Goodman, J. W.: 1985, Statistical Optics, Pure and Applied Optics, Wiley, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... remote imaging applications, the background illumination is incoherent and the optical image I (x; y; t) incident on the detector array can be expressed as the spatial convolution of the sensor s optical point spread function p(x; y) with the apparent radiance of the imaged scene I (x; y; t) [61, 30]. x; y; t) 3 p(x; y) 2:1) Although the point spread function may be spatially varying in general, it will be assumed in the following that p(x; y) is space invariant. The apparent scene radiance I (x; y; t) is a combination of the optical energy radiating from the image background and ....

....ideally imaged, constant intensity point source target in an additive independent, identically distributed Gaussian noise background will be developed as an illustration of the various approaches. It is generally accepted that the image formation process is fundamentally a statistical phenomenon [61]. The random fluctuations in detected energy are a function of discrete quantum interactions between light and matter which cannot in principle be perfectly predicted. This uncertainty leads to the development of probabilistic models for the received signal observations and places signal detection ....

Joseph W. Goodman. Statistical Optics. John Wiley and Sons, 1985.

....N 1 1 N j (k,n) k O (3.2) where 2r ( 3.3) Each exponential term in Equation (3.2) is uniformly distributed around the circle in the complex plane. Therefore, z(n) can be considered a sum of random phasors with unit magnitude, a quantity frequently considered in optics [8]. The real and imaginary components of Equation (3.2) are ( o( k O N 1 1 ( k, k O (3.4) By the central limit theorem, as N approaches infinity, Zr (n) and zi (n) become Gaussian . a = 1 Utilizing a transformation of distributed with mean zero and variance ar 2N ....

....a transformation of distributed with mean zero and variance ar 2N variables, z(n) can be shown to have a pdf approaching a Rayleigh distribution with V 2 1 r and variance lzl mean = 2 ] for sufficiently large N. The phase of the blurring kernel is uniformly distributed on ( r) [8]. A plot of the phase is displayed in Figure 3.2(b) The PGA and SSA algorithms were applied to the corrupted image in Figure 3.1(b) The resulting reconstructions are displayed in Figures 3.1(c) and (d) respectively. Ini tially, the reconstructions were circularly shifted with respect to the ....

J. W. Goodman, Statistical Optics. New York: John Wiley and Sons, 1985.

....of a spatially incoherent source. For both techniques, far field intensities are detected and processed phase information is not recorded. Because the far field mutual intensity of a spatially incoherent object is proportional to the Fourier transform of the object s intensity distribution [9], far field intensity measurements can be used to form an image; however, to do this, a difficult phase retrieval problem must be solved [7, 8, 10] Three factors generally determine the accuracy with which one can estimate the modulus or squared modulus of the mutual intensity. First, there is ....

....types of noise affect the estimation accuracy. Finally, the method used to process the data can also have a profound effect on the estimation accuracy. The signal to noise ratio (SNR) and variance for estimators of the modulus or squared modulus of the mutual intensity have been derived by others [1, 5,6,8, 9]; however, these analyses have been restricted to specific processing schemes. Alternative methods for processing the data may produce estimates with higher SNR or lower variance. In this paper, I present a fundamental limit on the accuracy with which the squared modulus of the mutual intensity ....

[Article contains additional citation context not shown here]

J. W. Goodman. Statistical Optics. John Wiley & Sons, New York, 1985.

....# l=1 H n,# (#) 3) B. Coherence Multiplexing The mux and dmux consist of Mach Zehnder interferometers (MZIs) with differential delays exceeding the coherence time of the light source # c . If the light source is assumed to have the rectangular spectrum with bandwidth ##, then # c # 1 ## [11]. The output of the MZI is a sum of two fields with time delay # m . This operation denotes Lm [c n (t) which splits, delays with # m , and recombines the encoded optical field cn (t) The analytical signal of the transmitted signal is expressed as s m,n (t) cn (t) cn (t # m ) 2 #L m ....

....k = #, reverting the polarity of the received data. The resulting photocurrent of the balanced detector is expressed as i B = ##I 0 L# 4## , 17) where # is the responsivity of the photodetector. Next, we consider the variance of the photocurrents. The variance of the photocurrent is given by [11] # 2 i = # 2 T # # # # # # T # # I (#) d# (# IT) 2 , 18) where I(t) and I are the instantaneous power and the average power, respectively, and #(x) # 1 x , for x #1 0, otherwise. 19) Since we assume that all the sources are thermal with the same spectrum G 0 ....

