Fachrichtung 6 . 1 – Mathematik Preprint Nr . 355 Physically Inspired Depth – from – Defocus

We propose a novel variational approach to the depth–from–defocus problem. The quality of such methods strongly depends on the modelling of the image formation (forward operator) that connects depth with out–of–focus blur. Therefore, we discuss different image formation models and design a forward operator that preserves essential physical properties such as a maximum–minimum principle for the intensity values. This allows us to approximate the thin–lens camera model in a better way than previous approaches. Our forward operator refrains from any equifocal assumptions and fits well into a variational framework. Additionally, we extend our model to the multi–channel case and show the benefits of a robustification. To cope with noisy input data, we embed our method in a joint depth–from–defocus and denoising approach. For the minimisation of our energy functional, we show the advantages of a multiplicative Euler–Lagrange formalism in two aspects: First, it constrains our solution to the plausible positive range. Second, we are able to develop a semi–implicit gradient descent scheme with a higher stability range. While synthetic experiments confirm the achieved improvements, experiments on real data illustrate the applicability of the overall method.