Extraction of 3D Transform and Scale Invariant Patches from Range Scans

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1 Extraction of 3D Transform and Scale Invariant Patches from Range Scans Erdem Akagündüz Dept. of Electrical and Electronics Engineering Middle East Technical University ANKARA 06531, TURKEY İlkay Ulusoy Dept. of Electrical and Electronics Engineering Middle East Technical University ANKARA 06531, TURKEY Abstract An algorithm is proposed to extract transformation and scale invariant 3D fundamental elements from the surface structure of 3D range scan data. The surface is described by mean and Gaussian curvature values at every data point at various scales and a scale-space search is performed in order to extract the fundamental structures and to estimate the location and the scale of each fundamental structure. The extracted fundamental structures can later be used as nodes in a topological graph where the links between the nodes can be defined as the spatial and geometric relations between the fundamental elements. 1. Introduction With the fast increase in the usage of 3D range scanners in many applications such as robotics and biometrics, the need to process 3D point cloud data in order to extract representative information from the scanned scene has become very important. The 3D points in the 3D point cloud are samples from the surface of the scanned scene and are usually uniform. There have been many studies performed for industrial parts which usually have flat surfaces [33] and for volumetric primitives such as cone and cylinder [5] but when the scanned scene includes free form shapes, arbitrary smooth surfaces are needed to be described. Especially when the task is face recognition from 3D scanner outputs, it is vital to extract descriptive surface information both to segment the face from the background and also to differentiate a face from another one. Reliable curvature estimation is an important goal in scene analysis to provide viewpoint independent cues for shape classification [12]. Concepts and techniques from differential geometry provide several measures of curvature including Gaussian and mean curvatures in describing the shape of arbitrary smooth surfaces arising in range images. Combination of Gaussian and mean curvature values enables the local surface types to be categorized that are invariant to rotations and translations [1]. Also, such local surface descriptions are better than global shape descriptors for shape matching schemes when occlusion exists. Although there are very successful global surface shape descriptors such as Extended Gaussian Images (EGI) [14] they have difficulty leveraging subtle shape variations, especially with large parts of the shape missing from the scene and they have to be processed further in order to be used for shape matching in occluded and cluttered scenes. There are many studies for local surface description. In [14] spin-image was used as a data level shape descriptor where the coordinates (alpha and beta) were computed for a vertex in the surface mesh that is within the support of the spin image. The support distance is a parameter which sets the maximum distance between the oriented point and a point contributing to the spin image and is analogous to window size in 2D template matching which has a direct relation to the scale of the analysis. Thus, spin image algorithm is sensitive to variation in data resolution. In [18] spin image was processed to become a scale invariant one. In [1] the signs of mean curvature (H) and Gaussian curvature (K) were used to determine a surface type for each pixel. An alternative for curvature estimation was defined in [17] where the shape (S) and the magnitude of the curvedness (C) are decoupled. Both HK and SC methods were compared in [4] in terms of their shape classification performance, their response to thresholding and their tolerance to noise. There are many other curvature estimation methods. In [8] Darboux frames which describe the orientation, principle curvatures and directions at a point on a surface were used. In [18] curves and surfaces were estimated using B-splines. In [30] the tensor voting formalism was used to infer the sign and direction of principal curvatures. Due to the need for second derivatives, surface curvature estimates are very sensitive to quantization noise. In [9] surface fitting and numerical curvature estimates were compared in terms of their tolerances to noise. They concluded that the numerical curvature estimation methods perform about as accurately as the analytical techniques and it is very important to apply surface smoothing before any type of curvature estimation /07/$ IEEE

