Gesture-Based Interaction with Time-of-Flight Cameras

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CHAPTER 4. SHADING CONSTRAINT 4.2.3 Computation of Surface Normals Both the Lambertian reflectance model and the shape prior for smooth surfaces re- quire the normals of the surface to be known; some care needs to be taken when computing these normals on a discretely sampled range map. An obvious way is to compute the cross product of two tangent vectors p(i + 1, j) − p(i − 1, j) and p(i, j + 1) − p(i, j − 1) (where p(i, j) are the three-dimensional coordinates of the point corresponding to pixel (i, j)), but surface normals calculated in this way can lead the minimizer astray: Because the normals of pixels with even indices de- pend only on the positions of pixels with odd indices, and vice versa, neighboring pixels are not constrained to have similar range values, and the minimizer may hap- pily compute a surface with a “checkerboard” pattern, where neighboring pixels are alternately displaced upwards and downwards by a certain offset, instead of forming a smooth surface. For this reason, care needs to be taken when evaluating the reflectance model and shape prior to obtain a formulation that is physically realistic and does not lead the minimizer astray. For any given pixel, the range and intensity measured by the camera are averages over the area covered by the pixel. Nevertheless, we will assume that these values correspond to the range and intensity of an individual point on the object’s surface. To obtain a continuous surface between this grid of points, we tessellate the grid into triangles. Of the many possible tessellations, we choose one where the diagonals dividing a quad of pixels into two triangles run parallel. To compute the intensity of a pixel, given a certain range map, we compute the average intensity over all triangles adjacent to it. (All triangles have the same pro- jected area in the image, hence they are all weighted equally.) Because the triangles in the mesh are small compared to their distance from the camera, we can assume that intensity is constant across a triangle. The intensity Ij for pixel j is thus obtained as follows: Ij = aj ∑ k∈Nj nk · lj R2 j |Nj| , ✞ ☎ ✝4.9 ✆ where aj is the albedo of the pixel, Rj is the range value of the pixel, lj is the unit vector from the surface point to the light source, nk is the surface normal of triangle k, and Nj is the set of triangles that are adjacent to pixel j; see Figure 4.1 for an illustration of the triangles that are adjacent to a pixel. When computing the shape prior, we must take care to count each edge exactly once. We do this by iterating over all pixels and, for each pixel, evaluating the shape prior only for those edges in the tessellation that lie below and to the right of the pixel 38
4.2. METHOD Figure 4.1: The intensity of a pixel is computed by averaging over the intensity of all triangles that are adjacent to it. Triangles that are adjacent to the central pixel are shaded in gray. (a) (b) Figure 4.2: To avoid directional artefacts, the shape prior is evaluated over two different tessellations, with the diagonals running in opposite directions. Figure 4.2a shows a tessellation with diagonals running top left to bottom right. For each pixel, the shape prior is evaluated for the edges (shown in bold) below and to the right of the pixel. Figure 4.2a shows a tessellation with diagonals running bottom left to top right. The shape prior is evaluated for the edges below and to the left of the pixel. 39
Page 1 and 2: Aus dem Institut für Neuro- und Bi
Page 3 and 4: Contents Acknowledgements vi I Intr
Page 5 and 6: CONTENTS 7.3.2 Sparse Features . .
Page 7: Acknowledgements There are a number
Page 11 and 12: 1 Introduction Nowadays, computers
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Page 15 and 16: 2.1 Introduction 2 Time-of-Flight C
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Page 40 and 41: CHAPTER 3. INTRODUCTION the class i
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7.2. SCALE INVARIANT FEATURES the o
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7.2. SCALE INVARIANT FEATURES exhib
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detection rate detection rate 1 0.8
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7.2. MULTIMODAL SPARSE FEATURES 7.3
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7.3. MULTIMODAL SPARSE FEATURES Fig
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7.3. MULTIMODAL SPARSE FEATURES Fig
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7.3. MULTIMODAL SPARSE FEATURES eac
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7.3. MULTIMODAL SPARSE FEATURES tem
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false positive rate 0.4 0.35 0.3 0.
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7.4. LOCAL RANGE FLOW FOR HUMAN GES
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Part III Applications 119
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CHAPTER 8. INTRODUCTION an alternat
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CHAPTER 9. FACIAL FEATURE TRACKING
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CHAPTER 9. FACIAL FEATURE TRACKING
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10.1 Introduction 10 Gesture-Based
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10.2. METHOD scenarios: One, where
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10.2. METHOD Figure 10.3: Segmented
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Table 10.1: Interpretation of point
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corner along the ray. 10.2. METHOD
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600 500 400 300 200 100 0 0 10 20 3
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10.4. DISCUSSION tention here was t
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CHAPTER 11. DEPTH OF FIELD BASED ON
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CHAPTER 12. CONCLUSION alization of
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Bibliography ARTTS 3D TOF Database.
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BIBLIOGRAPHY James R. Diebel and Se
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BIBLIOGRAPHY ence on Computer Visio
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BIBLIOGRAPHY Stefan Kunis and Danie
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BIBLIOGRAPHY Thierry Oggier, Michae
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BIBLIOGRAPHY Rudolf Schwarte, Horst
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BIBLIOGRAPHY ’06: Proceedings of
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modulation frequency, 8, 23, 26 Mon
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Gesture-Based Interaction with Time-of-Flight Cameras

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