[Article contains additional citation context not shown here]

J. W. Goodman, Statistical Optics, New York: John Wiley & Sons, Inc.

....negative values. Profiles through the center of the images are shown in Figure 3. where R is the amplitude ratio the amplitude of one wave field divided by the sum of amplitudes of the two wave fields, and k(x; y) is the amplitude PSF for transmitted light optics under coherent illumination [13]. The complex valued function h(x; y) is the amplitude PSF for DIC optics; a calculated h(x; y) is shown in Figure 2. It should be noted that the square magnitude of the DIC PSF, jh(x; y)j 2 , predicts the DIC image of a pinhole [11] i.e. an intensity point, rather than the DIC image of a delta ....

....approximation than temporal incoherence. We note that, for our expiremental parameters, the condition l c AE nt that ensures temporal coherence (see discussion in Section 3. 4) does not hold because the coherence length of the illuminating light, l c , is small; l c = 0:664 c= Delta = 2:9 m (see [13], p. 168) where c is the speed of light in air, and Delta = 6:8 Theta 10 7 MHz is the half power bandwidth of the excitation filter (Olympus IF550) which was placed in front of the light source. We note that using a filter with a narrower bandwidth would yield a larger l c which could ensure ....

J. W. Goodman. Statistical Optics. John Wiley and Sons, New York, 1984.

....considered in a similar way. The measurement vector for each pixel is X= y n , where y n =i n jq n . When speckle is fully developed, the (i n ,q n ) are statistically independent random processes. However, the y n are correlated complex Gaussian random processes with pdf given by Goodman, 1985 [6]: P(X C S ) exp( t X .C S 1 .X) p L C S ) 9) Separate complex looks case: In this case, the Complex Gaussian Distribution Entropy MAP (CGs DE MAP) filter for separate complex looks (L complex looks) is expressed as: X .C S 1 .X LR i R i 2 . log(R ik ) 1 Ln(10) ....

. J.W. Goodman, 1985: "Statistical Optics", J. Wiley & Sons, NY, 1985.

....z; z 0 ) 1 ( Rk) 2 ( J 2 n 1 (2 Rk) J n 1 [2 R (z)k]J n 1 [2 R (z 0 )k] z) z 0 ) Gamma 2 J n 1 [2 R (z)k]J n 1 (2 Rk) z) 25) where n is the radial order of the j th Zernike polynomial and J (x) is the th order Bessel function of first kind. Making, as usually[21, 22], the variable substitution ( 2j = z z 0 = z Gamma z 0 (26) and noting that where F (0) n 0 (k; j j) differs appreciably from zero G k; j 2 ; j Gamma 2 G(k; j; j) 27) Eq. 24) becomes oe 2 F A;j = 2 (n 1) 2 2 Z H 0 dzC 2 n 0 (z) Z k dk ....

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985), Subsec. 8.6.1, pags. 420-423.

.... in Radio Astronomy, by Thompson, Moran and Swenson (1986) A broader review of radio imaging and radio telescopes is given by Christiansen and Hogbom (the second, 1985, edition is recommended) A slightly different viewpoint (that of stochastic signal processing) is given in Statistical Optics by Goodman (1985). 1.2 Electromagnetic Field and Field Propagation I will start with a simple derivation of the fundamental equations of aperture synthesis. A number of simplifying assumptions will be introduced which are valid to a high order for the practical cases of special interest to us in VLBI observations ....

Goodman, J. W. 1985. Statistical Optics. New York: John Wiley and Sons.

....homogeneous, and isotropic, the method of small perturbations can be used to solve the wave propagation problem. The method of small perturbations yields the two dimensional spherical wave correlation functions (in a plane transverse to the direction of propagation) for the log amplitude and phase [12, 8, 13, 7], B (ae) 4 2 k 2 Z L 0 dz Z 1 0 d J 0 i aez L j sin 2 2 z(L Gamma z) 2kL Phi n ( z) 12) and B OE (ae) 4 2 k 2 Z L 0 dz Z 1 0 d J 0 i aez L j cos 2 2 z(L Gamma z) 2kL Phi n ( z) 13) where J 0 is a Bessel function of the first kind, ....

J. W. Goodman, Statistical Optics. New York: John Wiley & Sons, 1985.

.... find Gamma OD and Gamma DD using the image model given at Equation (6) First, consider the correlation between the uth spatial frequency of the object and vth spatial frequency of the detected image denoted Gamma OD (u; v) E[O(u)D (v) 16) Using standard correlation calculation methods [1, 22, 35], we note that (See Appendix A) Gamma OD = K) 2 H (v) Gamma OnOn (u; v) 17) where K is the average number of photoevents in the image, H is the mean OTF, and On is the normalized object spectrum defined as On (u) O(u) O(0) O(u) K : 18) Gamma OnOn (u; v) is the correlation ....