2 algorithm. However, although the selected mask (kernel or filter) size which defines the extent of smoothing and curvature estimation is very effective on the outcomes of the methods, they did not compare the methods with respect to various sizes. In [11] a method to infer surfaces while detecting intersections between surfaces and 3D juctions was proposed. Such qualitative segmentation of the range image into a set of volumetric parts not only captures the coarse shape of the parts, but qualitatively encodes the orientation of each part through its aspect. When the 3D scene is segmented into consistent surfaces then object recognition can be performed by matching these surfaces to the model (or database) based on their surface descriptors. There are many other methods which consider surfaces altogether with their boundaries and junctions [1], [29], [6]. However, in most of the surface segmentation studies, the main problem is that the curves and junctions among surfaces are not well localized. This problem is mostly related to the size of the mask used which has a direct link to the scale of the analysis. The scale problem that we mentioned above for the cases of curvature estimation and surface segmentation can be handled if a multi-scale analysis is performed on the surface structure. Most of the studies which consider multi-scale analysis perform scale-space processing which is concerned with obtaining more than one set of results for a single data, by processing the data at different resolutions [20]. Traditionally, a coarse to fine processing is performed in order to combine the results across scales. This also has some physiological and psychological roots since humans retrieve the coarse geometric shapes first and then more details are added to refine the judgment if needed. In [13] surface models from low to high details in scale space were used for modeling purposes where the aim was to model a 3D scene without losing any details but by including the least number of polyhedral surfaces. There are many other similar methods which use multiscale analysis for modeling purposes [35], [31]. In [33] multi-scale curvature computation was achieved by convolving local parameterizations of the surface iteratively with 2D Gaussian filters and by locating local maxima of Gaussian and mean curvatures at each scale which were then used as significant and robust feature points. Also, zero crossings of Gaussian and mean curvatures were used for segmenting surfaces into regions. Although significant points and their boundaries were extracted at different surface resolutions, information between different resolutions was not combined and these features were left for further processing for robust surface matching. In [2] a solution for matching using features at different scale levels was proposed but for CAD databases only. However, it is a promising approach where a graph structure is constructed for the object at various scales and those graphs are combined under a root. During matching, the multi-scale graph structure of the object was searched through the database. The most problematic assumption made here was that the graph structure of the object and the models in the database were available at every scale. Formulation of 3D object recognition problem as that of graph matching is an appreciated method in computer vision community. The nodes in the graph represent 3D features or their abstractions and edges represent spatial, geometric or hierarchical relations between the features. The relations can include other types of information as well. For example, segmented surfaces described by their curvature properties were used as nodes and the relation between them were viewed as links in attributed graphs for only industrial objects in [33] and for general objects in [7]. In [5] volumetric primitives (cone, cylinder, etc.) were extracted and their connectivity relations were used in a graph structure. The topic of graph matching in computer vision has been studied extensively, with both exact and inexact graph matching algorithms applied to object recognition, including [10], [16], [22], [24], [25], [26], [27], [28], [32] to name just a few. In this study, we also apply multi-scale analysis to 3D point cloud data but not as a coarse to fine approach (which was the main trend in earlier studies) but as a scale space where we can search for fundamental elements (significant locations) and their exact scales on the surface. Thus, it is a scale invariant approach, since we locate significant surface structures with their scale information. In doing this, we were inspired from the work of Lowe [21] and Mikolajczyk and Schmid [23] where they located significant points irrespective of their scales on 2D images. In our 3D case, the fundamental elements are taken as the peaks, pits and saddles which are more realistic structures for natural 3D objects such as human faces. For the extraction of local surface structures we use mean and Gaussian curvature values which are invariant to rotations and translations. We do not perform a surface segmentation in order to describe a scene as proposed by earlier studies where border localization was a very critical issue in segmentation but we describe