DOC PREVIEW
UCSD BENG 280A - 2D Fourier Transforms

This preview shows page 1-2-3-4-5-6 out of 18 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 18 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 18 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 18 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 18 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 18 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 18 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 18 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2004 Lecture 4 2D Fourier Transforms Thomas Liu BE280A UCSD Fall 2004 Topics 1 2 3 4 2D Signal Representations 2D Fourier Transform Transform Pairs FT Properties Thomas Liu BE280A UCSD Fall 2004 1 2D Signal a b c d a 0 0 0 0 0 c 0 b 0 0 0 0 0 0 d Thomas Liu BE280A UCSD Fall 2004 Image Decomposition a a b c d c 1 0 0 0 0 0 1 0 b d 0 0 1 0 0 0 0 1 g m n a m n b m n 1 c m 1 n d m 1 n 1 1 1 g k l m k n l k 0 l 0 1 1 c k l bk l m n k 0 l 0 Thomas Liu BE280A UCSD Fall 2004 2 Orthonormal Basis Functions Discrete bk l bk l b k l m n bk l m n m n k k l l Continuous b kx ky bkx ky bkx ky x y bkx ky x y dxdy kx k x ky k y Thomas Liu BE280A UCSD Fall 2004 Example Are these orthonormal 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 Thomas Liu BE280A UCSD Fall 2004 3 Example Are these orthonormal 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Thomas Liu BE280A UCSD Fall 2004 Discrete Expansion Coefficients The discrete expansion is g m n c b m n k l k l k l If the basis functions bk l m n are orthonormal then b k l c k l bk l g m n g m n m n b m n c m n c b k l k l b k l k l k l k l m n k l m n bk l m n m n c k k l l k l k l c k l Thomas Liu BE280A UCSD Fall 2004 4 Continuous Expansion Coefficients The continuous expansion is g x y c k k b x y kx ky x y dk x dky If the basis functions bkx ky x y are orthonormal then c kx k y bkx ky g b kx ky kx ky x y g x y dxdy b x y c k k b x y kx ky c k k b x c k k k c k k x y x y kx ky x y dkx dk y dxdy x y bkx ky x y dxdydk x dk y k x k y k y dkx dk y x y Thomas Liu BE280A UCSD Fall 2004 Separable Basis Functions Discrete bk l m n bk m bl n e g m k n l m k n l Continuous bkx ky x y bkx x bky y e g x x i y y i x x i y y i Thomas Liu BE280A UCSD Fall 2004 5 Separable Basis Functions b1 n 1 0 1 b1 m 1 0 0 0 1 0 0 b1 n 1 0 0 b2 m 0 0 1 0 b2 n 0 1 1 b1 m 0 1 0 0 b2 n 0 1 0 b2 m 1 0 0 0 1 Thomas Liu BE280A UCSD Fall 2004 Separable Basis Functions b1 n 1 1 2 1 1 1 2 1 2 b1 m 2 1 b1 m 1 1 1 2 1 2 2 1 1 2 1 2 b2 n 1 1 2 b1 n 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 b1 m b1 m 2 1 1 2 1 2 2 1 1 2 1 2 bk l m n bk m bl n 1 2 1 2 b2 n 1 1 2 bk m exp mk 2 bl n exp nl 2 Thomas Liu BE280A UCSD Fall 2004 6 Example x Basis Functions Coefficients Sum Object Thomas Liu BE280A UCSD Fall 2004 Fourier Basis Functions Recall that for 1D the basis functions are complex exponentials bkx x e j 2 kx x For 2D we use the separable 2D functions bkx ky x y bkx x bky y e j 2 kx x e j 2 ky y e j 2 kx x ky y Are they orthonormal Thomas Liu BE280A UCSD Fall 2004 7 Plane Waves e j 2 kx x ky y cos 2 k x x k y y j sin 2 k x x ky y 1 2 x k ky2 1 ky 1 kx cos 2 kxx cos 2 kyy cos 2 kxx 2 kyy Thomas Liu BE280A UCSD Fall 2004 Plane Waves 1 ky A ABC BDC AC AB BC BD D B BD AB C BC AC 1 1 k x ky 1 1 k x2 k y2 1 k x2 k y2 k arctan y kx 1 kx Thomas Liu BE280A UCSD Fall 2004 8 2D Fourier Transform Fourier Transform G kx k y F g x y e j 2 kx x ky y g g x y e j 2 kx x ky y dxdy Inverse Fourier Transform g x y G k k e x j 2 kx x ky y y dkx dk y Thomas Liu BE280A UCSD Fall 2004 Separable Functions g x y is said to be a separable function if it can be written as g x y gX x gY y The Fourier Transform is then separable as well G kx k y g x y e j 2 kx x ky y gX x e j 2 kx x dx gY y e GX kx GY k y Example g x y x y G kx k y sinc k x sinc k y dxdy j 2 ky y dy Thomas Liu BE280A UCSD Fall 2004 9 Example sinc rect Example g x y x y G kx k y sinc k x sinc k y y 1 2 1 2 1 2 x 1 2 Thomas Liu BE280A UCSD Fall 2004 Example sinc rect Thomas Liu BE280A UCSD Fall 2004 10 Examples Thomas Liu BE280A UCSD Fall 2004 Examples Thomas Liu BE280A UCSD Fall 2004 11 Examples Thomas Liu BE280A UCSD Fall 2004 Examples Thomas Liu BE280A UCSD Fall 2004 12 Examples g x y x y x y G kx k y 1 g x y x G kx k y k y Thomas Liu BE280A UCSD Fall 2004 Examples g x y 1 e j 2 ax G kx k y k x k y kx a k y g x y 1 e j 2 ay G kx k y k x k y kx k y a Thomas Liu BE280A UCSD Fall 2004 13 Examples g x y cos 2 ax by 1 1 G kx k y k x a k y b k x a k y b 2 2 Thomas Liu …


View Full Document

UCSD BENG 280A - 2D Fourier Transforms

Documents in this Course
Sampling

Sampling

23 pages

Lecture 1

Lecture 1

10 pages

Lecture 1

Lecture 1

22 pages

X-Rays

X-Rays

20 pages

Spin

Spin

25 pages

Lecture 1

Lecture 1

10 pages

Load more
Download 2D Fourier Transforms
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view 2D Fourier Transforms and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view 2D Fourier Transforms 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?