Image processing is the extraction of meaningful information about the subject of an image using manual or automated techniques. Medical image processing is a broad topic and here we will attempt to provide a framework which is relevant to this work. We discuss segmentation of medical images, non- convex fitting through conjugate gradient descent and non-linear regression using random forests.
5.1. Segmentation
The process of identifying meaningful anatomical structures within an image is known as segmentation. Segmentation problems vary widely based on the anatomy of interest and special considerations with each individual application. Segmentation has many applications including surgical planning, post-surgical assessment and abnormality detection [72]. Abnormality detection is of particular interest for this work and can include finding abnormal tissue (e.g. tumors) or detecting abnormal image
20
features whether they be structural, textural or intensity in nature. Segmentation is key to all of these tasks as the anatomy of interest must be localized to be analyzed.
Medical image segmentation has been applied to a wide variety of anatomies but the classic anatomy of interest in segmentation of the brain. The brain was one of the first structures to be segmented due to the relative regularity across subjects, the rigid container of the skull and the attractive applications. Moving to other anatomies, such as the ON which is surrounded by compressible fat and has a shape that varies widely between subjects, these constraints do not hold and complicate the problem.
Despite these challenges recent advances have been made in applying segmentation to a variety of anatomies [73].
Image segmentation has traditionally been performed manually by trained observers such as a radiologist. In the late 1990’s computer-assisted segmentation methods were developed. These methods increased the accuracy and speed by which an observer could segment images by utilizing image context information to inform the final segmentation [74, 75]. While these methods were a step in the right direction the ultimate goal was to develop robust automated methods to replace the human intervention.
This task proved difficult and to this day computer assisted segmentation is the de facto standard for many difficult to segment anatomies, including the ON [51, 60].
Automation of segmentation tasks yields large benefits in the scale at which data can be analyzed.
Computer assisted segmentation decreases the number of man hours required for each segmented volume while automated segmentation reduces it almost completely. This reduction in time facilitates the analysis of large data sets previously unreachable. Automated segmentation also has the benefit of reducing any bias introduced by human intervention, yielding more reliable segmentation results. Initial attempts at automated segmentation were focused on pattern recognition [76, 77]. Intensity models have also been used to segment images and correct for global inhomogeneities [78]. Today, there are many methods to automated segmentation including thresholding, region growing, clustering, neural networks, Markov Random Field models, deformable models and atlas-based approaches [79]. Atlas-based approaches including multi-atlas segmentation have had success in a variety of anatomies [73]. Canonical multi-atlas
21
segmentation involves registering a set of multiple atlas images to the target image to be segmented. In this context an atlas image is a previously labeled example. With all of the atlas images registered to the target space the segmentation of each atlas is propagated to the target through a process referred to as label fusion.
5.2. Conjugate Gradient Descent
There exists a multitude of methods for solutions to non-linear optimization problems [80]. If the gradient of the function to be optimized can be solved for analytically, that is the derivative of the objective function with respect to each parameter to be optimized, then gradient descent methods can be employed [81, 82]. Conjugate gradient methods are a good solution for unconstrained energy minimization problems iteratively when more direct solutions are prohibitive to implement. This method has the benefit of defining a search direction as the negative gradient direction of the objective function.
This ensures that the objective function should always be decreasing until a minimum is reached. The conjugate of this method denotes that descent directions at subsequent iterations be conjugate to each other. This has been shown to ensure faster convergence, derivation of this is beyond the scope of this document.
5.3. Random Forest Regression
Random forest regression and classification has gained popularity in medical imaging applications in recent years [83, 84]. Random forests are an ensemble learning method, meaning they use multiple models to learn a better solution than any one model, using an ensemble of decision trees.
Decision trees predict some output, either classification or a value for regression, based on a set of higher dimensional input. The root of the tree splits into multiple interior nodes, each of which contains a decision based on one of the input features. The nodes on the tree can be learned by repeatedly splitting the training data into subsets. This splitting procedure continues until the entire subset at a node contains the same output value. At this point a leaf node, or end node, is created which contains the output value.
22
Once a decision tree is learned, application is simple as an input data point can traverse each decision node until it reaches an end node which will contain the desired output value. Random forests are learned by training multiple decision trees on random subsets of the input features [85]. Training of a random forest intrinsically minimizes the error of regression of the training data. Validation of this error, to avoid overfitting, is important with the use of a yet unseen testing data set. The accuracy of the forest can then be evaluated on the testing data which was not used to train the model and should correspond to the generalization error of applying the model to any other yet unseen data.