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Digital Image Processing Lecture 7: Geometric Transformation

Digital Image Processing Lecture 7: Geometric Transformation. Prof. Charlene Tsai. Review. A geometric transform of an image consists of two basic steps Step1: determining the pixel co-ordinate transformation Step2: determining the brightness of the points in the digital grid.

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Digital Image Processing Lecture 7: Geometric Transformation

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  1. Digital Image Processing Lecture 7: Geometric Transformation Prof. Charlene Tsai

  2. Review • A geometric transform of an image consists of two basic steps • Step1: determining the pixel co-ordinate transformation • Step2: determining the brightness of the points in the digital grid. • In the previous lecture, we discussed brightness interpolation and some variations of affine transformation. T(x,y) (x,y)

  3. Affine Transformation (con’d) • An affine transformation is equivalent to the composed effects of translation, rotation and scaling. • The general affine transformation is commonly expressed as below: 0th order coefficients 1st order coefficients

  4. General Formulation • A geometric transform is a vector function T that maps the pixel (x,y) to a new position (x’,y’): • Tx(x,y) and Ty(x,y) are usually polynomial equations. • This transform is linear with respect to the coefficients ark and brk.

  5. Finding Coefficients • How to find ark and brk, which are often unknown? • Finding pairs of corresponding points (x,y), (x’,y’) in both images, • Determining ark and brk by solving a set of linear equations. • More points than coefficients are usually used to get robustness. Least-squares fitting is often used.

  6. Jacobian • A geometric transform applied to the whole image may change the co-ordinate system, and a Jacobian J provides information about how the co-ordinate system changes • The area of the image is invariant if and only if |J|=1 (|J| is the determinant of J). • What is the Jacobian of an affine transform?

  7. Variation of Affine (2D) • Translation: displacement • Euclidean (rigid): translation + rotation • Similarity: Euclidean + scaling • (for isotropic scaling, sx = sy)

  8. Types of transformations (2D) • Affine: Similarity + shearing • Some illustrations of shearing effects (courtesy of Luis Ibanez)

  9. Affine transform • Shearing in x-direction a01y y’ y y (x,y) ( x + a01y , y ) x’ x x

  10. Affine transform • Shearing in y-direction y’ ( x , y + b10x ) y y b10x x’ (x,y) x x

  11. Invariant Properties • Translation: • Euclidean: • Similarity: • Affine:

  12. Types of transformations (2D) • Bilinear: Affine + warping • Quadratic: Affine + warping

  13. Least-Squares Estimation • Because of noise in the images, if there are more correspondence pairs than minimally required, it is often not possible to find a transformation that satisfies all pairs. • Objective: To minimize the sum of the Euclidean distances of the correspondence set C={pi,qi}, where pi={xi,yi}, and qi is the corresponding point.

  14. Least-Squares Estimation - Affine • For affine model, • The Euclidean error , where • With the new notation, • To find which minimize we want

  15. Least-Squares Estimation - Affine • This leads to • Therefore,

  16. Some remarks • So far, we only discussed global transformation (applied to entire image) • It is possible to approximate complex geometric transformations (distortion) by partitioning an image into smaller rectangular sub-images. • for each sub-image, a simple geometric transformation, such as the affine, is estimated using pairs of corresponding pixels. • geometric transformation (distortion) is then performed separately in each sub-image.

  17. Real Application • Registration of retinal images of different modalities • Importance? Color slide Fluorocein Angiogram

  18. Mosaic Mosaicing

  19. Fusion of medical information

  20. Change Detection

  21. Computer-assisted surgery

  22. What is needed? • A known transformation, which is less likely OR • Computing the transformation from features. • Feature extraction • Features in correspondence • Transformation models • Objective function

  23. What might be a good feature? Color slide Fluorocein Angiogram

  24. Feature Extraction

  25. Features in Correspondence(crossover & branching points)

  26. Features in Correspondence(vessel centerline points)

  27. Transformation Models p 12-parameter quadratic transformation q

  28. Transformation Model Hierarchy Similarity Affine Reduced Quadratic Quadratic

  29. Feature point in image Set of correspondences Feature point in image Objective Function Transformation parameters Mapping of features from to based on

  30. Result

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