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Learn about geometric transformation steps, affine transformation, finding coefficients, and types of transformations in digital image processing. Discover the importance of Jacobian, invariant properties, and least-squares estimation techniques.
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Digital Image Processing Lecture 7: Geometric Transformation March 16, 2005 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. • In the previous lecture, we discussed brightness interpolation and some variations of affine transformation. T(x,y) (x,y) Lecture 7
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 Lecture 7
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. Lecture 7
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. Lecture 7
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? Lecture 7
Variation of Affine (2D) • Translation: displacement • Euclidean (rigid): translation + rotation • Similarity: Euclidean + scaling (for isotropic scaling, sx = sy) Lecture 7
Types of transformations (2D) • Affine: Similarity + shearing • Some illustrations of shearing effects (courtesy of Luis Ibanez) Lecture 7
Affine transform • Shearing in x-direction a01y y’ y y (x,y) ( x + a01y , y ) x’ x x Lecture 7
Affine transform • Shearing in y-direction y’ ( x , y + b10x ) y y b10x x’ (x,y) x x Lecture 7
Invariant Properties • Translation: • Euclidean: • Similarity: • Affine: Lecture 7
Types of transformations (2D) • Bilinear: Affine + warping • Quadratic: Affine + warping Lecture 7
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. Lecture 7
Least-Squares Estimation - Affine • For affine model, • The Euclidean error , where • With the new notation, • To find which minimize we want Lecture 7
Least-Squares Estimation - Affine • This leads to • Therefore, Lecture 7
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. Lecture 7
Real Application • Registration of retinal images of different modalities • Importance? Color slide Fluorocein Angiogram Lecture 7
Mosaic Mosaicing Lecture 7
Fusion of medical information Lecture 7
Change Detection Lecture 7
Computer-assisted surgery Lecture 7
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 Lecture 7
What might be a good feature? Color slide Fluorocein Angiogram Lecture 7
Feature Extraction Lecture 7
Features in Correspondence(crossover & branching points) Lecture 7
Features in Correspondence(vessel centerline points) Lecture 7
Transformation Models p 12-parameter quadratic transformation q Lecture 7
Transformation Model Hierarchy Similarity Affine Reduced Quadratic Quadratic Lecture 7
Feature point in image Set of correspondences Feature point in image Objective Function Transformation parameters Mapping of features from to based on Lecture 7
Result Lecture 7
