Advanced Computer Graphics Spring 2008

1. Advanced Computer Graphics Spring 2008 K. H. Ko Department of Mechatronics Gwangju Institute of Science and Technology

2. Today’s Topics • Deformable Models in Computer Graphics • Control Point Deformation

3. Deformable Models in Computer Graphics: Survey • Non-Physical Models • Purely geometric techniques • They are generally computationally efficient. • They rely on the skill of the designer rather than on physical principles. • Splines and Patches • Bezier curves/surfaces, B-spline, NURBS, etc. • Support interactive modification of shape. • Subtle control of object shape is possible. • But precise specification or modification of curves or surfaces can be laborious.

4. Deformable Models in Computer Graphics: Survey • Non-Physical Models • Free-Form Deformation (FFD) • It is a general method for deforming objects that provides a higher and more powerful level of control than adjusting individual control points. • FFD changes the shape of an object by deforming the space in which the object lies though mapping. • Ex.Twist about the z-axis • More complex deformations can be constructed by composing mappings.

5. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • Non-physical methods for modeling deformation are limited by the expertise and patience of the user. • Deformations must be explicitly specified and the system has no knowledge about the nature of the object being manipulated. • Modeling an object as complex as the human face is a daunting task. • Therefore, physics is considered in the modeling.

6. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • They are physically based technique that has been used widely and effectively for modeling deformable objects. • An object is modeled as a collection of point masses connected by springs in a lattice structure. • The spring forces are linear/nonlinear.

7. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • The equation of motion for the entire system are assembled from the motions of all of the mass points in the lattice. • The system is evolved forward through time by re-expressing the equations as a system of first-order differential equations

8. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • They have been used widely in facial animation. • Tension nets: static versions of mass-spring systems. Kx = f. • The face is modeled as a two-dimensional mesh of points warped around an ovoid and connected by linear springs. • Muscle actions are represented by the application of a force to a particular region of nodes.

9. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • Dynamic mass-spring systems to facial modeling • A three-layer mesh of mass points based on three anatomically distinct layers of facial tissue: • The dermis • A layer of subcutaneous fatty tissue • The muscle layer

10. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • Dynamic mass-spring systems to facial modeling • Different spring constants were used to model the different layers based on tissues properties. • Facial models are created • Manually • Using a radial laser-scanned image data • Computer Tomography (CT) • Prediction of the post-operative appearance of patients whose underlying bone structure has been changed during cranio-facial surgery. • Spring stiffness for the system is derived from tissue densities obtained by CT image data.

11. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • Mass-spring models combined with free-form deformations are used to animate muscles in human character animation. • A mass-spring model for deformable bodies is used to model a transition change from solid to liquid. • Mass-spring systems can be used to generate “artificial fish”.

12. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • Advantages • Simple, well understood dynamics • Easy to construct. • Can be animated at rates not possible with other techniques • Interactive and real-time simulation is possible. • Well suited to parallel computation.

13. Deformable Models in Computer Graphics: Survey • Mass-Spring Models • Drawbacks • The discrete model is a significant approximation of the true physics that occurs in a continuous body. • Proper values of spring constants may not be easily obtained. • “Stiffness” issue: numerical instability would occur. • Large spring constants to model objects that are nearly rigid.

14. Deformable Models in Computer Graphics: Survey • Continuum Models • Accurate physical models treat deformable objects as a continuum. • bodies with mass and energies distributed throughout. • Modeling itself can be derived based on the assumption of continuum. But ultimately computation is discrete.

15. Deformable Models in Computer Graphics: Survey • Continuum Models • The full continuum model of a deformable object considers the EQUILIBRIUM of a general body acted on by external forces. • The object deformation is a function of the acting forces and the object’s material properties. • The object reaches equilibrium when its total energy is at a minimum. • П=Λ-W

16. Deformable Models in Computer Graphics: Survey • Continuum Models • To determine the equilibrium shape of the object, • Both Λ and W are expressed in terms of the object deformation. • Λis the total strain energy of the deformable object • W is the work done by external forces • The total potential reaches a minimum when the derivative of the total potential with respect to the material displacement function is zero. • This approach leads to a continuous differential equilibrium equation.

17. Deformable Models in Computer Graphics: Survey • Continuum Models • A closed-form analytic solution of the differential equation is not always possible. • We instead find an approximate solution to the equation. • FEM method.

18. Deformable Models in Computer Graphics: Survey • The use of FEM in computer graphics has been limited because of the computational requirements. • In real-time applications, it has proven difficult to use FEM. • The force vectors and the mass and stiffness matrices are computed by integrating over the object, they must, in theory, be re-evaluated as the object deforms. • The re-evaluation is very costly.

19. Deformable Models in Computer Graphics: Survey • FEM Methods • Advantage • Provide a more physically realistic simulation than mass spring methods with fewer node points. • Disadvantages • Significant pre-processing time. • If the topology of the object changes during the simulation, or if the object shape changes beyond small deformation limits, the mass and stiffness matrices must be re-evaluated during the simulation. • Meshless approach???

