Applying texture correlation

A pure translation is the simplest possible transformation to map. Applying tex-ture correlation here does not give a particular advantage, as every interrogation window in the grid will yield the same vector. Instead, texture correlation is a powerful tool when used to map complex transformations that involve de-formations as well as translations (and possibly rotations). Here the similarity with Particle Image Velocimetry is evident, as PIV was conceived as a tool to map displacements within fluids, where inter-particle distances are typically not conserved.

PIV has been also applied to study deformations of elastic substrates in trac-tion force microscopy, which is a technique developed to measure the mechanical forces exerted by living cells[124, 125]. In a typical experiment, cells are grown on transparent polymer substrates of known elastic properties that are seeded with fluorescent microspheres. The displacements of the sphere are recorded using a microscope and successively mapped using a PIV algorithm. From these displacements the cellular traction forces can be calculated.

In fact, PIV algorithms used for traction force microscopy can also be used for measuring tissue displacements with texture correlation. The difference lies only in the fact that traction force microscopy analyzes actual particle displacements and is therefore entitled to the name of PIV, while texture correlation applies the very same algorithms to a broader spectrum of microscopy images.

Compared to the simple explanation provided in fig. 2.4 and in fig. 2.5, experimental data is usually analyzed using more sophisticated implementations of the texture correlation method. An important addition in this respect is that of a search window larger than the interrogation windows of the grid.

Consider a texture correlation algorithm that subdivides the target image into N_i×N_jinterrogation windows termed{A_{i j}}. In a basic implementationN_i×N_j displacement vectors are obtained, one centered in each interrogation window.

These vectors are obtained by computing corr(A_{i j},A^∗_{i j}), where the starred letter indicates the same interrogation window but taken from the transformed image.

As previously discussed and as can be understood from fig. 2.4, displacements larger than the size ofA_{i j} will not be correctly tracked. A search window reduces this risk by introducing an expanded boxB_{i j}^∗ concentric withA^∗_{i j}. The correlation is now performed betweenA_{i j} and B^∗_{i j}, thus allowing the algorithm to take into account large displacements.

Another feature of advanced texture correlation algorithms is multi-pass analysis. In this case, correlation analysis is performedntimes, with grids of decreasing interrogation window size. The vectors yielded by the larger grids are taken into account in the analysis performed on the finer grids. This is to ensure that large displacements, which would not be detected by the finer grid, are not lost. This is particularly useful when combined with a search window for each of thenpasses. Multi-passes and search windows are especially important in those cases where small local deformations and large scale translations or rotations coexist in the same dataset. Finally, in order to maximize the number of computed vectors and obtain a denser sampling of the displacement field, the N_i×N_j grids can be constructed with overlapping interrogation windows rather than an edge-to-edge tiling.

A simple graphic in fig. 2.6 outlines the functioning of the texture corre-lation/PIV algorithm used to analzye the data presented in this work. This algorithm was developed and is maintained by Qingzong Tseng as a plugin for the image analysis software Fiji originally desgined as a tool for PIV applications rather than texture correlation[124, 126].

Precision and limitations

When defining the precision and accuracy of texture correlation algorithms different aspects need to be taken into account. First of all the intrinsic limits of the starting dataset need to be understood. The information available from images generated by any optical system is limited by the resolution of the system employed, and by the means of detection. There will be a limit for the smallest distance that can be resolved, and also the smallest possible displacement that can be extracted from the dataset. The discrete nature of a pixel-based image also poses some limits. Displacements smaller than one pixel in principle can not be detected. Due to the averaging which is intrinsic in the texture correlation method, it is possible for displacements smaller than one pixel to be interpolated as the average displacement of an interrogation window.

Importantly, a consequence of texture correlation, is that the precision of the displacement vectors can be very high. Namely, the uncertainty on the lengths of the components of the vectors is subpixel, typically 0.1 of the pixel size[123]. The coordinates of the displacement vectors are determined by the location of the crosscorrelation maximum, which can be interpolated with high accuracy. Lastly, the density of the displacement vectors depends on the grid of

Figure 2.6: A schematic overview of how a texture correlation/PIV algorithm works, implementing overlapping grids, search windows and multi-pass.

Figure 2.7: aA colour the white frame inais shown here with arrows depicting the vector field.

The length of the arrows is multiplied by5for clarity.

interrogation windows used. The spacing of the interrogation windows can be minimized at the cost of increasing computation time.

2.3.4 Studying the tendon-bone insertion with texture