CS 335 Graphics and Multimedia. Geometric Warping

Transcription

1 CS 335 Graphics and Multimedia Geometric Warping

2 Geometric Image Operations Eample transformations Straightforward methods and their problems The affine transformation Transformation algorithms: Forward mapping Reverse mapping Interpolation schemes to reduce aliasing

3 Geometric Transformations Treat image as a set of points (piels) Treat source image as input, destination image as output Transform each point in source image to get its new location in the destination image Set the intensit of the destination point to be the same as the source image

4 Rotation Y f(,) origin

5 Scale in a Single Direction

6 Scale: Two Directions Constant Image Size

7 Straightforward Methods Double in size (scale b a factor of 2) Shrink b /2 Rotate 9 degrees a a a a a a a a

8 More General Transformations Transformed piel location ma not be integer Destination image size ma be larger than original image Destination image ma not have all piels mapped: Source Destination Not Mapped

9 The Affine Transformation ), ( ), ( T T (, ) is a point in the image Affine: the transformations T are linear, i.e., 2 2 b b b a a a

10 The Affine Transformation Write as a matri using Homogenous Coordinate sstem: 2 2 b b b a a a

11 Translation (, ) t (,) t

12 Matri Transformations Translation of a point: t t + + Matri form: T t t [ ] t For eample,

13 Scaling (, ) (,) (shown w.r.t. origin)

14 Matri Transformations Scaling of a point: s s Matri form: S s s [ ] s For eample,

15 Rotation (, ) φ (,) (rotation w.r.t. origin)

16 Matri Transformations Rotation of a point: cos sin sin cos θ θ θ θ + Matri form: R θ cos sin sin cos θ θ θ θ [ ] sin cos θ θ For eample,

17 Appling Transformations to Images Transformations applied to all image piels Transformed point gets intensit from its corresponding source piel Transformation applied to objects are based on the defining points of the object: circle (move center, circle follows) lines (endpoints) polgons (vertices)

18 The Affine Transformation Combines scale, translation, and rotation: s s 2 2 b b b a a a

19 Affine Transformation t t Combines scale, translation, and rotation: 2 2 b b b a a a

20 Affine Transformation cos sin sin cos θ θ θ θ Combines scale, translation, and rotation: 2 2 b b b a a a

21 Transforms and Images Coordinates origin f(,) Y

22 Transforms and Images Coordinates (- value) origin Y f(,)

23 Converting to an Image New Image Dimensions origin Y

24 The transformed image origin New Image Dimensions Y

25 Creating the new image Forward Mapping Inverse Mapping Sampling

26 Mapping Piels (,) (,N) 2 Forward Mapping (,) (,N) (,M) (M,N) (,M) (M,N) Source Image [u,v,s] T A [,,] T Destination Image Transform

27 Forward Mapping Draw backs Source piels do not map directl to a single piel in the destination space Possibilit for holes in the destination image

28

29 Reverse Mapping Reverse Mapping (,) (,N) (,M) (M,N) 4 black [,,s] T A - [u,v,] T

30 Inverse Mapping Advantages We assign an intensit to each piel in the destination (no holes) Affine/projective transforms have inverses (not a problem) just reverse direction of the point correspondences We still don t have piel to piel mapping

31 Sampling the source How do we sample the source to determine the intensit for the destination?

32 Sampling the source How do we sample the source to determine the intensit for the destination?

33 Mapping Source 2 2 piels Option : Pick the piel nearest to our center.

34 Mapping Source 2 2 piels Small change results in big difference Option : Pick the piel nearest to our center.

35 Tr different sampling Source 2 2 piels 2 What if we assign an intensit to each verte and then average? Pick the intensit which the verte lies New Sample

36 Sampling Eample Source 2 2 piels 2 Move the destination slightl. Pick the intensit which the verte lies New Sample

37 No Difference Source 2 2 piels Source 2 2 piels 2 4 3

38 Tr different Sampling Source 2 2 piels 2 3 What if we had more samples? 4

39 Tr different Sampling Source 2 2 piels 2 What if we sampled a larger area? 4 3

40 Tr different Sampling Source 2 2 piels How should we sample?

41 Common Sampling Approaches Nearest Neighbor Sample Take closest piel value Bi-linear Interpolation 22 (4) Samples Interpolate from these samples Slower Bi-Cubic 44 (6) samples Construct a new sample using a non-linear interpolation Slower

42 Common Sampling Approaches Nearest Neighbor Bi-linear Interpolation Bi-Cubic

43 Common Sampling Approaches What do these approaches mean? How can evaluate what the are doing? Nearest Neighbor Bi-linear Interpolation Bi-Cubic

44 Nearest Neighbor Source 2 2 piels

45 First-Order Interpolation Bilinear weighting of 4 piels: ), ( ), ( f f [ ] f f + ), ( ), ( [ ] f f f f + + ), ( ), ( ), ( ), ( [ ] f f + ), ( ), ( where

46 Bilinear Interpolation: Derivation, ) (, ) ( f, ) f, ) ( ( f (, ) f (, )

47 Bilinear Interpolation: Derivation f, ) f, ) ( ( ( unit) f (, ) f (, )

48 Bilinear Interpolation: Derivation ( )( ) ( )( ) f, ) f, ) ( ( ( unit) ( )( ) f (, ) f (, ) ( )( )

49 Bilinear Interpolation: Derivation f (, ) f (, ) [( )( ) ]+ f (, ) [( )( ) ]+ f, )[( )( )]+ f ( (, ) [( )( ) ] Rearrange terms: f, ) f, ) ( ( f (, ) f (, ) + [ f ( ], ) f (, ) + [ f (, ) f (, )] + f, ) + f (, ) f f (, ) f (, ) [ (, ) f (, )] (

50 Comparison

51 Take Home Eercise Suppose an output piel is mapped to (58.3, 45.9) as shown below, calculate the output piel value using bilinear interpolation. (58.3, 45.9)