PG2011 Talk by Fatih Calakli

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Slides used by Fatih Calakli to present his paper at Pacific Graphics 2011

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  • Creating digital models of real world objects always attracts a lot of attention due to numerous applications :
  • Creating digital models of real world objects always attracts a lot of attention due to numerous applications :
  • Creating digital models of real world objects always attracts a lot of attention due to numerous applications :
  • Creating digital models of real world objects always attracts a lot of attention due to numerous applications :
  • Digitization procedure starts off with shape acquisition.
  • Even more convenient, it is even possible to scan objects and scenes with a set a photographs taken from different views. In this case, only equipment you need would be a single camera and a robust multiview stereo algorithm. 1. These devices provide typically provide point samples. 2. We assume each point sample is associated with a surface normal vector at the sample’s location.3. In this work, we address the problem of reconstructing watertight surfaces from oriented point sets. 4. This problem is still challenging due to various complications.
  • Even more convenient, it is even possible to scan objects and scenes with a set a photographs taken from different views. In this case, only equipment you need would be a single camera and a robust multiview stereo algorithm. 1. These devices provide typically provide point samples. 2. We assume each point sample is associated with a surface normal vector at the sample’s location.3. In this work, we address the problem of reconstructing watertight surfaces from oriented point sets. 4. This problem is still challenging due to various complications.
  • Even more convenient, it is even possible to scan objects and scenes with a set a photographs taken from different views. In this case, only equipment you need would be a single camera and a robust multiview stereo algorithm. 1. These devices provide typically provide point samples. 2. We assume each point sample is associated with a surface normal vector at the sample’s location.3. In this work, we address the problem of reconstructing watertight surfaces from oriented point sets. 4. This problem is still challenging due to various complications.
  • Even more convenient, it is even possible to scan objects and scenes with a set a photographs taken from different views. In this case, only equipment you need would be a single camera and a robust multiview stereo algorithm. 1. These devices provide typically provide point samples. 2. We assume each point sample is associated with a surface normal vector at the sample’s location.3. In this work, we address the problem of reconstructing watertight surfaces from oriented point sets. 4. This problem is still challenging due to various complications.
  • 1.The prior-art in surface reconstruction is extensive. 2. One family of algorithms produces interpolating polygon meshes where all or some of the input points become vertices of the polygons .3. Most of these algorithms are combinatorial in nature. As a result, are not scalable to large amount of data. 4. It is often more desirable to produce approximating surfaces especially in presence of measurement noise. 5. There has been extensive work in this family as well. We refer the interested audience to a survey paper by Schall and Samazino summarizing two decades of work. 6. However, the area is still active. With recent developments, it is possible to reconstruct surfaces from large amount of data in great detail.
  • 1.The prior-art in surface reconstruction is extensive. 2. One family of algorithms produces interpolating polygon meshes where all or some of the input points become vertices of the polygons .3. Most of these algorithms are combinatorial in nature. As a result, are not scalable to large amount of data. 4. It is often more desirable to produce approximating surfaces especially in presence of measurement noise. 5. There has been extensive work in this family as well. We refer the interested audience to a survey paper by Schall and Samazino summarizing two decades of work. 6. However, the area is still active. With recent developments, it is possible to reconstruct surfaces from large amount of data in great detail.
  • 1.The prior-art in surface reconstruction is extensive. 2. One family of algorithms produces interpolating polygon meshes where all or some of the input points become vertices of the polygons .3. Most of these algorithms are combinatorial in nature. As a result, are not scalable to large amount of data. 4. It is often more desirable to produce approximating surfaces especially in presence of measurement noise. 5. There has been extensive work in this family as well. We refer the interested audience to a survey paper by Schall and Samazino summarizing two decades of work. 6. However, the area is still active. With recent developments, it is possible to reconstruct surfaces from large amount of data in great detail.
  • Let’s look at the general problem of fitting an implicit function to oriented points. Given oriented points, samples from an unknown surface S, the problem is to compute a scalar field f whose zero level set is the estimate for the true surface S. Since multiplying the implicit function with a non zero function doesn’t change the zero-level set, researchers tried a variety of functions.
  • Let’s look at the general problem of fitting an implicit function to oriented points. Given oriented points, samples from an unknown surface S, the problem is to compute a scalar field f whose zero level set is the estimate for the true surface S. Since multiplying the implicit function with a non zero function doesn’t change the zero-level set, researchers tried a variety of functions.
  • Let’s look at the general problem of fitting an implicit function to oriented points. Given oriented points, samples from an unknown surface S, the problem is to compute a scalar field f whose zero level set is the estimate for the true surface S. Since multiplying the implicit function with a non zero function doesn’t change the zero-level set, researchers tried a variety of functions.