20. Deformable Models in Computer Graphics: Survey • Approximate Continuum Models • Physically motivated, but adhere less strictly to the laws of physics than the FEM methods. • Snakes • One-dimensional deformable curves that are often used to deform or define edges or contours or to tract motion in a moving image. • Discretized deformation energy • Hybrid models

21. Deformable Models in Computer Graphics: Survey • Low Degree of Freedom Models • Discretization of the physically based models leads to systems with many degrees of freedom. • A large number of node points • The systems are slow to simulate, limiting their use in interactive and real time settings. • Alternative approximate continuum models • They restrict the deformable object to many fewer degrees of freedom, sacrificing generality for speed.

22. Deformable Models in Computer Graphics: Survey • Low Degree of Freedom Models • Modal Analysis • Dynamic Global Deformation • Minimal Energy Surfaces

23. Control Point Deformation • A deformable body can be modeled as a parametric surface with control points that are varied according to the needs of an application. • This approach is not physically based. But a careful adjustment of control points can make the surface deform in a manner that is convincing to the viewer.

24. Control Point Deformation • Curves and Surfaces • Bezier, B-Spline, NURBS • Cylinder Surfaces and Generalized Cylinder Surfaces • Revolution Surfaces • Surfaces Built From Curves: Skinning or Lofting

25. Bezier Curves • Mathematical Formulation • Bi,n(u) is called, the Bernstein Polynomial. • The polygon joining Ri’s is called the control polygon.

26. Properties of Bezier Curves • Geometry Invariance Properties • Bezier curves are invariant under translation and rotation. • We transform Bezier curves by transforming their control polygons only. • Relative position of control vertices is important in the generation of a Bezier curve.

27. Properties of Bezier Curves • Endpoint Geometric Properties • The first and last control points are the endpoints of the curve. • The curve is tangent to the control polygon at the endpoints.

28. Properties of Bezier Curves • Convex Hull Property

29. Properties of Bezier Curves • Convex Hull Property is useful in • Intersection problems • Detection of absence of interference. • Providing the approximate position of curve through simple bounds

30. Properties of Bezier Curves • Variation Diminishing Property • A Bezier curve oscillates less than its polygon. • The polygon’s segments exaggerate the oscillation of the curve. • Useful in detecting the fairness of Bezier curves

31. Algorithms for Bezier Curves • Evaluation and subdivision algorithm • A Bezier curve can be evaluated at a specific parameter value t0 and the curve can be split at that value using the de Casteljau algorithm. • The values bi0 are the original control points of the curve. • The value of the curve at parameter value t0 is bnn. • The curve can be represented as two curves, with control points and .

32. Algorithms for Bezier Curves

33. Bezier Surfaces • A tensor product surface is formed by moving a curve through space while allowing deformations in that curve. • A Bezier surface with degrees m and n in u and v is defined by • bij is called the control net. • A Bezier surface inherits three properties • Geometry invariance, Endpoints geometric property and convex hull property. • No variation diminishing property is known.

34. Bezier Surfaces

35. Non Uniform B-splines • Definition • Pi are n+1 control points. • Ni,k(u) are piecewise polynomial B-spline basis functions of order k with k-1 ≤ n. • The parameter u obeys the inequality

36. Non Uniform B-splines • Knot vector • For open (non-periodic) curves, it is usual to define a set T of non-decreasing real numbers which is called the know vector. • The total number of knots is n+k+1.

37. Non Uniform B-splines • Properties and definition of basis function

38. Non Uniform B-splines • Properties and definition of basis function

39. Non Uniform B-splines • 2nd order B-spline basis function

40. Non Uniform B-splines • 3rd order B-spline basis function

41. Non Uniform B-splines • 4th order B-spline basis function (Cubic B-spline)

42. Non Uniform B-splines

43. Non Uniform B-splines • Properties • Local support • Convex hull (stronger than Bezier) • Each span is in the convex hull of the k vertices contributing to its definition • Consequence: k consecutive vertices are collinear • Span is a straight line segment • Variation diminishing property as for Bezier curves • Exploit knot multiplicity to make complex curves.

44. Non Uniform B-splines • Special Case : n = k - 1 • The B-spline curve is also a Bezier curve in this case. • Derivatives

45. Non Uniform B-splines • Design with B-spline curves • Procedure A • Designer chooses knot vector and control points. • Designer displays a curve and tweaks control points to improve the curve. • Procedure B • Designer starts with data points on or near curve. • Construct an interpolating/approximating B-spline curve. • Display curve and tweak control points to improve the curve.

46. Non Uniform B-splines • Evaluation and subdivision of B-splines • De Boor Algorithm for B-spline curve evaluation

47. Non Uniform B-splines • Evaluation and subdivision of B-splines • De Boor Algorithm for B-spline curve evaluation

48. Non Uniform B-splines • Knot Insertion: Boehm’s algorithm

49. Tensor Product Polynomial Surface Patches • Tensor product surface • Let be 3D curve. • Let this curve sweep a surface by moving and possibly deforming by letting each Ri trace a curve. • The resulting surface is a tensor product surface.

50. Tensor Product Polynomial Surface Patches