Noise Model

The first, and possibly most significant difficulty in working with coherent illumination is the introduction of speckle noise as described in chapter III. Because nearly all surfaces are rough on the scale of an optical wavelength, the wave reflected from the object will have a spatially random phase, that when propagated will cause the various field components to interfere and produce the grainy speckle patterns characteristic of reflected laser light. As shown previously, speckle noise follows exponential statistics such that for a given pixel

pI(I) = _{E [I]}1 exp ·

− I

E [I] ¸

(5.1)

where it has been assumed that I ≥ 0.

One of the biggest consequences of laser speckle is its extremely noisy nature as demon- strated by the fact that the mean and standard deviation are equal. Defining the signal-to- noise ratio (SNR) as the ratio of the mean signal with the noise standard deviation, speckle noise maintains a SNR of 1 regardless of the signal strength. Unlike photon or detector noise sources, SNR doesn’t improve with higher signal levels. As a result of the noisy nature of active illumination, a multi-frame approach will be required to help mitigate noise effects.

While any one pixel in the image plane follows the statistics of (5.1), the joint statistics of the detected intensity are required to develop a reconstruction algorithm. Given that the complex field is Gaussian distributed in all planes, one could ideally perform variable transformations on the joint-Gaussian distribution as was done for (3.40), and integrate out all phase dependence. However, beyond the second-order distribution derived in Chapter II the applicable integrations and transformations become intractable. As a result, an approximate form for the joint-distribution must be developed.

The simplest approximation is to assume statistical independence between pixels, and form the joint distribution by multiplying the marginals. For this simplification to be valid, the correlation between data points must be negligibly small. In the active imaging case, the required correlation can be found by using the mutual intensity in accordance with the complex Gaussian moment theorem of (3.34). As shown in chapter III, the mutual intensity in the PP of the imaging system can be described by a scaled Fourier transform of the

object. This mutual intensity can then be propagated to the FP and DP of the imaging system using the more general (3.24). However, the required integrals become prohibitively difficult. Fortunately, it was shown by Zernike that in nearly all cases of interest, the light in the PP of a coherent imaging system could be approximated as a constant intensity incoherent source [24]. Under this assumption, the Van Cittert-Zernike theorem can be used to determine the FP and DP mutual intensities.

Substituting the modulus squared of the imaging system’s pupil-function in for the intensity in (3.31) and normalizing

µ_c(∆u, ∆v) = Z Z _∞ −∞ |P (x, y)|2exp ½ −j2π λf (∆ux + ∆vy) ¾ dxdy (5.2)

where µc(∆u, ∆v) is the complex correlation coefficient for points separated by (∆u, ∆v). For a clear circular aperture, this correlation is given by the scaled point-spread function of the un-aberrated imaging system where the spatial variables (x, y) have been replaced by (∆x, ∆y). Assuming a clear circular pupil, the modulus of the mutual intensity falls off rapidly with increasing point separation as shown in Fig. 5.2. Assuming that the pixel spac- ing of the detector is small compared with the width of the PSF, measured pixel intensities are hereafter assumed statistically independent.

One major failure of the assumptions used to generate Fig. 5.2 is that they provide no insight into the average detected intensities. An alternative approach for finding these quantities stems from the description of the mutual intensity in the object plane given by (3.29). This description is identical to that of an incoherent source, and the effect of averaging across many speckle realizations will be equivalent to that obtained for incoherent illumination. Because of this, the modeled noiseless images representing the mean of the statistical distributions are described by the incoherent image formation model of (4.13).

An additional consideration is the temporal correlation between speckle realizations. In the absence of any relative motion between the object, illumination source, and imaging system; the phase relationships that give rise to speckle will remain constant, and speckle observed in the observation planes will be stationary. However, even small relative changes on the order of a wavelength are sufficient to significantly change the speckle pattern. For this work it’s assumed that the time over which speckle realizations become de-correlated

−150 −100 −50 0 50 100 150 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Clear Aperture 1m F10 Telescoop at 1µm

Point Separation (µm)

Normalized Correlation Magnitude |

µc

| (au)

Figure 5.2: Approximate normalized correlation magnitude as a function of point sep-

aration for a 1m F10 telescope operating at 1µm. The correlation falls off rapidly with increasing point separation.

is small compared with the sampling time of the imaging system. Therefore, temporal independence of speckle realizations is assumed.

Finally, to form the approximate joint probability distribution, independence between observation planes is assumed. Though not rigorously justified, this assumption is necessary to form a tractable reconstruction algorithm. Combining these simplifications and assump- tions to form a joint distribution similar to (4.56), the joint probability distribution for a multi-frame active PD data set is given by

pD(D) = Y f,m,x,y 1 gmf(x, y) exp " −d f m(x, y) gfm(x, y) # (5.3)

where gmf(x, y) is the ensemble average intensity in the FP or DP for the fth frame and is as given in (4.13).