Page - 168 - in Proceedings of the OAGM&ARW Joint Workshop - Vision, Automation and Robotics

a 2D line fit which greatly reduces the computational effort. While not being as straight forward as any of the presented direct algorithms it is capable of finding planes where noise is dominant or in rough outdoor environments [9]. To improve results of certain plane segmentation algorithms it is vital to reduce noise of the input data by proper filtering. While a Gaussian Blur might be sufficient to remove noise from some intensity images it is not fit to be applied to depth maps. Besides not being able to handle areas where thesensorwasunable tocapturedata thisfilterwoulddestroy any information on discontinuities. Bilateral filtering [7] therefore is more selective and reduces over-smoothing along discontinuities. It is therefore a possible candidate for point clouds but has serious issues regarding our task since this filter introduces a bending along edges of tilted planes. This so called ski effect can be tackled by restraining the filter from working on edges [2], or by applying a bilateral filter, that is specifically designed for 3D geometry [4]. Approaches as the Total Variation [6] and the Total Gen- eralized Variation [1] based algorithms do not inherit their principle from convolution. Instead they minimize a cost function to fulfill a tradeoff of being close to the input and minimizing a smoothness measure. The Total Variation image denoising algorithms work well for intensity images but do have downsides as a tendency to frontoparalell planes when applied to depth-maps. This tendency in particular can be countered by using the Total Generalized Variation algorithm which allows for more refined regularization with higher order derivatives but at the cost of increased compu- tational complexity. III. FILTER The quality of depth images obtained from the Kinect is moderate, especially in distances bigger than 3 meter (see Figure 2). To reduce the noise and other artifacts like quantization, the raw data has to be filtered. A. Bilateral Filter The bilateral filter is a suitable candidate. It preserves dis- continuities and smooths out noise. D∗p= 1 Wp ∑ q∈Sp Gσs(‖p−q‖)Gσc(|Dp−Dq|)Dq (1) Wp=∑ q∈Sp Gσs(‖p−q‖))Gσc(|Dp−Dq|) (2) p: the coordinate of the resulting pixel. q: the coordinate of a surrounding pixel. Gσ: Gauss function. Dp,q: depth values of p or q. Sp: the neighborhood of pwhere |p−q|<rSth. Wp: a normalization term. σs: standard deviation for difference in depth. σc: standard deviation for pixel distance. This filter unfortunately introduces the unpleasant ski effect as shown in Figure 1. (a) Unfiltered kinect image of a desk. (b) After filtering the edge of the plane is bent upwards. Fig. 1: The ski effect (red) is introduced by Bilateral Filter- ing. B. Sigma Adaptive Bilateral Filter Andreas Deutschmann [2] introduced the Sigma Adaptive Bilateral Filter which got rid of the ski effect and is contain- ing edges, by reducing sigma around corners and edges. D∗p= 1 Wp ∑ q∈Sp Gσs,p(‖p−q‖)Gσc,p(|Dp−Dq|)Dq (3) Wp=∑ q∈Sp Gσs,p(‖p−q‖))Gσc,p(|Dp−Dq|) (4) Where σs,c,p=σs,c,max+msat,p∗(σs,c,min−σs,c,max) (5) is depending on the depth-maps curvature msat,p=      1 ifm>(1−kth) 0 ifm<kth m else . (6) With mp= m˜p−m˜min m˜max−m˜min (7) and m˜p= ∥∥∥∥∥ 1|Rp|∑q∈pR(Pp−Pq) ∥∥∥∥∥. (8) The terms are described the following: m˜: is the raw curvature. m: is the normalized curvature of the surface see Figure 5a. msat: is a curvature that is saturated by kth and (1−kth). σs,p: standard deviation of the Gauss filter. Weighing depend- ing on the difference in depth. σc,p: standard deviation of the Gauss filter. Weighing depend- ing on the pixel distance. Pp,q: thepoint at porq, givenasvectorPp,q= [ xp yp zp ]T Rp: likeS a neighborhood around pwhere ∥∥Pp−Pq∥∥<rRth. This filter essentially is a Bilateral Filter which is suppressed in critical regions like edges (see Figure 3). 168