As mentioned previously, the deformation model used in a registration application can vary significantly in its complexity, mainly due to the number of parameters or DOF of the trans- formation model. The aim of the optimization is to find the transformation parameters that maximize the similarity (or minimize the distance) of the two images. In general, the more complex the transform, the harder it is for the optimization to find the ideal set of parame- ters. The most general form of the objective function that is optimized in image registration is
Here the first term (also known as data term) characterizes the similarity between the source and target image and the second term (regularization, penalty or smoothness term) associates a cost to a particular deformation. When simple transformations are considered, e.g. rigid or affine transformation, the regularization term is often omitted.
The optimization procedure in image registration generally requires iterative methods that gradually minimize a cost function. In the case of point-based rigid registration, the process of finding the optimal solution has a closed formed solution and it is generally known as the ”Orthogonal Procustes“ problem [106]. Another way to estimate the parame- ters of a rigid or affine transformation model in a point-based registration procedure is using RANSAC [67]. This permits an estimation that is robust against outliers (point localization errors). In broad terms, optimization techniques can be subdivided in to two categories: Continuous and discrete. Figure 2.4 shows a summary of some of the available optimization methods. Optimization methods Gradient descent Conjugate gradient Quasi-Newton Levenberg-Marquardt Graph-based Message passing Linear programing Continuous methods Discrete methods
Figure 2.4: Diagram of optimization methods.
2.3.1
Continuous optimization methods
Continuous optimization methods are generally used when the registration/transformation parameters are assumed to be continuous and the associated cost function is differentiable. Assuming thatθ is the vector of the transformation parameters, t denotes an iteration index, αt is a step size and gt defines a search direction, then continuous optimization iteratively
looks for the ”best“ solution using an update rule of the form: θt+1= θt+ αtgt(θt). The αt
and gt parameters can be defined in several ways. The choice in setting these parameters dif-
ferentiates between continuous optimization methods. For example, the search direction g can be specified using first-order information. The step sizeαtcan be constant, change with
each iteration or be such that minimizes the objective function. Variations of the behavior of these parameters is what distinguishes between optimization methods. Some continu- ous optimization approaches commonly used in medical image registration are: Gradient descent, conjugate gradient descent, quasi-Newton and Levenberg-Marquardt methods, to name a few. For a comprehensive overview we refer the reader to [167].
Gradient descent methods traverse the search space in the direction of the negative gra- dient of the function g= −∇θ(θ). Gradient descent optimization is often used in registration algorithms like LDDMM [16], FFD [182] or [30, 137], to name a few. A drawback is that the gradient direction is local, and while it locally decreases the value of the function, it may not be moving in the best direction. For parameter spaces with two or more dimensions, this can lead to a slow convergence to the optimum.
Conjugate gradient descent algorithms guarantee convergence in a finite number of steps for quadratic functions. Instead of moving in the direction of the gradient, it does in the direction of the conjugate gradient. This linearly combines directions of previous steps with the current as g= f (∇θ(θt), gt−1). Some examples of registration methods that use conjugate gradient optimization are [145, 86, 166]. In general conjugate gradient methods converge faster, although computing the conjugate direction is slightly more complicated.
Similar to conjugate gradient descent, quasi-Newton methods accumulate information from the previous iterations in order to achieve better convergence. Their goal is to estimate the inverse Hessian matrix H−1(θ) to define the search direction. Thus, the search direction is defined as g= − ˆH−1(θ)∇θ(θ) where the ˆ denotes the type of approximation that is used. Quasi-Newton optimization methods have been tested in several registration applications [45, 135, 214].
Another optimization method is the Levenberg-Marquardt algorithm. In this method the search direction is given by g= − ˆH−1(θ) + ζI ∇θ(θ) where I is the identity matrix
andζ is a computational cost vs. stability weight factor. Smaller ζ values achieve greater speeds while higher values increase stability. Ifζ equals to zero, then g is the same as in the quasi-Newton algorithm. Some applications of the Levenberg-Marquardt optimization can be found in [234, 128, 11].
2.3.2
Discrete optimization methods
One of the shortcoming of continuous optimization methods is that they usually perform a local search in the parameter space for the optimal solution. This makes such methods sen- sitive to the initial estimate of the transformation parameters and they can easily get trapped in local minima [195]. The fact that the methods require the computation of the gradient of the cost functions limit their use to cost functions which are differentiable. Furthermore, the assumption that the design parameters are continuous does not always hold. Discrete methods are less sensitive to the initial conditions and often converge faster compared with continuous methods [88]. In many practical problems in engineering the design parameters can be modelled as discrete variables [170], and image registration is no exception. The main limitation of discrete methods is their lack of precision due to the fact that they quan- tize the search space. Therefore, a trade-off between computational speed and precision exists in discrete methods. If precision is desired a denser sampling of the parameter space is required, however, higher computational costs will be incurred. An additional trait of discrete methods is the possibility to introduce knowledge about the expected location of the solution through the quantification of the parameter space [195]. Discrete optimization techniques can be separated into three classes: Graph-based, message passing and linear programing methods.
Graph-based methods are based on the max-flow min-cut principle [73] that states that the maximum amount of flow that can pass from the source to the sink is equal to the
minimum cut that separates the two terminal nodes. The α-expansion optimization tech-
nique [22] is a multi-label extension of the maximum a posteriori estimation [90] algorithm.
Some applications of medical image registration based α-expansion optimization include
Message passing methods are based on belief propagation [155] where messages are locally exchanged between the nodes of a graph and then backtracking is used to recover the best solution to the problem. Each message conveys the belief of a node to its neighboring node regarding each solution. Belief propagation methods can provide an exact inference for graphs with a chain or tree-shaped topology, however this not the case for graphs that contain loops. In this case loopy belief propagation [77, 151] must be used. Image registration techniques that use message passing optimization can be found in [191, 133].
Another class of discrete optimization methods are based on linear programing, which provides better theoretical properties. Linear programing methods avoid solving the origi- nal, generally N-P hard problem, in favor of an LP relaxation solution of the problem. Some examples of linear programing based optimization methods in image registration can be found in [125, 126, 87].