# Multi-Dimensional Signal Alignment

Local All-Pass Filter Framework

The estimation of a geometric transformation that aligns two or more signals is a problem that has many applications in signal processing. The problem occurs when signals are either recorded from two or more spatially separated sensors or when a single sensor is recording a time-varying scene. Examples of fundamental tasks that involve this problem are shown in the figure below. In this project we estimate the transformation between these signals using a novel **local all-pass (LAP) filtering framework**.

The underlying principle in our LAP framework is that, on a local level, the geometric transformation between a pair of signals can be approximated as a rigid deformation which is equivalent to an all-pass filtering operation. Thus, efficient estimation of the all-pass filter in question allows an accurate estimation of the local geometric transformation between the signals. Accordingly, repeating this estimation for every sample/pixel/voxel in the signals results in a dense estimation of the whole geometric transformation. This processing chain can be performed efficiently and achieve very accurate results. We have applied this framework to image registration [1], [2], [3], motion correction [4], [5] and time-varying delay estimation [6].

To overcome these challenges, we assume the deformation field is slowly varying such that locally it is equivalent to a rigid deformation and apply our local all-pass filter framework.Remark:This problem is both restrictive, as the brightness consistency is unlikely to be satisfied exactly, and ill-posed as, for \(n>1\) many deformations many satisfy the equation and most are meaningless. However, in many applications, it is important to determine a meaningful deformation field.

To allow for a non-rigid deformation, we limit this estimation to a small local region, estimate a local all-pass filter and extract a local estimate of the deformation. This local estimate corresponds to the centre of the region. Accordingly, a dense, per-sample, deformation estimate is obtained by repeating this process for all the samples in the signal using a sliding-window mechanism, see figure below. Importantly, as discussed in [1] this can be performed very efficiently.

To allow estimation of both slowly and quickly varying deformations, we use an iterative poly-filter LAP framework that starts with large filters estimating the deformation, aligning the images and then repeating with a smaller filter [1].

##### Parametric Registration

For parametric image registration, we introduce a quadratic parametric model for the deformation and iteratively estimate the parameters of the model [3], [8], [9]. This parametric extension is robust to model mis-match (noise, blurring, etc), very accurate and capable of handling very large deformations. Furthermore, by modeling intensity variations, the parametric LAP is capable of handling multi-modal registration problems.*in-vivo*MRI dataset is shown below.

To allow estimation of both slowly and quickly varying deformations, we use an iterative poly-filter 3D LAP framework that starts with large filters estimating the deformation, aligning the images and then repeating with a smaller filter [10].