This section reviews possible directions of future research related to this dissertation. 7.2.1 Confidence in multi-modal registration
The coupled dictionary learning based image analogy usually cannot get perfect image synthesis results. Therefore the reconstruction error can be used to estimate the local confidence of the image
analogy result (while the prediction error is not known). If the reconstruction error is large which means that the local region is difficult to reconstruct the confidence of the image analogy result is low. This confidence estimate could then be used to weight the registration similarity metrics to focus on regions of high confidence.
7.2.2 Validation on large dataset from different modalities
The proposed method for multi-modal registration in Chapter 3 is only tested on a limited dataset. To further evaluate the benefit of the proposed method, a larger dataset is required. Also tests on different modalities would be interesting to see the limits of the proposed method.
7.2.3 Iterative deformation estimation
The proposed deformation estimation method is a one-pass algorithm, i.e. after learning the dictionary, the deformation is estimated in one run of the algorithm. However, there is no guarantee that the estimated result is good enough. It is possible to do this in an iterative approach, i.e. each time after obtaining an estimated deformation, I transform the image based on the obtained deformation and use the deformed image as an input of the algorithm. The process continues until converge.
7.2.4 Validation of cell velocities prediction on real cell data
The common patterns of GTPase activities and cell velocities can be captured by the learned coupled dictionary. However, the method is only validated on a limited dataset. It would be useful to do real experiments to validate these patterns, i.e. by creating activation patterns in a cell, I could test if they indeed result in the predicted cell boundary changes.
7.2.5 Robust dictionary learning for other noise models
The current robust dictionary learning method is based on the Gaussian noise model. Although I validated it on real data where the noise model may not be Gaussian, it would be nice to directly model the noise in the training data. The framework would be the same for different noise models, the only difference is how to compute the confidence in the EM algorithm.
APPENDIX A: APPENDIX FOR CHAPTER 5 A.1 Boundary Tracking
For a marker pi = (xpi, ypi) on boundaryΓt,∇φt(xp, yp) is the gradient of the level setφt at pointpi, thus the direction of∇φt(xp, yp)points in the direction with greatest increase of the
distance of φt. Let Dt,t+1(pi)represent the distance from pi on Γt to the closest point onΓt+1,
Dt,t+1(pi) =−φt+1(xpi, ypi). IfDt,t+1(pi)is positive, there is a boundary protrusion fromttot+ 1 (the boundary pointpi is inside of the cell at time t+1), thus the search direction should point in the
gradient direction∇φt(xp, yp)(based on the definition of protrusion and retraction). Similarly, if
Dt,t+1(pi)is negative, there is a boundary retraction and the search direction should point in the
negative gradient direction−∇φt(xp, yp). Thus the search distance forpi at time t is defined as
follows, Spi = (Dt,t+1(pi) ∇φt+1(xpi, ypi) |∇φt+1(xpi, ypi)| )· ∇φt(xpi, ypi) |∇φt(xpi, ypi)| , (A.1)
where ‘·’ represents the inner product operator of two vectors, pi is the tracked ith boundary
point at timet,Dt,t+1(pi)is the distance of pi att to the closest point onΓt+1,∇φt(xpi, ypi)and
∇φt+1(xpi, ypi)are the gradients ofpionφtandφt+1respectively. Equation (A.1) defines the search distance as the projection of the closest distance frompi attto theΓt+1 onto the gradient direction
ofpionφt. A positive search distance corresponds to cell protrusion. Figure A.1 explains the search
distance in Equation (A.1).
The tracking process can be achieved by computing search distances iteratively. The velocityvt i
at markerpi is defined as the distance betweenpi onΓtandΓt+1on the gradient direction∇φt(pi)
computed from the tracking process and divided by the number of frames between timetandt+ 1 (which is one for my data). Positive velocity points to the direction of cell protrusion. The tracking algorithm is described in Algorithm 7. After boundary tracking, the velocities are extracted for each marker at different framesvi ={vi1, . . . , v
k−1
Figure A.1: Illustration of search distanceSpiin Equation (A.1). p0is the location of a marker onΓt, whileqis the corresponding location of the marker onΓt+1. The goal is to estimate the location of
q. In this example, I usep2 as the estimation the locationq. Green curve and blue curve represent
the cell boundariesΓtandΓt+1 at timetandt+ 1respectively;pis the sampled boundary point;
∇φtand∇φt+1are the gradient directions of level setsφtandφt+1 at pointprespectively;Spis the
search distance at pointpwhich is a projection ofDt,t+1(p)onto the unit normal∇φt;θis the angle
between∇φt(pi)and ∇φt+1(pi)and cosθ can be computed from the inner product of
∇φt+1(pi) |∇φt+1(pi)| and ∇φt(pi)
|∇φt(pi)|.