  • In poisson reconstruction, an indicator function is used. It is a binary valued function which takes 1 in the inside, 0 on the outside of the surface. Instead, we present to use smooth function, so we can compute the gradient, and directly compare it with the normal vector data. Smooth signed distance function becomes Euclidean signed distance function near data samples
  • Mention that the gradient of f is forced to be normal near data, so its length is also forced to be 1. As a result, it behaves like an Euclidean signed distance function near data.
  • Mention that the gradient of f is forced to be normal near data, so its length is also forced to be 1. As a result, it behaves like an Euclidean signed distance function near data.
  • Mention that the gradient of f is forced to be normal near data, so its length is also forced to be 1. As a result, it behaves like an Euclidean signed distance function near data.
  • Mention that the gradient of f is forced to be normal near data, so its length is also forced to be 1. As a result, it behaves like an Euclidean signed distance function near data.
  • Mention that the gradient of f is forced to be normal near data, so its length is also forced to be 1. As a result, it behaves like an Euclidean signed distance function near data.
  • Minimizing the Hessian of f, is the same as minimizing the jacobian of the gradient of f.
  • Minimizing the Hessian of f, is the same as minimizing the jacobian of the gradient of f.
  • Minimizing the Hessian of f, is the same as minimizing the jacobian of the gradient of f.
  • Minimizing the Hessian of f, is the same as minimizing the jacobian of the gradient of f.
  • 1. The energy may or may not have a unique solution, so we constrain f belong a linear family of functions around a fixed set of basis function phi. 2. Phi can be one of the popular smooth basis functions. 3. With this restriction, our energy E(f) becomes a non-homogeneous and quadratic energy in the vector F. 4. Thematrix A is typicallysymmetric and positive definite, and the resulting minimization problem has a unique minimum. 5. The global minimum is determined by simply solving the system of linear equations, AF = b.
  • 1. The energy may or may not have a unique solution, so we constrain f belong a linear family of functions around a fixed set of basis function phi. 2. Phi can be one of the popular smooth basis functions. 3. With this restriction, our energy E(f) becomes a non-homogeneous and quadratic energy in the vector F. 4. Thematrix A is typicallysymmetric and positive definite, and the resulting minimization problem has a unique minimum. 5. The global minimum is determined by simply solving the system of linear equations, AF = b.
  • 1. The energy may or may not have a unique solution, so we constrain f belong a linear family of functions around a fixed set of basis function phi. 2. Phi can be one of the popular smooth basis functions. 3. With this restriction, our energy E(f) becomes a non-homogeneous and quadratic energy in the vector F. 4. Thematrix A is typicallysymmetric and positive definite, and the resulting minimization problem has a unique minimum. 5. The global minimum is determined by simply solving the system of linear equations, AF = b.
  • 1. The energy may or may not have a unique solution, so we constrain f belong a linear family of functions around a fixed set of basis function phi. 2. Phi can be one of the popular smooth basis functions. 3. With this restriction, our energy E(f) becomes a non-homogeneous and quadratic energy in the vector F. 4. Thematrix A is typicallysymmetric and positive definite, and the resulting minimization problem has a unique minimum. 5. The global minimum is determined by simply solving the system of linear equations, AF = b.
  • PG2011 Talk by Fatih Calakli

    1. 1. SSD: Smooth Signed Distance Surface Reconstruction Fatih Calakli and Gabriel Taubin School of Engineering, Brown University
    2. 2. Motivation• Industry – Reverse engineering – Fast metrology – Physical simulations
    3. 3. Motivation• Industry – Reverse engineering – Fast metrology – Physical simulations• Entertainment – Animating digital clays for movies or games
    4. 4. Motivation• Industry – Reverse engineering – Fast metrology – Physical simulations• Entertainment – Animating digital clays for movies or games• Archeology and Art – Digitization of cultural heritage and artistic works
