Mathematical Foundations and Generalisations of the Census Transform for Robust Optic Flow Computation

Abstract

In recent years, the popularity of the census transform has grown rapidly. It provides features that are invariant under monotonically increasing intensity transformations. Therefore, it is exploited as a key ingredient of various computer vision problems, in particular for illumination-robust optic flow models. However, despite being extensively applied, its underlying mathematical foundations are not well-understood so far. The main contributions of our paper are to provide these missing insights, and in this way to generalise the concept of the census transform. To this end, we transfer the inherently discrete transform to the continuous setting and embed it into a variational framework for optic flow estimation. This uncovers two important properties: the strong reliance on local extrema and the induced anisotropy of the data term by acting along isolines. These findings open the door to generalisations of the census transform that are not obvious in the discrete formulation. To illustrate this, we introduce and analyse second-order census models that are based on thresholding the second directional derivatives. Last but not least, we constitute links of census-based approaches to established data terms such as gradient constancy, Hessian constancy, and Laplacian constancy, and we confirm our findings by means of experiments.

Line-Constrained Camera Location Estimation in Multi-Image Stereomatching

“…The image transformation should preserve the geometry of the input views while mitigating lighting effects; generally speaking, it is a local transform of the pixel values. Good choices are histogram equalisation, a census-transform [ 26 , 27 ], a correlation transform [ 3 ], or a conversion to the CIEL colorspace with removal of the luminosity component [ 7 , 28 ].…”

Most Likely Separation of Intensity and Warping Effects in Image Registration

“…This type of approach has become increasingly popular in image matching in recent years. Examples include the rank transform and the census transform [47,22,10,11], and more recently the complete rank transform [7]. While both robust and invariant data terms have been shown to give very good results in a wide array of applications, they induce a fixed measure of variation that does not directly model variation in the data.…”