Geometric modeling 1

Transcription

1 Geometric Modeling 1

2 Look around the room. To make a 3D model of a room requires modeling every single object you can see. Leaving out smaller objects (clutter) makes the room seem sterile and unrealistic (e.g. similar to an IKEA furniture store)

3 The same is true for outdoor scenes

4 Movies

5 Movies

6 Games

7 Games

8 MeSH Construction

9 A polygonal mesh is represented on the computer using various primitives: vertices, edges, and faces (triangles are the most common) Problem: Manually specifying every vertex, edge, and face is cumbersome (even if it is slightly faster than chipping a model out of stone ) Solution: software packages can help automate some vertex placement (but it can still take a long time) Process: Mesh Construction Model a low resolution mesh by hand Refine that mesh with an *automatic* method/algorithm Edit the refined mesh with a semi-*automated* method/algorithm Refine more, Edit more, rinse, repeat, etc. etc.

10 Refinement via subdivision

11 Subdivision Given a coarse input mesh, generate a finer output mesh via subdivision For a given mesh, various smooth limit surfaces exist the exact smooth limit surfaces depends on the subdivision algorithm used After just a few refinements, additional changes become too small to see/matter

12 Subdivision Curves

13 Subdivision Surfaces

14 LOOP subdivision

15 Charles Loop 1824 citations

16 Loop Subdivision Subdivide each triangle into 4 triangles Move vertices to new positions Repeat the above two steps until you get the desired resolution Generates a C 2 continuous limit surface almost everywhere except at some extraordinary vertices where the limit is C 1 continuous

17 Subdivide Each Triangle into 4 Triangles

18 Subdivide Each Triangle into 4 Triangles

19 Move the Vertices Compute perturbed locations for new vertices (black) Move the original vertices too (grey) Original Subdivided

20 Move the Vertices Compute perturbed positions of new vertices (black) using a weighted average of the four adjacent original vertices (grey) Change the original vertex (grey) positions using a weighted average of six adjacent original vertices (grey) Repeat until converged (or a few times)

21 Extraordinary Points Most vertices are regular (degree 6), but not all If a mesh is topologically equivalent to a sphere, not all the vertices can have degree 6 Extraordinary point

22 Extraordinary Points Find weights for extraordinary points that generate a smooth surface (tangent plane continuous) Want the surface normal to be continuous Complex math problem Warren weights

23 An Example

24 Starting Mesh

25 Add New Vertices

26 New Vertex & Stencil

27 Move New Vertex

28 Original Vertex & Stencil

29 Move Original Vertex

30 Extraordinary Vertex & Stencil

31 Move Extraordinary Vertex

32 Subdivided Surface

33 Subdivide Again

34 And Again

35 And Again

36 Limit surface

37 Mesh editing

38 Mesh Editing Artist manipulates a few control points or mesh points and the system *automatically* deforms the mesh E.g., twist, bend, stretch, etc. Fast, intuitive, preserves details Widely used in CAD software such as Blender, Maya, etc. Spline (e.g. B-spline, NURBS) mesh editing uses control points Laplacian mesh editing allows one to directly move mesh points Mesh editing operations in Blender Spline mesh editing Laplacian mesh editing

39 B-splines

40 The i-th B-spline basis function of order jj is recursively defined as NN ii,0 uu = 1, iiii uu ii uu < uu ii+1 0, ooooooooooooooooo NN ii,jj uu = uu uu ii NN uu ii+jj uu ii,jj 1 uu + uu ii+jj+1 uu NN ii uu ii+jj+1 uu ii+1,jj 1 uu ii+1 where uu 0, uu 1, uu mm are the knots B-Spline NN 0,0 NN 1,0 NN 2,0 NN 0,1 NN 1,1 NN 0,2 Basis functions with different orders (pp = 0,1,2) for knots 0,1,2,3 Then the B-spline is defined as: nn CC uu = NN ii,pp uu PP ii ii=0 where PP 0, PP 1,, PP nn are the 2D/3D control points nn + 1 = mm pp determines how many control points are needed - based on the number of intervals m and the order p For example: = 6 3 (for the figure to the right) the basis functions of an order 3 B-spline

