GEOMETRICAL IMAGING First-order image is perfect object (input) scaled (by magnification) version of object optical system magnification = image distance/object distance no blurring object distance image (output) image distance ECE/OPTI533 Digital Image Processing class notes 96 Dr. Robert A. Schowengerdt 2003
Real image is blurred by two optical effects: aberrations e.g. spherical aberration, astigmatism, coma Optical systems can be designed with little aberration diffraction due to wave nature of light and finite apertures of optical systems cannot be avoided or reduced High quality, aberration-free system is diffractionlimited ECE/OPTI533 Digital Image Processing class notes 97 Dr. Robert A. Schowengerdt 2003
DIFFRACTION IMAGING Physical Basis Even a perfect system, from a geometrical optics viewpoint, will not form a point image of a point source Describes the wave nature of light, unlike geometrical optics Optical system apertures truncate radiation wavefronts physical aperture ( stop ) entrance and exit pupils are virtual apertures, i.e. images of the physical aperture ECE/OPTI533 Digital Image Processing class notes 98 Dr. Robert A. Schowengerdt 2003
Imaging System as a Linear System Consider an optical imaging system as a black box, with an input wavefront from the object, an entrance pupil, exit pupil, and an output wavefront forming the image for incoherent light, system is linear in intensity (square of complex field amplitude) point source entrance pupil focal length f image plane spherical wavefronts exit pupil (diameter D) optical system truncated spherical wavefronts image of point source system f-number N = f/d ECE/OPTI533 Digital Image Processing class notes 99 Dr. Robert A. Schowengerdt 2003
Impulse response called the Point Spread Function (PSF) image of a point source (delta function input) General LSI imaging equation for optical diffraction gxy, ( ) = f x y (, ) PSF ( x, y ) where f(x,y) = geometrical image g(x,y) = diffraction-limited image geometrical image diffractionlimited image ECE/OPTI533 Digital Image Processing class notes 100 Dr. Robert A. Schowengerdt 2003
2-D Fourier transform of the PSF called the Optical Transfer Function (OTF) OTF u, v ( ) = PSF x y (, )e j2π ( xu + yv ) x d d y Always a Low-Pass Filter (LPF) ECE/OPTI533 Digital Image Processing class notes 101 Dr. Robert A. Schowengerdt 2003
Diffraction-Limited PSF Incoherent light, circular aperture with diameter D exit pupil function r tr ( ) = cyl --- D PSF is Fourier-transform, scaled and squared PSF ( r' ) 2 J 1 ( r' = ------------- ) r' 2 where J 1 is the Bessel function of the first kind and the normalized radius r is, r' πd = ------- r = λf πr ------- λn where D = exit pupil diameter f = focal length N = f-number λ = wavelength of light r = 1.22 λf ----- = 1.22λN r' = 1.22π D The first zero occurs at, or ECE/OPTI533 Digital Image Processing class notes 102 Dr. Robert A. Schowengerdt 2003
ECE/OPTI533 Digital Image Processing class notes 103 Dr. Robert A. Schowengerdt 2003 0 2 4 6 8 10 normalized radius r' 0 0.2 0.6 0.4 radial profile PSF 0.8 1 central bright region, to firstzero ring, is called the Airy disk Airy pattern 2-D view (contrast-enhanced) OPTICAL IMAGE FORMATION
Example calculation of PSF size system specs: D = 1cm f = 50mm N = f/d = 5 λ = 0.55µm (green) radius of PSF = 1.22λN = 3.36µm very small! ECE/OPTI533 Digital Image Processing class notes 104 Dr. Robert A. Schowengerdt 2003
Diffraction-Limited OTF Incoherent light, circular aperture with diameter D OTF ρ' ( ) = 2 -- [ acos ( ρ' ) ρ' 1 ρ' 2 ] π where the normalized radial spatial frequency is given by, ρ' = ρ ρ c and the cutoff frequency is given by, D ρ c = ----- = λf 1 ------- λn where D = aperture diameter f = focal length N = f-number λ = wavelength of light ECE/OPTI533 Digital Image Processing class notes 105 Dr. Robert A. Schowengerdt 2003
General LSI imaging equation for optical diffraction gxy, ( ) = f x y (, ) PSF ( x, y ) where f(x,y) = geometrical image g(x,y) = diffraction-limited image Take Fourier transform Guv (, ) = Fuv (, ) OTF ( u, v ) where F(u,v) = spatial frequency spectrum of geometrical image G(u,v) = spatial frequency spectrum of diffraction image OTF(u,v) = Optical Transfer Function Note, all three quantities are functions of spatial frequency in the image space, i.e. magnification between the object and image has been included ECE/OPTI533 Digital Image Processing class notes 106 Dr. Robert A. Schowengerdt 2003
ECE/OPTI533 Digital Image Processing class notes 107 Dr. Robert A. Schowengerdt 2003 u ρ = ρ c v OTF 0 0 0.2 0.4 0.6 0.8 1 normalized spatial frequency (ρ/ρ c ) 0.2 0.4 By convention, always normalized to 1 at zero frequency OTF 0.6 Nearly a cone function of (u,v) 0.8 1 OTF is a low-pass filter of spatial frequencies OPTICAL IMAGE FORMATION
