifold. The
manifold distance (MD) is the minimum distance between
the query image p and the manifold generated by the reference
image s, see Figure 1.
At the same time, a theory of sparse signals has recently
emerged, often referred to as compressed sensing (CS). According
to this theory, a few random projections of a... [continues]

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