.... between real and imaginary components [30] Our analysis does not explicitly consider the correlation of the real and imaginary components but treats the detected image spectral elements as single complex numbers [1] The resultant expression is attained using the same methods as above [1, 22, 35] such that (See Appendix B) Gamma DD (u; v) K) 2 Gamma OnOn (u; v) Gamma HH (u; v) K H(u Gamma v)On (u Gamma v) P oe 2 det ffi (u Gamma v) 21) where Gamma HH (u; v) is the correlation between the uth and vth spatial frequencies of the OTF, On denotes the normalized mean object ....

J. W. Goodman, Statistical Optics. John Wiley and Sons: New York, 1985.

....one must solve the wave equation for a random media, r 2 E k 2 n 2 E = 0; 2.58) where n is a random field of index variations over the region of propagation. If the turbulence is weak, homogeneous, and isotropic, the method of small perturbations can be used to solve Eq. 2. 58) [2, 5, 10, 25]. The method of small perturbations yields the two dimensional spherical wave correlation functions (in a plane transverse to the direction of propagation) for the log amplitude and phase, B (ae) 4 2 k 2 Z L 0 dz Z 1 0 d J 0 aez L sin 2 2 z(L Gamma z) 2kL # Phi n ....

Goodman, J. W. Statistical Optics. New York: John Wiley & Sons, 1985.

....y] T 2 S is a set of discrete positions in image plane corresponding to the detector pixel locations. The photon count detected for each pixel is modeled by a Poisson PDF. The probability of detecting d( x) counts from the pixel located at x 2 S given knowledge of the average count is given by [11] f dj (d( x) j ( x) x) d( x) expf Gamma( x)g d( x) x 2 S; 2) where d( x) is a random variable characterizing the number of photo events at location x 2 S and ( x) is the average count. The average count ( x) is proportional to the subaperture image irradiance given in Eq. 1) We ....

..... The joint PDF of d and x s is given by f d; x s (d; x s ) f dj x s (dj x s ) f x s ( x s ) 5) where f x s ( x s ) is PDF of the subaperture irradiance centroid. For atmospheric turbulence induced wavefront perturbations, the PDF of x s is modeled as circular Gaussian random variable [1, 11]: f x s ( x s ) 1 2 oe 2 s exp ae Gamma j x s j 2 2oe 2 s oe (6) where oe 2 s is the variance of the x and y directed components of x s . The notation j x s j 2 = x T s x s = x 2 s y 2 s . 2.2 Overall SH WFS joint statistics The joint PDF for the single subaperture ....

J. W. Goodman, Statistical Optics (John Wiley & Sons, New York, 1985).

....of N 0 when the phases are evenly spaced by T=N . If the transmitters of all channels use independent clocks, each random bit skew OE i can be modeled with a random variable distributed uniformly over [ GammaT =2; T=2] In that case, the characteristic function of oe 2 x in (9) is given by [9] J N 0 (4f j 1 j)e j2fN 0 , where J 0 (x) is a Bessel function of the first kind, order zero. A Fourier transform of its characteristic function yields the probability density function of oe 2 x . When many channels are interfering with each other, the distribution of oe 2 x approaches a ....

....j2fN 0 , where J 0 (x) is a Bessel function of the first kind, order zero. A Fourier transform of its characteristic function yields the probability density function of oe 2 x . When many channels are interfering with each other, the distribution of oe 2 x approaches a Gaussian distribution [9] with mean (N Gamma 1) 0 and variance 2(N Gamma 1)j 1 j 2 . 6 Averaging (8) over oe 2 x yields a BER as follows: P e = Z (N Gamma1) 0 2j 1 j) N Gamma1) 0 Gamma2j 1 j) 8 : 1 Gamma p)Q 0 z th Gamma z 0 q oe 2 x oe 2 n0 1 A pQ 0 z 1 Gamma z th q oe ....

J. W. Goodman, Statistical Optics, John Wiley & Sons, pages 145--151, 1985.

No context found.

Goodman, J. W.: 1985, Statistical Optics, Pure and Applied Optics, Wiley, New York.

No context found.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 206.

No context found.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), p. 206.

No context found.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

No context found.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5, p. 208.

No context found.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 5, p. 157.

No context found.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

No context found.

J. W. Goodman. Statistical Optics. Wiley, 1985.

No context found.

J. W. Goodman, Statistical Optics, John Wiley & Sons, 1985.

No context found.

J. W. Goodman. Statistical Optics, pages 286324. John Wiley and Sons, New York, 1984.

No context found.

J.W. Goodman, Statistical Optics, (John Wiley & Sons, New York, 1985), Chap. 5, p. 181.

No context found.

J. W. Goodman. Statistical Optics. John Wiley & Sons, New York, 1985.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC