Analysis of Descent-Based Image Registration

TL;DR

This paper analyzes how smoothing and noise affect the accuracy of gradient descent methods in image registration, providing mathematical insights into the trade-offs involved in multiscale approaches.

Contribution

It offers the first detailed analysis of descent-based image registration, quantifying how smoothing enlarges the well-behaved region while increasing noise-induced errors.

Findings

The well-behaved neighborhood size increases quadratically with filter size.

Alignment error grows at least linearly with filter size in noisy conditions.

Hierarchical multiscale strategies are justified by the trade-off between robustness and accuracy.

Abstract

We present a performance analysis for image registration with gradient descent methods. We consider a typical multiscale registration setting where the global 2-D translation between a pair of images is estimated by smoothing the images and minimizing the distance between them with gradient descent. Our study particularly concentrates on the effect of noise and low-pass filtering on the alignment accuracy. We adopt an analytic representation for images and analyze the well-behavedness of the image distance function by estimating the neighborhood of translations for which it is free of undesired local minima. This corresponds to the neighborhood of translation vectors that are correctly computable with a simple gradient descent minimization. We show that the area of this neighborhood increases at least quadratically with the smoothing filter size, which justifies the use of a smoothing step in image registration with local optimizers such as gradient descent. We then examine the effect of noise on the alignment accuracy and derive an upper bound for the alignment error in terms of the noise properties and filter size. Our main finding is that the error increases at a rate that is at least linear with respect to the filter size. Therefore, smoothing improves the well-behavedness of the distance function; however, this comes at the cost of amplifying the alignment error in noisy settings. Our results provide a mathematical insight about why hierarchical techniques are effective in image registration, suggesting that the multiscale coarse-to-fine alignment strategy of these techniques is very suitable from the perspective of the trade-off between the well-behavedness of the objective function and the registration accuracy. To the best of our knowledge, this is the first such study for descent-based image registration.