the scene by its fundamental elements. The configuration of these fundamental elements may later be formulated as a graph where the nodes can be the fundamental structures and the links can be directed and carry relative positional or geometric relations between the nodes. Graph structure and graph matching are out of the scope of this paper and may be done by using one of the successful algorithms suggested in the literature listed in the previous paragraph and left for our future work. The outline of the paper is as follows: In section 2, we commence by detailing steps of scale space analysis including HK map calculation, Gaussian pyramiding and fundamental element labeling. Section 3 discusses the scale estimation in UVS space where u and v are used for surface dimensions and s is for scale. Section 4

3 demonstrates the results on natural objects. Finally, Section 5 offers some conclusions and directions for future research. 2. Surface Representation We begin by introducing our method for finding the fundamental elements on the input surface. Then we explain Gaussian pyramid construction which is computed by Gaussian filtering and down sampling. Finally, we introduce our scale space Extraction of Fundamental Surface Elements In their original work [1], Besl and Jain calculated mean and Gaussian curvatures on the surface and used tolerance signum functions sgn ЄH (H(i,j)) and sgn ЄK (K(i,j)) with preselected zero thresholds to label the fundamental elements on the surface. We use exactly the same method to detect peaks, pits and saddle regions with the threshold values very close to the ones defined by them (We used Є H =0.03 and Є K =0.005). In order to calculate the Gaussian and mean curvatures we fit third order 3x3 B-Spline surfaces on each point, which is the closest analytical technique to using a numerical estimate, as it approximates the derivation with difference operations. In Figure 1, HK map for a facial surface is shown where blue regions are the peaks, cyan regions are the pits, red regions are the saddle ridges and yellow regions are the saddle valleys Gaussian Pyramiding Estimating surface labels at the given resolution of the original surface, restricts us to find the fundamental elements only at the lowest scale. In order to detect other curvatures which have higher scales, a scale space representation must be introduced. For this purpose, we first generate a Gaussian pyramid of the input surface using the Reduce function from the original work of Burt and Adelson [3] where a Gaussian pyramid is constructed by smoothing and down sampling the data into its half at each Reduce operation. Therefore the size of each fundamental element on the surface is halved between successive scales. Then we calculate the HK maps for each scale in the pyramid and obtain a new pyramid of HK maps where 3x3 window size is used at every scales (Figure 2). As can be seen from Figure 2, at the higher levels of the pyramid, the smaller surface elements vanish and bigger elements reside. Finally, we expand the higher levels of the HK pyramid to the original size by using Expand function. After this expansion any label on the s th level will widen 2 S times. Figure 1: Peaks (blue), pits (cyan) and saddles (red) on the surface of a face UVS Space Figure 2: Gaussian Pyramid of the HK maps. Putting each level of the expanded HK pyramid on top of each other, we obtain a 3D volume where u and v are the surface dimensions and s is the scale dimension. In order to extract fundamental surface elements we follow several steps. First, we check each voxel of the 3D volume for its similarity within its 10 neighbors (8 at the same level, 2 up and down through scale). If all labels of the neighbors are the same, the center voxel continues to carry its label. Otherwise it becomes a blank voxel. After voxel labeling, we apply erosion and dilation operations respectively to extract connected labels in the 3D volume. Finally, each connected component inside the u-v-s volume represents a fundamental element on the surface and becomes a node of our topology graph. Figure 3 illustrates the u-v-s space for the face surface given in Figure 2. It is easily seen that the connected components of a fundamental element, for example the nose, has components at a number of successive layers. Remember that the points in the components are found by applying some threshold to H and K values at each scale level. In Figure 4 the points are given with their actual values above the threshold. The bigger the value is the lighter the color becomes. For example, the change in the value of the curvature for the pits of the eyes can be observed from its color which changes from light cyan at the smaller scales to dark cyan at the larger scales. Examining the color change for the eye pits, one can decide that the scale of the eye pit is between level 2 and 3 of the pyramid. Localization of scale information in scale-space is explained in the next section.

