1 / 20

Geometric Transformations (Chapter 2)

Geometric Transformations (Chapter 2). CS474/674 – Prof. Bebis. Geometric Processes. Transformation applied on the coordinates of the pixels (i.e., relocate pixels). A geometric transformation has the general form (x,y) = T{(v,w)} where (v,w) are the original pixel coordinates and

jana
Download Presentation

Geometric Transformations (Chapter 2)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Geometric Transformations (Chapter 2) CS474/674 – Prof. Bebis

  2. Geometric Processes • Transformation applied on the coordinates of the pixels (i.e., relocate pixels). • A geometric transformation has the general form (x,y) = T{(v,w)} where (v,w) are the original pixel coordinates and (x,y) are the transformed pixel coordinates.

  3. Geometric Transformations affine transformation y=v sinθ + w cosθ

  4. Forward mapping • Transformed pixel coordinates might not lie within the bounds of the image. • Transformed pixel coordinates can be non-integer. • There might be no pixels in the input image that map to certain pixel locations in the transformed image No one-to-one correspondence!

  5. Forward mapping (cont’) • An example of holes due to image rotation, implemented using the forward transformation.

  6. Inverse Mapping • To guarantee that a value is generated for every pixel in the output image, we must consider each output pixel in turn and use the inverse mapping to determine the position in the input image. • To assign intensity values to these locations, we need to used some form of intensity interpolation.

  7. Interpolation • Interpolation — process of using known data to estimate unknown values. • Interpolationcan be used to increase (or decrease) the number of pixels in a digital image. e.g., some digital cameras use interpolation to produce a larger image than the sensor captured or to create digital zoom http://www.dpreview.com/learn/?/key=interpolation

  8. Interpolation (cont’d) Zero-order interpolation: nearest-neighbor

  9. Interpolation (cont’d) First-order interpolation:average

  10. Interpolation (cont’d) Bilinear interpolation I(x,y) = ax + by + cxy + d The 4 unknowns (a,b,c,d) can be determined from 4 equations formed by the 4 nearest neighbors.

  11. Linear Algebra Review

  12. Interpolation (cont’d) Bilinear interpolation http://en.wikipedia.org/wiki/Bilinear_interpolation

  13. Interpolation (cont’d) Bicubic interpolation • It involves the sixteen nearest neighbors of a point (i.e., 4x4 window). • The 16 unknowns aij can be determined from sixteen equations formed by the 16 nearest neighbors.

  14. Examples: Interpolation

  15. Examples: Interpolation

  16. Examples: Interpolation

  17. Examples: Interpolation

  18. Image Registration • Goal: align two or more images of the same scene. • How: estimate a transformation that aligns the two images.

  19. Image Registration (cont’d) • Under certain assumptions, an affine transformation can be used to align two images. • There are 6 unknowns (i.e., t11, t12, t21, t22, t31, t32) • We need at least 6 equations. • Three correspondences are enough, more are better. Unknowns? Equations? Correspondences?

  20. Image Registration Example apply affine transformation Error when comparing the transformed and original images.

More Related