Page - 169 - in Proceedings of the OAGM&ARW Joint Workshop - Vision, Automation and Robotics

Fig. 2: Unfiltered depth data. (a) Bilateral Filter. Introduces ski effect (green). (b) But shows good results at planes. (c) Sigma Adaptive Filter. Pre- serves details like edges, but can’t filter noise at discontinu- ities (red). (d) Delivers good results when applied to planar regions. Fig.3:FilteringresultsofBilateralFilterandSigmaAdaptive Filter. C. Bilateral Tangential Filter The promoted filter is based on the Bilateral Mesh Denoising algorithm [4] which is used for meshes but not raw depth data. The idea behind this filter is to correct each point along its normal by a value composed by the deviation of surrounding points to its tangent plane. We adapt this principle to depth data by not correcting the points along their normal as in [4] but along the camera view rays. The filter is written as Cp= 1 Wp ∑ q∈Sp Gσs(‖p−q‖)Gσc(dp,q)dp,q (9) Wp=∑ q∈Sp Gσs(‖p−q‖)Gσc(dp,q) (10) where the correction termCp,k is used to correct the depth values D∗p=Dp+Cp. (11) dp,q is the distance of the point q to the tangent plane of p dp,q=np ·(Pp−Pq). (12) It is not implied in this equation, but this filter is meant to be used iteratively. 0 5 10 15 20 25 0510 152025 −4 −2 0 2 4 x 10−3 (a)Kq,big. 1 2 3 4 5 1 2 3 4 5 −0.5 0 0.5 (b)Kq,small. Fig. 4: Filtering kernels, to calculate horizontal and vertical derivation of x, y and z. Sizes for these kernels are 23x23 and 5x5. The quality of this filter strongly depends on the normal vectors np which tend to be difficult to obtain, especially along discontinuities and in noisy data. Incorrect normal values can make the algorithm locally unstable. Figure 6 shows a good example for how normal vectors affect the result. The normal vector is calculated by the vertical and horizontal derivation of x, y and z coordinates by the image coordinates u and v. np= n˜p ‖n˜p‖ n˜p=    dxpdu0 dzp du    ×    0dypdv dzp dv    (13) To obtain the needed derivatives we can not rely on a Canny Edge detection like approach since this would lead to wrong normals along discontinuities. We therefore have to mix the Canny Edge detection with the idea of the Bilateral Filter. To reduce the impact of discontinuities on the normals, points which are further away from the center point contribute less or not at all. This is achieved by an other Gaussian termGσn. d(x,y,z)p du,v =∑ q∈Sp Kq,pGσn(Dp−Dq)((x,y,z)p−(x,y,z)q) (14) Since the depth data along edges of objects is often distorted, it is necessary to compensate for that by locally extending the kernel: Kq,p= { Kq,big if cp>cth Kq,small else (15) The filter kernels itself are shown in Figure 4. As basis to decide we are using a measure for how erratic the image is (Figure 5b). cp=∑ q∈R′p ∣∣Dp−Dq∣∣ (16) One example for proper filtering kernels are shown in Figure 4. Note thatR′p is in this case the neighborhood of p where |p−q|<rR′th. IV. RESULTS OF FILTERING The standard Bilateral Filter introduces the unpleasant ski effect [2] and therefore does not preserve information on edges (see Figure 3). On said edges the ski effect refers to a 169