4 Figure 3: UVS volume after morphological operations printed above the surface. connected component using the 2 nd norm of the absolute differences of the curvature values: 1 2 ( H ) + ( ) ) 2 2 w (3) i, j = i, j H K i,j K Figure 4: A closer view of the UVS space in Figure 3. The colors label the surface labels (cyan for pits, red for peaks and blue for saddles). The lighter color means that the absolute difference between the curvature value and the threshold is bigger. 3. Scale-Space Localization Since we use Gaussian smoothing to construct the scalespace volume, the relation between the smoothing kernel size and the scale dimension of the volume is as follows: σ B ( S A S B ) = 2 (1) σ A This relation is verified on synthetic Gaussian surfaces at different scales. Figure 5 shows UVS spaces for three synthetically generated unit volume Gaussian surface models. Each Gaussian surface has the half standard deviation of the one on its right. Then according to (1) the scale values for these Gaussian surfaces must have the following relation: σ B ( S A S B ) = 2 = 2 S A SB = 1 σ A (2) The location and scale of a fundamental element in the UVS volume are estimated by computing the weighted average of the curvature values of the elements (voxels) covered by the connected component defining that fundamental element (peak for synthetic Gaussians). A weight value is assigned to each voxel inside the After computing the weighted averages we observe that the estimated locations are the centers of the synthetic Gaussians and scales satisfy the relation given in (2) where S A = 3.86, S B = 2.80 and S C = Each connected component with its surface label, location and scale information may be represented as a node in a topology graph which is out of the scope of this paper. But we suggest that if the links between the nodes carry some relative information (for example ratio of the bigger node scale to the smaller node scale contributing to the link) the graph will become a directed one and this may help graph matching which is usually a hard problem for fully connected graphs. Figure 5: Three unit volume Gaussian surfaces at different scales (σ A = σ K, σ B = 2*σ K, σ C = 4*σ K ) and their respective UVS volumes. 4. Results Initially we have tested the algorithm on objects belonging to the same class and having nearly the same size and orientation. We have used range maps of three different facial surfaces for which the UVS volumes are shown above the facial surfaces in Figure 6. The nodes (fundamental elements) extracted for each of these surfaces are given in Table 1. Since they are similar surface structures, the extracted components have similar

5 Figure 6: UVS volumes and surfaces for three different facial models. Table 1: Extracted nodes from the surfaces in Figure 6. type U V S type U V S type U V S Pit 18,81 38,82 2,76 Pit 23,55 45,35 2,58 Pit 22,44 40,13 2,69 Left Eye Pit Pit 19,96 85,53 2,86 Pit 22,43 82,73 2,50 Pit 23,45 82,85 2,63 Right Eye Pit Peak 55,19 62,00 2,67 Peak 57,43 62,70 2,72 Peak 53,52 64,42 2,71 Nose Peak Peak 101,15 72,17 1,59 S. Rd. 19,59 64,30 2,83 S. Rd. 21,31 63,59 2,54 Nose Saddle Peak 28,30 102,36 1,53 Peak 104,70 72,04 1,90 Peak 98,38 71,52 1,66 Pit 20,15 3,97 2,00 Pit 30,53 4,54 2,29 Pit 24,02 3,89 2,00 Peak 28,93 22,47 2,00 Peak 105,58 54,38 2,36 Peak 98,65 54,00 2,14 Pit 97,76 32,53 2,53 Peak 88,87 63,00 2,00 Peak 85,74 55,69 2,25 S. Vl. 34,98 102,03 2,00 Peak 85,97 67,98 2,00 Peak 85,70 71,46 2,30 Pit 9,38 93,56 3,00 S. Vl. 116,52 77,21 2,00 Peak 29,93 101,48 2,00 Peak 100,76 53,62 1,60 S. Vl. 97,01 57,00 3,00 Pit 98,88 93,89 2,44 S. Vl. 116,96 63,99 3,00 Pit 19,19 51,00 1,00 S. Vl. 97,01 66,93 3,00 a) b) c) coordinates and scales. For the sake of clarity we placed some selected facial fundamental components such as the nose peak and the eye pits on the top rows in Table 1 where u, v and scale values for each component are given in the corresponding columns. As it can be seen from this table, these components have very similar scales and coordinates on the surfaces. Also, scale space of each face has some different fundamental structures from the others. For example, the face given in the middle of Figure 6 has an emphasized nose saddle (3 layers of red between the eye pits). Next, we have tested our algorithm against transformations and scaling. Figure 7 shows a facial surface, which has been scanned at different resolutions and poses. The frontal view of the original face is shown on the left. In the middle, the face is given at a different resolution and while it is rotated with respect to the scanner s viewing direction. Finally, on the right, the face scanned from a different viewing angle is given. Since this face has an out of plane rotation some facial surfaces are occluded. These surfaces are generated artificially from the original face scan. Likewise, Table 2 lists the nodes extracted from these surfaces and again important facial components are given on the top rows. The u-v coordinates and scales of these surfaces follow the transformations and scaling applied to the original face which also shows that the algorithm is invariant to scale and rotation.

