Interpolation and Basis Fns

Transcription

1 CS148: Introduction to Computer Graphics and Imaging Interpolation and Basis Fns Topics Today Interpolation Linear and bilinear interpolation Barycentric interpolation Basis functions Square, triangle,, Hermite cubic interpolation Interpolating random numbers to make noise Thursday Splines and curves Catmull-Rom splines Bezier curves Page 1

2 Interpolation Fill in between values Convert discrete (finite) to continuous (infinite) Examples: Interpolating across a triangle Interpolating between vertices Filtering and reconstructing images Interpolating between pixels/texels Creating random functions Noise Generating motion Interpolating inbetween frames from keyframes Curves and surfaces Interpolating between control points Interpolation Page 2

3 Linear Interpolation Linear Interpolation - lerp One of the most useful functions in computer graphics! lerp(t, v0, v1) {! }!!return (1-t)*v0 + t*v1;! lerp(t, v0, v1) {!!return v0 + t*(v1-v0);! }!!!!!!! Page 3

4 Bilinear Interpolation bilerp(s, t, v0, v1, v2, v3) {!!!!!!v01 = lerp(s, v0, v1);!!!!!!v23 = lerp(s, v2, v3);!!!!!!v = lerp(t, v01, v23);!!!!!!return v;!!!!! }!!!!!!! Bilinear Interpolation Page 4

5 Ruled Surfaces Surfaces made with lines Barycentric Coordinates Given masses: m1, m2; Given distances: d1, d2 Balance condition (torques equal): m1 d1 = m2 d2 Or: m1 x m2 = m2 x m1 => d1 m2 and d2 m1 Page 5

6 Barycentric Coordinates Suppose d1 = m2 / (m1 + m2); d2 = m1 / (m1 + m2) These distances define the center of mass For this reason, d1 and d2 are called barycentric coordinates of the center point Barycentric Interpolation Edge Page 6

7 Barycentric Interpolation Edge Triangle Barycentric Interpolation - Triangle Center of mass: Page 7

8 Drawing Triangles Barycentric interpolation used for values inside the tri. Parameters define points inside the triangle If all parameters positive, then inside the triangle Can be used to interpolate z or depth z = a0 z0 + a1 z1 + a2 c2 Can be used to interpolate colors c = a0 c0 + a1 c1 + a2 c2 Can be used to interpolate texture coordinates uv = a0 uv0 + a1 uv1 + a2 uv2 Basis Functions Page 8

9 Linear Interpolation = Triangle Basis Triangle Basis Page 9

10 Linear Interpolation = Triangle Basis Constant Interpolation = Square Basis Nearest- Neighbor Interpolation Page 10

11 Basis Functions Basic formula Basis functions Often i th functions are shifted versions of 0 th Basis functions are like unit vectors Interpolating Function Necessary conditions: True for triangle and square basis functions Page 11

12 Cubic Hermite Interpolation Smooth Using box functions causes discontinuity in value Using triangle functions causes discontinuity in slope How can we smoothly interpolate values? Mathematically, we say a function is C k continuous if the first k derivatives match. Hence, we want to be C 1 continuous Page 12

13 Faceted Shading Smooth Shading Page 13

14 Cubic Hermite Interpolation Given: values and derivatives at 2 points Hermite Basis Function Formulation Page 14

15 Cubic Hermite Interpolation Assume cubic polynomial Why? 4 coefficients need 4 degrees of freedom Cubic Hermite Interpolation Assume cubic polynomial Solve for coefficients: Page 15

16 Matrix Representation Matrix Representation Inverse Page 16

17 Matrix Representation Matrix Representation of Polynomials Page 17

18 Hermite Basis Matrix Matrix Representation of Polynomials Page 18

19 Matrix Representation of Polynomials Matrix Representation Transpose Page 19

20 Matrix Representation of Polynomials Hermite Basis Matrix Page 20

21 Hermite Basis Functions Ease A very useful function In animation, start and stop slowly (zero velocity) Page 21

22 Fractal Interpolation Ken Perlin Noise Idea: Interpolate random slopes Page 22

23 Code double noise1(double x)! {! double t = x + N;! // compute integer locations! int i0 = (int)t % BITSN;! int i1 = (i0+1) % BITSN;! // compute fractional parts! double f = t - (int)t;! double f0 = f;! double f1 = f - 1.;! double g0 = rand[ perm[ i0 ] ];! double g1 = rand[ perm[ i1 ] ];! double u = f0 * g0;! double v = f1 * g1;! double s = f0 * f0 * ( * f0); // hermite spline! return(lerp(s, u, v));! Noise and Turbulence Functions Ken Perlin, An Image Synthesizer Page 23

24 Things to Remember Interpolation Widely used in graphics: image, texture, noise, animation, curves and surfaces Nearest neighbor, bilinear, cubic interpolation Basis functions Square Triangle Hermite (smooth) Noise (random) Many others: sines, cosines, sinc, wavelets, Page 24