Total Euler Characteristic as a Noise Measure to aid Transfer Function Design (English)

For image-based, semi- or fully automated design of transfer functions for direct volume rendering, it is beneficial to have an estimation of the noise level in the rendered image. A topology-motivated noise measure is suggested based on the Euler Characteristic, i.e., on the number of islands and holes in the rendering. The effectiveness of this concept lies in the observation that the Euler Characteristic of the real objects of interest in a rendered scene is usually very low, typically orders of magnitude lower than the Euler Characteristic caused by noise. Interestingly, the Euler Characteristic, which is defined for binary images, can be computed simultaneously for all possible super level sets of an integer-valued gray-scale image in a single raster scan. An efficient algorithm for 2D and 3D is presented with computation cost as low as total variation. The Total Euler Characteristic of the super level sets is proposed as a noise level estimation, i.e., the L1-norm of the evolution curve of the Euler Characteristics for all possible isovalues. The superiority of the Total Euler Characteristic to other common noise estimation methods such as total variation and total second derivatives is demonstrated on renderings of the Marschner-Lobb phantom.