Chirp images in -D fractional Fourier transform domain Presenter: Ming-Feng Lu lumingfeng@bit.edu.cn Beijing Institute of Technology July 0, 016
目录 Introduction of chirp images Chirp images in FRFT domain Applications Conclusions
Chirp signals Introduction of chirp images Also called Linear frequency modulation (LFM) signals Very common in radar, communication, sonar, and etc. Radar Communication Sonar
Chirp signals Introduction of chirp images 1-D Chirp signals (LFM signals) f x A cos Kt f t 1 0 A cos k t k t k 0 0 A amplitude f0 start frequency K chirp rate φ0 fixed phase
Chirp signals Introduction of chirp images -D chirp signals: what are they? f x, y Acos( k x k x k k y k y k ) x x1 x0 y y1 y0 usually circular concentric fringes relative spacing is wider near the center and progressively narrower toward the exterior.
Chirp signals -D chirp signals: Introduction of chirp images f x, y Acos( k x k x k k y k y k ) x x1 x0 y y1 y0 I x, y I A x, y cos( k x k x k k y k y k ) DC x x1 x0 y y1 y0 Newton s rings ESPI (closed) ESPI (unclosed) Electronic speckle pattern interferometry (ESPI)
Chirp signals Introduction of chirp images -D chirp signals Treated as images, and can be called chirp images. Also called fringe patterns with quadratic phase. many of chirp images are like concentric circular images. Newton s rings are classical chirp images. Often encountered in physics, optics and medical science.
Introduction of chirp images Chirp images Newton s rings Light O R Object C d Flat plate r Dn Dm Traditional Newton Interferometer and its schematic diagram
Chirp images in FRFT domain Fractional Fourier transform (FRFT) The continuous-time FRFT is defined as: F( u) K( u, x) f ( x)d x where K ( u, x) which is defined as: is the kernel function, u x jux B expj cot, n, sin K ( u, x) u x, n, u x, n, p 0 p nz B 1 j cot
Chirp images in FRFT domain Fractional Fourier transform (FRFT) The continuous-time FRFT of the 1-D chirp signal is shown as: is rotation angle in the time-frequency plane. p 0 p nz When, FT domain ( ) ( ) jux d / f 1-D chirp signal F u f x e x u t
Chirp images in FRFT domain Fractional Fourier transform (FRFT) As a generalization of the Fourier transform. Understood as chirpbased decomposition. Received much attention in digital signal processing of communication, radar, and etc. FT domain f 1-D chirp signal u t
Chirp images in FRFT domain 1-D Chirp signal in (FRFT) domain 1-D Chirp signals FT
Chirp images in FRFT domain 1-D Chirp signal in FRFT domain 1-D Chirp signals matched α unmatched α
F ( u) K ( u, x) f ( x)d x Chirp images in FRFT domain u x ux B exp j cot Aexp jkx jk1x jk0 dx sin cot cot AB exp j u jk0 exp j k x exp j csc u k xdx when 1 cot k cot F u ABexp j u jk0 exp jk1x exp j csc uxdx cot AB exp j u jk0 csc u k1
Chirp images in FRFT domain Chirp Image in FRFT domain -D chirp signal -DFRFT matched α Ideal Newton s rings,, I x y I A x y DC cos( k x k x k k y k y k ) x x1 x0 y y1 y0
Chirp images in FRFT domain Chirp Image in FRFT domain -D chirp signal, or chirp image I x, y I I cos( k x k x k k y k y k ) 0 1 x x1 x0 y y1 y0 Each row (or column) of the image has the form of a 1-D chirp signal m 0 1 cos x x1 x0 I x rect x r I I k x k x k where r, r m m is the duration.
Chirp images in FRFT domain Chirp Image in FRFT domain For simplification, considering the following complex chirp signal I x rect x r I jk x jk x jk ( m) 1 exp x x1 x0 when the rotation angle α matches the relation : F u x cot k x, the FRFT of the complex chirp signal is a sinc function, i.e., cot F ux I1Br m exp j ux jkx0 sin c csc ux kx1rm
Chirp images in FRFT domain Chirp Image in FRFT domain when F u x csc ux kx 0 1 reaches its peak at the location u x 0 The same procedure is performed on each row and column of the image with the matched rotation angle.
Chirp images in FRFT domain Chirp Image in FRFT domain The FRFT of the chirp image reaches its peak at the position u, u in the FRFT domain, as shown in x0 y0 the following figure. The chirp image in FRFT domain.
Applications Estimate physical parameters E.g. estimate curvature radius R, and center of Newton s rings (x0, y0) Light Object O C R d D-FRFT matched α Flat plate r Actual Newton s rings I I I Kr N cos I I cos K x x0 K y y0 I I cos Kx Kx x Kx Ky Ky y Ky where K / ( R) 0 0 0 0 0 x, y 0 0 Dn Dm
Applications Estimate physical parameters E.g. estimate curvature radius, center of Newton s rings x 0 y R 0 cot 0 u x 0 u y 0 csc K csc K 0 u, x0 y0 incident wavelength u location of peak matched rotation angle The estimation accuracy of curvature radius is 1.9% for analyzing actual image, and the error of the retrieved center is approximately 3 to 4 pixels. The curvature radius is related to the matched rotation angle α. The center is related to the location of peak in FRFT domain.
Applications Estimate physical parameters E.g. measuring deformation, displacement out plane D-FRFT matched α ESPI fringe pattern of out -of-plane displacement ESPI fringe pattern in FRFT domain
Applications Advantages of estimating physical parameters Directly retrieve the curvature radius without the fringe center, which can not be fulfil using other methods. In addition: Insensible to obstacles, in which cases the accuracy of other methods will decrease greatly. Insensible to noise, SNR is up to -10dB,0 db can be achieved at most using other methods. convenient and commercial: without complex and expensive optical design and equipment
Applications Denoising in -D FRFT domain In optical measurement, denoising plays an important role in retrieving the accurate information on the measured physical quantities from fringe patterns. FRFT matched α Blurred Newton s rings (SNR=10) Blurred Newton s rings in FRFT domain Filtered fringe pattern by fractional Fourier filter (PSNR=0.4567) Filter in FRFT domain
Applications Denoising in -D FRFT domain Filtered fringe pattern by arithmetic mean filter (PSNR=15.5836) Filtered fringe pattern by geometric mean filter (PSNR=15.9045) Filtered fringe pattern by median filter (PSNR=15.5384) Filtered fringe pattern by adaptive filter (PSNR=15.9331)
Applications Dealing with artifacts in -D FRFT domain Besides acting as Newton s rings that include information on optical measurement, concentric rings can also serve as artifacts blurring other images. Black and white scanned image Color scanned image Interference pattern Computed tomography(ct) image
Applications Dealing with artifacts in -D FRFT domain Original aircraft image (51 51 8bit) Blurred aircraft image
Applications Dealing with artifacts in -D FRFT domain Blurred aircraft image in the FT domain Blurred aircraft image in the FRFT domain
Applications Dealing with artifacts in -D FRFT domain Filtered aircraft image by fractional Fourier filter. (PSNR=67.541) Filtered aircraft image by Fourier filter. (PSNR= 67.1113)
Conclusions Chirp images have good properties in the -D FRFT domain, like 1-D chirp signals in the 1-D FRFT domain. The properties can be used for measurement, denosing, and removing artifacts.
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