    5. 5. Motivation• Industry – Reverse engineering – Fast metrology – Physical simulations• Entertainment – Animating digital clays for movies or games• Archeology and Art – Digitization of cultural heritage and artistic works• Medical Imaging – Visualization – Segmentation• …
    6. 6. Data acquisitionLaser range scanning devices Multi-camera systems Structured lighting systems
    7. 7. [Snavely et. al. 2006][Furukawa and Ponce 2008]
    8. 8. [Snavely et. al. 2006] MVSsoftware [Furukawa and Ponce 2008]
    9. 9. [Snavely et. al. 2006][Furukawa and Ponce 2008]
    10. 10. [Snavely et. al. 2006][Furukawa and Ponce 2008]
    11. 11. ChallengesUniform sampling
    12. 12. Challenges Uniform samplingNon-uniform sampling Noisy data Misaligned scans
    13. 13. Outline• Introduction• Related Work• Approach• Evaluation• Conclusion
    14. 14. General Approaches• Interpolating polygon meshes -Boissonnat, 1984 -Edelsbrunner, 1984 -Amenta et al., 1998 -Bernardini et al., 1999 -Dey et al., 2003, 2007, …• Implicit function fitting -Hoppe et al., 1992 -Curless et al., 1996 -Whitaker, 1998 -Carr et al., 2001 -Davis et al., 2002 -Ohtake et al., 2004 -Turk et al., 2004 -Shen et al., 2004
    15. 15. General Approaches• Interpolating polygon meshes -Boissonnat, 1984 -Edelsbrunner, 1984 -Amenta et al., 1998 -Bernardini et al., 1999 -Dey et al., 2003, 2007, …• Implicit function fitting -Hoppe et al., 1992 -Curless et al., 1996 -Whitaker, 1998 -Carr et al., 2001 -Davis et al., 2002 -Ohtake et al., 2004 -Turk et al., 2004 -Shen et al., 2004 Surface from Scattered Points: A Brief Survey of Recent Developments [Schall and Samozino, 2005]
    16. 16. General Approaches• Interpolating polygon meshes -Boissonnat, 1984 -Edelsbrunner, 1984 -Amenta et al., 1998 -Bernardini et al., 1999 -Dey et al., 2003, 2007, …• Implicit function fitting -Hoppe et al., 1992 -Curless et al., 1996 -Whitaker, 1998 -Carr et al., 2001 -Davis et al., 2002 -Ohtake et al., 2004 -Turk et al., 2004 -Shen et al., 2004 -Kazhdan et al., 2006 -Manson et al., 2008
    17. 17. Implicit function framework Oriented Points, D(samples from unknown surface S) Find a scalar valued function f : D , whose zero level set Z(f) = S’ is the estimate for true surface S
    18. 18. Implicit function framework <0 0 f >0 Oriented Points, D(samples from unknown surface S) Find a scalar valued function f : D , whose zero level set Z(f) = S’ is the estimate for true surface S
    19. 19. Implicit function framework Z( f ) Oriented Points, D Computed Implicit Surface, S’(samples from unknown surface S) Find a scalar valued function f : D , whose zero level set Z(f) = S’ is the estimate for true surface S
    20. 20. Poisson reconstruction [Kazhdan ‘06] 0 1 0 Indicator Function Smooth Signed Distance Function
    21. 21. Outline• Introduction• Related Work• Our Approach• Evaluation• Conclusion
    22. 22. Continuous formulation• Oriented point set: D = { ( pi, ni ) } sampled from a surface S
    23. 23. Continuous formulation• Oriented point set: D = { ( pi, ni ) } sampled from a surface S• Implicit surface: S = { x | f (x) = 0 } such that
    24. 24. Continuous formulation• Oriented point set: D = { ( pi, ni ) } sampled from a surface S• Implicit surface: S = { x | f (x) = 0 } such that f (pi) = 0 and ∇f (pi) = ni ∀(pi,ni) ∈ D
    25. 25. Continuous formulation• Oriented point set: D = { ( pi, ni ) } sampled from a surface S• Implicit surface: S = { x | f (x) = 0 } such that f (pi) = 0 and ∇f (pi) = ni ∀(pi,ni) ∈ D• Least squares energy: N N 2 2 ED ( f ) f (p i ) 1 f (p i ) n i i 1 i 1 2 ER ( f ) 2 Hf (x) dx V
    26. 26. Continuous formulation• Oriented point set: D = { ( pi, ni ) } sampled from a surface S• Implicit surface: S = { x | f (x) = 0 } such that f (pi) = 0 and ∇f (pi) = ni ∀(pi,ni) ∈ D• Least squares energy: N N 2 2 ED ( f ) f (p i ) 1 f (p i ) n i i 1 i 1 2 ER ( f ) 2 Hf (x) dx V E( f ) ED ( f ) ER ( f )
    27. 27. Role of each energy term N N 2 2E( f ) f (p i ) 1 f (p i ) n i 2 || Hf (x) || 2 dx V i 1 i 1• In the vicinity of the data points, – the data terms dominate the total energy – make the function to approximate the “signed Euclidean distance function”.
    28. 28. Role of each energy term N N 2 2E( f ) f (p i ) 1 f (p i ) n i 2 || Hf (x) || 2 dx V i 1 i 1• In the vicinity of the data points, – the data terms dominate the total energy – make the function to approximate the “signed Euclidean distance function”. N 2 Q (v ) 1 v(p i ) n i 2 || Dv(x) || 2 dx, V i 1 where Dv denotes the Jacobian of the vector field v. If ∇f = v, then Hf = Dv.
    29. 29. Role of each energy term N N 2 2E( f ) f (p i ) 1 f (p i ) n i 2 || Hf (x) || 2 dx V i 1 i 1• In the vicinity of the data points, – the data terms dominate the total energy – make the function to approximate the “signed Euclidean distance function”.
    30. 30. Role of each energy term N N 2 2E( f ) f (p i ) 1 f (p i ) n i 2 || Hf (x) || 2 dx V i 1 i 1• In the vicinity of the data points, – the data terms dominate the total energy – make the function to approximate the “signed Euclidean distance function”.• Away from the data points, – the regularization term dominates the total energy – tends to make the gradient vector field ∇f(x) constant.
    31. 31. Linear families of functions
    32. 32. Linear families of functions• Popular Smooth Basis Functions – Radial basis functions [Carr et al., ‘01], – Compactly supported basis functions [Othake et al. ‘04], – Trigonometric polynomials [Kazhdan et al. ‘05], – B-splines [Kazhdan et al., 06], – Wavelets [Manson et al. ‘08],
    33. 33. Linear families of functions• Popular Smooth Basis Functions – Radial basis functions [Carr et al., ‘01], – Compactly supported basis functions [Othake et al. ‘04], – Trigonometric polynomials [Kazhdan et al. ‘05], – B-splines [Kazhdan et al., 06], – Wavelets [Manson et al. ‘08], Non-homogenous, t t Quadratic energy E(F ) F AF 2b F c
    34. 34. Linear families of functions• Popular Smooth Basis Functions – Radial basis functions [Carr et al., ‘01], – Compactly supported basis functions [Othake et al. ‘04], – Trigonometric polynomials [Kazhdan et al. ‘05], – B-splines [Kazhdan et al., 06], – Wavelets [Manson et al. ‘08], Non-homogenous, t t Quadratic energy E(F ) F AF 2b F c Global minimum AF b
    35. 35. Independent Discretization• Hybrid FE/FD discretization – Trilinear interplant for the function f(x) – Finite differences for the gradient ∇f(x) – Finite differences for the Hessian Hf(x)
    36. 36. Independent Discretization• Hybrid FE/FD discretization – Trilinear interplant for the function f(x) – Finite differences for the gradient ∇f(x) – Finite differences for the Hessian Hf(x)• As long as f(x), ∇f(x), and Hf(x) are written as a linear combination of parameter vector F, Non-homogenous, t t Quadratic energy E(F ) F AF 2b F c Global minimum AF b
    37. 37. Implementation• Iterative solver (conjugate gradient) with cascading multi-grid approach – Solve the problem on a much coarser level – Use the solution at that level to initialize the solution at the next level – Refine with the iterative solver• Iso-surface extraction (crack-free) – Dual marching cubes [Schaefer 2005]
    38. 38. Outline• Introduction• Related Work• Our Approach• Evaluation• Conclusion
    39. 39. Input point MPU Poisson D4 Waveletscloud [Othake ‘03 ] [Kazhdan ‘06 ] [Manson ‘08 ] Our SSD
    40. 40. Input point MPU Poisson D4 Waveletscloud [Othake ‘03 ] [Kazhdan ‘06 ] [Manson ‘08 ] Our SSD
    41. 41. Input point MPU Poisson D4 Waveletscloud [Othake ‘03 ] [Kazhdan ‘06 ] [Manson ‘08 ] Our SSD
    42. 42. Input point MPU Poisson D4 Waveletscloud [Othake ‘03 ] [Kazhdan ‘06 ] [Manson ‘08 ] Our SSD
    43. 43. Accuracy• Hausdorff distance between real surfaces and reconstructed surfaces using Metro tool [Cignoni ’96]. (Lower is better!)
    44. 44. Performance characteristicsMPU Poisson D4 Wavelets Our SSD[Othake ‘03 ] [Kazhdan ‘06 ] [Manson ‘08 ]
    45. 45. Outline• Introduction• Related Work• Our Approach• Evaluation• Conclusion
    46. 46. Conclusion• Theoretical contributions: – Oriented point samples samples of signed Euclidean distance function – Reconstruction as global minimization problem – Yet sparse system• Empirical advantages: – Robustness to noise – Adaptive to sampling density• Future work: – Streaming out-of-core implementations – Parallel GPU implementations
    47. 47. Ongoing work
    48. 48. mesh.brown.edu/ssd Thank you! Any questions? This material is based upon work supported bythe National Science Foundation under Grant No. IIS-0808718, CCF-0729126, and CCF-0915661.

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