Optimization

Fitting a forward model to the data always involves the definition of a cost function which measures the distance between the measured signal, and the prediction by the forward model. In most cases the cost function is the mean-square-difference between data and prediction. The problem of optimization then involves finding the parameters that minimize this cost function.

The most common approach is iterative optimization, where starting values are defined for the free parameters, and these are then adjusted iteratively to bring the prediction closer to the data. The adjustment is done by a gradient-descent method, where each parameter is first tested to measure its effect on the cost function. Then parameters are modified in proportion to that effect.

dcmri performs iterative optimization by default in end-to-end applications and uses scipy.optimize.curve_fit. Iterative optimization is flexible and convenient but also has some downsides - notably the risk of returning local optima where the solution depends on the choice of initial values.

The simplest alternative to iterative optimization is a brute-force approach where the forward model is used to predict signals for all possible combinations of tissue parameters. For each prediction the fit to the data is measured, and the prediction with the best fit is identified. The brute-force approach is reliable and robust, but it is often computationally prohibitive. Brute force optimization is available in scipy through scipy.optimize.brute.

In some cases a dictionary-based approach can be used where signals are computed up front for all possible combinations of parameters, and these are retained in a dictionary and saved. For any given signal it then suffices to load the dictionary and find the prediction closest to the measured signal. This is fast but requires different dictionaries to be generated if, for instance, experimental parameters are modified.

An alternative data-driven method uses deep learning to approximate the inverse relationship that maps the signals to the tissue parameters. Just like in a dictionary-based method, the forward model is used to generate signals for many combinations of tissue parameters. Rather than storing them in a dictionary these are then used to train a neural network to perform the inverse transformation.