Statistical Shape Analysis & Modeling Group

Blurring-Invariant Riemannian Metrics for Comparing Signals and Images

The point spread function of the imaging device introduces some level of blurring in the captured images which can be further exaggerated due to techniques for storage and processing. Standard metrics for image comparisons provide results that are affected by the amount of blurring present in images. A common solution has been to deblur the images using one of the many techniques available for deblurring and then compare the deblurred images. Another approach is to define a set of features that are invariant to blurring and then to compare those features.

A natural mathematical representation of a gray-scale image or a signal is simply as a real-valued function on an appropriate domain. We will consider them as real-value functions on R. (The domain is chosen to be R or R^2 since we can easily extend any signal on an interval, say [0, 1], as a periodic signal on the full real line. Traditional analysis of signals is performed in the original time (spatial) domain or the frequency domain using Fourier transforms. However, using the standard metrics, the action of the (Gaussian) blurring group is not by isometries. In contrast, in the log-Fourier space, the orbits of the blurring group are given by straight lines and the action is by isometries. So, we formulate a framework for blurring-invariant comparisons of signals and images under their log- Fourier representations. We suggest a set of Riemannian metrics in that representation and derive a framework for computing geodesics and geodesic distances between orbits of given signals.

Since the orbits of blurring are straight lines in the log-Fourier of signals and images, we try to seek the quantification of blurring in this space. Let's call it Q function. It is a projection on the unit vector in the direction of log-Fourier transform of standard Gaussian.

With the Q function, the following steps define an algorithm to compute a geodesic path between two signals f1 and f2.

- Normalize f1 and f2 and compute the Fourier transforms. Then take the natural logarithms of the Fourier transforms, note as log-F(f1) and log-F(f2).
- Compute the c1 = Q[log-F(f1)] and c2 = Q[log-F(f2)] and let c2>c1.
- Find a d such that Q [log-F(f3)] = c2, where f3 is the blurred function of f1 with d. Now f2 and f3 is in the same blurring level. The distance between the two signals is given by ||log-F(f3)-log-F(f2)||
- Geodesics between log-F(f2) and log-F(f3) is t*(log-F(f3)) + (1-t)log-F(f2), where t is between 0 and 1.

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