Differentiable ray-tracing is an exciting new development in computer graphics to approach all sorts of 3D scene design problems by obtaining gradients of renders produced by ray-tracing with respect to parameters that define the scene. These gradients can then be incorporated in a gradient-descent type optimization pipeline.

One such type of design problem is caustic design with free-form lenses. This is the process of obtaining the geometry of lenses in an optical system such that the light that passes through this system forms a desired illumination distribution on a target screen. A typical application of this is distributing the light from streetlights or car headlights in a pleasing and efficient way over the street.

The non-imaging optics literature offers many methods to design free-form lenses for caustic design, but differentiable ray-tracing is largely unknown in this field. Therefore this thesis proposes a method of optimizing free-form lenses for caustic design with differentiable ray-tracing.

The optimization pipeline starts with a multi-layer perceptron neural network which outputs parameters that define one free-form lens side in the form of a B-spline surface. A specially implemented ray-tracer then produces a caustic render by tracing through this lens from either a plane wave or a (grid of) point source(s). Back-propagation and optimization takes place using automatic differentiation in PyTorch and the Adam optimizer.

The results, which are verified with LightTools, show great promise for this technique, especially for the plane wave optimizations. The point source grid optimizations proved to be more challenging, but also here the optimization was able to achieve improvement. This shows that the proposed technique also has potential in positive etendue optimizations.

In this thesis we examine various aspects of the output signal produced by the electrical pickup of an electric guitar. We do this by formulating a model of this output signal based on the one dimensional damped wave equation for the motion of one guitar string and (initially) a simple linear model for the output signal based on this motion. This model is compared to measurements of the output signal of a real guitar and their corresponding spectra. From this comparison it is concluded that the damping term in the damped wave equation is sufficient to roughly describe the decrease in amplitude over time of the envelope in the output signal of an electrical guitar.
The measured guitar signal also displays beating which has been considered to be caused by the interference of parallel and perpendicular modes on the guitar string with slightly differing frequencies. It is shown that the order of magnitude of this difference in frequencies between parallel and perpendicular modes is neither caused by different effective values of the damping coefficient nor by the influence of co-vibrating string boundaries. The last chapter gives a very simple method to introduce beating into the predicted signal. This is done by describing the motion of the string as occurring in a rotating plane in which the radial direction is given by the above mentioned model.