6 Figure 7: UVS volumes and surfaces for scaled and rotated versions of a face. a. Original face, b. Smaller and in plane rotated face, c. Out of plane rotated face. Table 2: Extracted Nodes from the surfaces in Figure 7. type U V S type U V S type U V S Pit 79,20 35,17 2,64 Pit 45,16 68,88 2,45 Pit 64,39 40,68 3,00 Left Eye Pit 50,99 34,96 2,69 Pit 27,97 74,67 2,43 Pit 41,55 40,83 3,00 Right Eye Peak 66,25 64,56 2,90 Peak 41,79 86,39 3,04 Peak 60,95 62,84 2,93 Nose Peak 78,13 120,89 2,65 Peak 54,46 115,52 2,56 Peak 53,71 107,23 2,68 Chin Peak 56,42 121,63 2,46 Peak 44,85 57,48 2,62 Peak 35,34 25,76 2,42 Peak 35,51 17,26 2,36 Peak 18,91 66,34 2,73 Peak 21,98 26,02 2,00 Peak 98,81 20,12 2,70 Peak 59,28 82,70 2,00 Peak 29,66 38,00 2,00 Peak 35,05 32,15 2,00 Peak 41,10 102,08 2,00 Pit 81,20 20,49 3,00 S. Vl. 12,49 1,00 3,00 Pit 10,71 34,14 3,00 Peak 70,74 27,38 3,00 Pit 16,99 106,00 3,00 Pit 56,88 37,26 3,00 Pit 81,10 45,27 3,00 Pit 105,56 116,11 3,00 Pit 65,08 48,87 3,00 Peak 54,00 83,54 3,00 Pit 23,89 116,12 3,00 Pit 69,02 56,99 3,00 Pit 76,70 101,08 3,00 Pit 8,99 95,53 3,00 Pit 7,71 106,61 3,00 Pit 12,97 103,10 3,00 S. Vl. 57,00 127,01 3,00 Pit 20,80 116,85 3,00 S. Vl. 80,52 1,00 4,00 a) b) c) When examined in detail, one can see that the nose of the face given in Figure 7.b. has bigger scale than that of the noses given in Figure 7.a and 7.c. Even though this is something not expected, it has a reasonable explanation. In Figure 7.b, as the surface is processed for higher scales also, the face itself becomes a peak lying on the plane the face belongs to. This bump is at higher scales and has coordinates very close to that of the nose, since nose is in the middle of a face. However if they are taken as a single connected component, the scale of the peak at the coordinates of the nose becomes bigger. This phenomenon is expected for data with lower resolution. 5. Conclusion In this paper we have defined a method to extract transform and scale invariant elements from 3D surfaces using a multi-scale analysis technique. We have estimated mean and Gaussian curvature values for a given surface at various scales. Then, we have constructed Gaussian Pyramid of the surface in order to estimate the scale and coordinates of fundamental surface elements on the surface. We have built a UVS volume space where fundamental elements can be extracted by morphological operations and connected component search. Pyramid techniques are fast as the scale increases exponentially. However inspection of the UVS space by increasing the scale linearly might give a better localization in scale space. By obtaining such transform and scale invariant fundamental surface elements, registration of scanner outputs could be achieved easily. Also, by constructing topological graphs based on the fundamental surface elements and their relations, some very well known exact or inexact graph matching techniques could be applied for object recognition purposes. The features to be used for the nodes and the links in the graph structure and the graph matching methods form our future study topics.

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