41 B-Spline Use knots to subdivide the parametric domain of the curve into intervals Use control points to locally control the shape of the curve e.g., interval ss 0 is controlled by control points PP 0, PP 1, PP 2, PP 3 control point parametric domain: ss 0 ss 1 ss 2 ss 3 uu = knot

42 NURBS

43 NURBS Curve NURBS (Non-Uniform Rational B-Spline) curve is defined by: CC uu = ii=0 nn NN ii,pp uu ww ii PP ii nn ii=0 NN ii,pp uu ww ii where NN ii,pp is a B-spline basis function, PP ii is a control point, and ww ii is the weight of PP ii. Increasing ww ii pulls the curve closer to the corresponding control point PP ii, while decreasing ww ii pushes the curve farther from PP ii Demo app:

44 Extending the idea from curves to surfaces, a NURBS surface is defined as SS uu, vv = ii=0 NURBS Surface mm nn jj=0 mm ii=0 nn jj=0 NN ii,pp uu NN jj,qq vv ww ii,jj PP ii,jj NN ii,pp uu NN jj,qq vv ww ii,jj Now we have a 2D array of control points PP ii,jj with the associated basis function NN ii,pp uu NN jj,qq vv.

45 Subdividing splines

46 Subdividing Spline Curves An spline curve can be subdivided by adding more knots and control points First, insert new knots without changing the shape of the curve Whenever inserting a knot, add a control point as well The positions of the control points need to be updated to preserve the shape of the curve E.g., given a spline curve of degree pp with knots {uu 0, uu 1, uu mm } and control points PP 0, PP 1,, PP nn, we insert a new knot uu between uu kk and uu kk+1, and add one more control point obtaining the new set of control points PP 0, PP 1,, PP nn+1. The positions of these control points are computed using linear interpolation: PP ii = (1 α ii )PP ii 1 + α ii PP ii where α ii = 1 ii < kk pp ii > kk uu uu ii uu ii+pp uu ii kk pp + 1 ii kk

47 Original Curve

48 Insert a Knot uu

49 Add/Adjust Control Points PP 7 PP 1 PP 2 PP 6 PP 3 PP 4 PP 5 PP 0 uu

50 Laplacian Mesh editing

51 Differential Coordinates For each vertex, we represent its differential coordinate by the difference between its position x i and the average position of its neighbors: d i = L( x Differential coordinates approximate the local shape The direction of d i approximates the normal The magnitude of d i approximates the mean curvature i ) A mesh/model can be described by a vector of differential coordinates of all its vertices D={d i } = x i D can be calculated by multiplying a constant coefficient sparse matrix L (each vertex only interacts with its local neighbors) with a position vector X (a vector of positions of all the vertices X={x i }) i.e., D = LX 1 n i j N i x j

52 Laplacian Mesh Editing User selects some control points x i, i = {1,..., n} in the region of interest (ROI), and sets the target position for each control point x i p i Solves for for all vertices in the region of interest in a least squares sense LX = D with (soft) constraints x i = p i for i { 1,..., n} Equivalent to minimizing the least squares error E 2 n * ( X ) = di L( xi ) + i ROI i= 1 Since differential coordinates are sensitive to local deformations, we need to compute an * appropriate transformation matrix ( X ) for each vertex obtaining the modified error function E T i 2 n ' * * ( X ) = Ti ( X ) di L( xi ) + i ROI i= 1 T i is expressed as a function of X by solving the following least squares problem analytically min( T x 2 i i xi + Ti j N i T x i x j i x p j i x 2 i 2 ) p i 2 [Sorkine et.al. 2004]

53 TUTORIALS & ADVICE

54 Modeling: Tutorials & Advice