Example calculation of OTF cutoff frequency system specs: D = 1cm f = 50mm N = f/d = 5 λ = 0.55µm (green) cutoff frequency = 1/λN = 0.363cycles/µm = 363cycles/mm very high! (the human vision system has a cutoff frequency of about 10cycles/mm (object scale) at normal viewing distance) ECE/OPTI533 Digital Image Processing class notes 108 Dr. Robert A. Schowengerdt 2003
Modulation Transfer Function (MTF) Amplitude of complex OTF MTF ( u, v ) OTF ( u, v ) Guv (, ) -------------------- Fuv (, ) = = = output signal modulation ----------------------------------------------------------- input signal modulation signal modulation signal modulation = max min -------------------------- max + min modulation is a measure of signal contrast A+ Bsin( 2πu o x ) for sinewave,, modulation = B/A ECE/OPTI533 Digital Image Processing class notes 109 Dr. Robert A. Schowengerdt 2003
Use MTF to predict image contrast, given object and system output modulation(u,v) = input modulation(u,v) x MTF(u,v) for sinewave input to LSI system input ( x ) = A+ Bsin( 2πu o x ) corresponds to the geometrical image formed by the optics the output is output ( x ) = A + MTF ( u o )B sin( 2πu o x ) corresponds to the diffraction image formed by the optics and the image modulation at spatial frequency u o is B image modulation = MTF ( u o ) --- A ECE/OPTI533 Digital Image Processing class notes 110 Dr. Robert A. Schowengerdt 2003
MTF 1 0.8 0.5 typical MTF for imaging system 0.1 u u 0 2u 0 4u 0 u = u 0 u = 2u 0 u = 4u 0 input output ECE/OPTI533 Digital Image Processing class notes 111 Dr. Robert A. Schowengerdt 2003
DIFFRACTION IMAGING EXAMPLE Input pattern: chirp function spatial frequency increases linearly with distance 1 1 + -- cos( 2πx 2 ) 2 maps spatial frequency into a spatial representation spatial frequency, u greyscale pattern profile What is going on here? ECE/OPTI533 Digital Image Processing class notes 112 Dr. Robert A. Schowengerdt 2003
Diffraction-limited optics with a circular aperture convolution System response (PSF) Output pattern (image) envelope of ouput pattern amplitude is the MTF (because of pattern used) cutoff-frequency, u c ECE/OPTI533 Digital Image Processing class notes 113 Dr. Robert A. Schowengerdt 2003
LINE AND EDGE SPREAD FUNCTIONS PSF is often difficult to measure because of insufficient energy Use a line source (slit) or edge (step) source to increase energy for measurement ECE/OPTI533 Digital Image Processing class notes 114 Dr. Robert A. Schowengerdt 2003
ECE/OPTI533 Digital Image Processing class notes 115 Dr. Robert A. Schowengerdt 2003 If PSF is not rotationallysymmetric, LSF is different in different directions LSF does not have zeros like PSF LSF y ( y ) = PSF x y (, ) d x distance (arbitrary units) 0-10 -5 0 5 10 LSF x ( x ) = PSF x y (, ) d y 0.01 irradiance (watts/m 2 ) 0.02 0.03 0.04 Integrate PSF along direction of the line source 0.05 0.06 Line Spread Function (LSF) diffraction-limited LSF OPTICAL IMAGE FORMATION
ECE/OPTI533 Digital Image Processing class notes 116 Dr. Robert A. Schowengerdt 2003 ESF is a monotonic function of distance Equivalent to step response in electronic system distance (arbitrary units) 0-20 -15-10 -5 0 5 10 15 20 ESF y ( y ) = LSF y α y ( ) d α 0.2 irradiance (watts/m 2 ) 0.4 0.6 ESF x ( x ) = LSF x α x ( ) d α 0.8 1 Integrate LSF up to location of ESF measurement Edge Spread Function (ESF) OPTICAL IMAGE FORMATION
Relations among PSF, OTF, LSF and ESF (x-direction) PSF(x,y) one-sided 1-D integration ESF x (x) 2-D Fourier Transform 1-D integration one-sided 1-D integration 1-D derivative OTF(u,v) OTF(u,0) LSF x (x) profile 1-D Fourier Transform ECE/OPTI533 Digital Image Processing class notes 117 Dr. Robert A. Schowengerdt 2003
DEFOCUS image is out-of-focus if receiving plane is not at the focal plane By geometry, diameter of blur circle, d, is: D --- f d = --------- d 2 = ------- 2N or, where N = f-number aperture diameter D focal plane f blur circle diameter d /2 ECE/OPTI533 Digital Image Processing class notes 118 Dr. Robert A. Schowengerdt 2003
Filter Model PSF defocus r ( ) = 4 --------cyl πd 2 r -- d OTF defocus ρ ( ) = somb ( ρd ) MTF defocus ( ρ ) = somb ( ρd ) defocus = defocus = 2 ECE/OPTI533 Digital Image Processing class notes 119 Dr. Robert A. Schowengerdt 2003
Spatial Phase Shift π spatial phase shift at spatial frequencies where OTF is negative defocus = defocus = 2 first-zero in OTF defocus ECE/OPTI533 Digital Image Processing class notes 120 Dr. Robert A. Schowengerdt 2003