Image Registration: A . . . Existing Image . . . Towards Formulating . . . Utility theory: a . . . Image Registration: Utility for image . . . Utility for image . . . An Overview with an Why Fourier Methods Why Fourier Methods . . . Possibility of Further . . . Emphasis on Geometry, Case of Rotation Only General Case: Shift . . . Foundations, and Acknowledgments Title Page Computational Complexity ◭◭ ◮◮ Vladik Kreinovich ◭ ◮ Department of Computer Science Page 1 of 13 University of Texas at El Paso 500 W. University, El Paso, TX 79968, USA Go Back vladik@utep.edu Full Screen Joint work with Chandrajit Bajaj and Roberto Araiza Close Quit
Image Registration: A . . . 1. Image Registration: A Practical Problem Existing Image . . . Towards Formulating . . . • In many areas of science and engineering, we have two images I 1 ( � x ) and I 2 ( � x ) Utility theory: a . . . of the same 2-D or 3-D object. Utility for image . . . • Since I 1 and I 2 represent the same object, I 2 ( � x ) ≈ I 1 ( λ · R� x + � a ) for some Utility for image . . . scaling λ , rotation R , and shift � a . Why Fourier Methods Why Fourier Methods . . . • Often, we do not know the relative orientation of I 1 ( � x ) and I 2 ( � x ). Possibility of Further . . . • In such situations, we must register images, i.e., find λ , R , and � a after which Case of Rotation Only the images match. General Case: Shift . . . Acknowledgments • Similar problem: images of different objects that should match – e.g., protein docking. Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 13 Go Back Full Screen Close Quit
Image Registration: A . . . 2. Existing Image Registration Techniques Are Mainly Existing Image . . . Heuristic Towards Formulating . . . Utility theory: a . . . • Good news: there exist many different image registration techniques. Utility for image . . . Utility for image . . . • Problem: most existing methods are heuristic. Why Fourier Methods • Specifically: Why Fourier Methods . . . Possibility of Further . . . – There is often no precise formulation of the problem. Case of Rotation Only – Even when there is such a formulation, there is no clear relation between General Case: Shift . . . the formulation and the method. Acknowledgments • Resulting practical problems: Title Page – sometimes methods do not work; it is not clear when they are applicable; ◭◭ ◮◮ – it is not clear which method is better – or whether a yet better method is possible. ◭ ◮ • Our main objectives: Page 3 of 13 – provide a general formalization of the problem; Go Back – use this formalization to explain existing techniques – thus providing explanations of when they are applicable; Full Screen – use this formalization to compare existing techniques; – if possible, design new, optimal techniques. Close Quit
Image Registration: A . . . 3. Towards Formulating the Image Registration Prob- Existing Image . . . lem in Precise Terms Towards Formulating . . . Utility theory: a . . . • Ideal no-noise case: Utility for image . . . Utility for image . . . – we know images I 1 ( � x ) and I 2 ( � x ); Why Fourier Methods – we want to find � a , R , and λ after which the images match exactly: Why Fourier Methods . . . I 2 ( � x ) = I 1 ( λ · R� x + � a ). Possibility of Further . . . • In practice: perfect macth is not possible: Case of Rotation Only General Case: Shift . . . – there is noise, Acknowledgments – there are measurement errors, Title Page – change in imaging conditions (or in the image itself) between the time when these two images were taken. ◭◭ ◮◮ • Objective: select transformations for which, according to the user’s prefer- def ◭ ◮ ences, the difference between I 2 ( � x ) and I T ( � x ) = I 1 ( λ · R� x + � a ) is the “most acceptable” (or, equivalently, the “least unacceptable”. Page 4 of 13 • Problem: often, we do not have a clear description of user preferences. Go Back • Solution: we must provide the exact description of what “most acceptable” means. Full Screen Close Quit
