Analysis of Descent-Based Image Registration

Abstract

Abstract. We present a performance analysis for image registration with gradient descent. 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 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 is the neighborhood of translations 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. 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 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.

Non-invasive single-cell biomechanical analysis using live-imaging datasets

“…The image similarity measure is taken as the driving force of the morphing process (Modersitzki, 2004), the applied force field required to morph (transform) one image onto another; analogous to the force required to deform an object from one shape (deformation) to another. Image morphing is approached through minimizing the similarity measure (SSD) between two images (Modersitzki, 2004;Vural and Frossard, 2013). We calculated force fields from detectable changes in intensity values of consecutive image pairs; this takes into account both the shape of the cell as well as information within the cytosolic region.…”

“…where ν(x) is the velocity field, f(x, u(x)) is the force field that is used to drive the viscous flow; here, we define f(x, u(x)) as derivative of the sum of squared difference between the images (Modersitzki, 2004;Vural and Frossard, 2013): f ðx; uðxÞÞ ¼ À½M ðx À uðx; tÞÞ À SðxÞrM j xÀuðx; tÞ :…”

“…The responsibility of the cell to maintain or adjust cell shape or motility is dependent on cytoskeletal kinetics and requires an exertion of internal cellular forces. The intimate coupling between exogenous and cell-generated forces (Chen, 2008) implies the existence of an opposite external force, calculated through minimization of sum of squared differences (SSD) between images (Modersitzki, 2004;Vural and Frossard, 2013). As there is a point-wise correlation between pixel points in the images and material points of the cell during cellular deformation, movement in the pixels can be translated into cellular strain.…”

“…Image morphing aims to bring the SSD to zero, iteratively moving pixels until two consecutive images of the dataset are identical. The minimization of the SSD is interpreted as the force required for pixel displacement (Modersitzki, 2004;Vural and Frossard, 2013). The two variables (force and displacement) are linked using the Navier-Stokes equation for non-rigid viscous fluid flow and the resultant is a strain field (ε) (M. Bro-Nielsen, Medical image registration and surgery simulation, PhD thesis, Technical University of Denmark, 1996; I. Yanovsky, Unbiased nonlinear image registration, PhD thesis, University of California, Los Angeles, 2008;Zitová and Flusser, 2003).…”

On the Link between Gaussian Homotopy Continuation and Convex Envelopes

“…Examples include image segmentation [49], image alignment [67], image completion [46], dictionary learning [44], part-based models [25], and optical flow [62]. Unfortunately, a severe limitation of nonconvex problems is that finding their global minimum is generally intractable.…”