Image Registration: A . . . 4. Utility theory: a known way to describe users’ pref- Existing Image . . . erences Towards Formulating . . . Utility theory: a . . . • The need to describe user preferences is important in decision making in Utility for image . . . general. Utility for image . . . Why Fourier Methods • To describe these preferences, a special utility theory has been developed. Why Fourier Methods . . . • Main idea: sometimes, instead of choosing one of the alternatives A 1 , . . . , A n , Possibility of Further . . . a user may choose A i with probability p i (e.g., flip a coin). Case of Rotation Only General Case: Shift . . . • We thus consider preference relation ≻ between such “lotteries” L i . Acknowledgments • Reasonable conditions: e.g., if for a user, A is preferable to B and B is preferable to C , then for this user A should be preferable to C . Title Page • Main result: under reasonable conditions, there exists a function u from the ◭◭ ◮◮ set L of all possible lotteries into the set IR of real numbers for which: ◭ ◮ – L 1 ≻ L 2 if and only if u ( L 1 ) > u ( L 2 ), and – for every lottery L , in which each alternative A i appears with probability Page 5 of 13 p i , we have u ( L ) = p 1 · u ( A 1 ) + . . . + p n · u ( A n ) . Go Back This function u is called a utility function . Full Screen • Uniqueness: if u 1 ( L ) and u 2 ( L ) describe the same ≻ , then there exist a > 0 and b for which u 2 ( L ) = a · u 1 ( L ) + b for all lotteries L . Close Quit
Image Registration: A . . . 5. Utility for image registration: reasonable require- Existing Image . . . ments Towards Formulating . . . Utility theory: a . . . • Objective: describe a utility function v ( I 1 , I T ) that describe the quality of Utility for image . . . matching. Utility for image . . . • Signal-to-noise ratio: Why Fourier Methods Why Fourier Methods . . . – if high, we can match images exactly; Possibility of Further . . . – is low, i.e., if signals are weak, then we can expand the scoring function Case of Rotation Only v ( I 2 , I T ) in Taylor series in I 2 ( � x ) and I T ( � x ), and keep only the lowest General Case: Shift . . . non-zero terms. Acknowledgments def • Comment: we want to find T for which I T ( � x ) ≈ I 2 ( � x ), i.e., ∆ I ( � x ) = I T ( � x ) − I 2 ( � x ) is small. Title Page • To make this smallness explicit, we describe v in terms of I 2 ( � x ) and ∆ I ( � x ). ◭◭ ◮◮ • For perfect match ∆ I = 0 we must have v → min, i.e., ∂v/∂ ∆ I = 0 – hence ◭ ◮ v cannot have linear terms in ∆ I ; hence, v ( I 2 , ∆ I ) is quadratic. • In general, a quadratic function can have terms independent on ∆ I , terms Page 6 of 13 linear in ∆ I , and terms which are quadratic in ∆ I . Go Back • Terms that are independent on ∆ I do not depend on the choice of � a , R , and λ and thus, do not affect the choice of registration. Full Screen • Conclusion: v does not depend on I 2 , i.e., Close � � x ) 2 d� v (∆ I ) = a 1 ( � x ) · ∆ I ( � x + a 2 ( � x, � x ′ ) · ∆ I ( � x ) · ∆ I ( � x ′ ) d� xd� x ′ . Quit
Image Registration: A . . . 6. Utility for image registration: reasonable require- Existing Image . . . ments (cont-d) Towards Formulating . . . Utility theory: a . . . • A quadratic scoring function must be non-degenerate : for a function ∆ I ( � x ) Utility for image . . . which is equal to a finite number inside the bounded region and to 0 every- Utility for image . . . where else, we get a finite value of v (∆ I ). Why Fourier Methods • Invariance: Why Fourier Methods . . . Possibility of Further . . . – We want to find the shift, rotation, and scaling after which the images Case of Rotation Only match as much as possible. General Case: Shift . . . – It is therefore reasonable to assume that the relative quality of two Acknowledgments possible matches does not change if we simply shift, rotate, and/or scale both images. Title Page – Two utility functions v 1 ( A ) and v 2 ( A ) lead to the same preference re- ◭◭ ◮◮ lation if and only if they can be obtained from each by using a linear transformation: v 2 ( A ) = a · v 1 ( A ) + b . ◭ ◮ – Conclusion: for every � a , R , and λ there exist real numbers a ( � a, R, λ ) and b ( � a, R, λ ) for which, for every function ∆ I ( � x ), we have v (∆ I T ) = Page 7 of 13 a · v (∆ I ) + b. Go Back • Result: Every non-degenerate invariant scoring function has the L 2 -form x ) 2 d� v (∆ I ) = c · � ∆ I ( � x for some real number c . Full Screen Close Quit
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