Evaluation of a multi-frame blind deconvolution algorithm using Cramér-Rao bounds

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1 Evaluation of a ulti-frae blind deconvolution algorith using Craér-Rao bounds Charles C. Beckner, Jr. Air Force Research Laboratory, 3550 Aberdeen Ave SE, Kirtland AFB, New Mexico, USA Charles L. Matson Air Force Research Laboratory, 3550 Aberdeen Ave SE, Kirtland AFB, New Mexico, USA Abstract: Saple statistics fro a ulti-frae blind-deconvolution (MFBD) algorith are copared with Craer- Rao bounds (CRB) in order to evaluate the noise reduction perforance of the MFBD algorith. In this paper, CRB theory is eployed as a etric to evaluate the perforance of a MFBD iaging algorith developed at the Air Force Maui Optical and Supercoputing Site, a site operated by the Air Force Research Laboratory. Saple variances fro the MFBD algorith naed PCID (physically constrained blind deconvolution) and CRB lower bounds to variances are copared for a baseline odel iaging scenario that eploys an object, blurring, photon noise and read noise. The variance reduction effects produced by iposing support constraints on the object and on the point spread function (PSF) are analyzed. Pixel-by-pixel saple variance aps are copared to CRB aps for the case of perfect and loose object support constraints. The PCID saple variance aps are evaluated against CRBs both to deterine the relative agnitude of these variances as opposed to CRB lower bounds and to assess overall orphology differences. For the baseline odel iaging scenario, the PCID pixel-by-pixel saple variance agnitudes atch their associated CRBs, and the PCID saple variances and CRBs share the sae overall orphology. Additionally, PCID saple variance results are presented for cases where the baseline odel iaging and post-processing scenario above is extended beyond where CRB theory has been developed. The odel iaging scenario is extended to include the use of positivity in the iaging algorith. 1. Introduction Recently, Craér-Rao bound (CRB) theory for support-constrained ulti-frae blind deconvolution was developed. 1 CRB theory can be used to generate algorith-independent lower bounds to either unbiased or biased estiators of a set of paraeters. In this paper, the CRB theory for unbiased estiators is eployed as a etric to evaluate the perforance of a ulti-frae blind-deconvolution (MFBD) iaging algorith developed at the Air Force Maui Optical and Supercoputing Site, a site operated by the Air Force Research Laboratory. Saple variances fro our MFBD algorith naed PCID (Physically Constrained Blind Deconvolution) are copared to their corresponding CRBs for a odel iaging scenario. PCID eploys a conjugate gradient search algorith in order to iniize a cost function that expresses the difference between the given easureent and estiates of the easureent based on convolving the estiated object and blurring point spread functions (PSF). It is shown that the PCID saple variances and the CRBs have siilar orphology, and that the PCID saple variances approach their respective CRBs, i.e., they nearly achieve the theoretical perforance bounds of noise reduction. Then, the iaging scenarios eployed in the PCID/CRB coparison are once again post-processed using PCID, this tie enforcing a positivity constraint. The associated positivity constrained CRBs are not presented. The positivity enforced PCID results are evaluated against the unconstrained PCID and CRB results both to deterine the nuerical effect of positivity on reconstructions and to assess orphological differences. The reainder of this paper is structured as follows: the basic iaging odel and an explanation of the PCID algorith are given in Section 2, a brief explanation of how Craer-Rao bound theory is eployed is this research is provided in Section 3, results are presented in Section 4, then a conclusion and rearks about future work are given in Section 5.

2 2. Iaging odel and PCID The iaging odel used in this research is () x o() x h () x n () x i = + (1) where i is the th easured iage, x specifies locations in the iage, * denotes convolution, h is the th PSF, o is the object, and n is the th zero ean, spatially-uncorrelated noise function In a typical iaging application, the exposure tie of i (x) is on the order of 1-10 s so that high frequency content out to the diffraction liit is retained. 2 Then, our PCID algorith can be eployed to extract uch of the inforation content concealed by the blurring and noise. The PCID algorith jointly estiates both the object and the blurring function fro the blurred and noisy iage data by iniizing the following cost function: J ( ô, ĥ,..., ĥ M ) = [ î ( x) i ( x) ] 2 1 (2) x where J( ) is the cost function to be iniized with respect to estiates of the object and all the PSFs, using all M easured iages, and the î ( x) are the estiates of the M iages generated by convolving the estiated object and the M blurring functions. 3 It should be noted that both positivity and support constraints can be iposed on the algorith. 3. Craer-Rao Bounds Craer-Rao bound theory provides an algorith-independent analysis approach to explore how the cobination of the odel iaging scenario and support constraints ipact reconstructed iage quality. This approach is eployed because it produces gold standard results that characterize the benefits of prior knowledge constraints without the concerns that the results are algorith dependent and that the results ay change if a ore clever algorith is eployed. Craér-Rao lower bound (CRB) theory relies upon the Fisher inforation atrix (FIM) to generate lower bounds to the variances of any estiate of a collection of rando variables. As a result, it can be viewed as characterizing how the inforation content in a prior knowledge constraint reduces noise in an iage, rather than how a specific algorith uses the inforation in a prior knowledge constraint to reduce noise Results The easured data consisted of the satellite object shown in Fig.1 convolved with M atospheric blurring point spread functions corresponding to D/r 0 = 8 and corrupted by either zero-ean Gaussian white noise or by Poisson distributed photon noise and zero-ean Gaussian white noise (read noise), where D is the telescope diaeter and r 0 is the Fried paraeter. The PSFs were ade invertible by first low-pass-filtering the with a square low-pass filter and then ebedding these low-pass-filtered versions in an array of the sae size as the low-pass filter. They were then ade to have finite support by applying a circular support constraint in the iage doain. These odifications to the PSFs were ade to siplify the coparison between saple variances and CRBs by aking possible unbiased estiates of the object. A large set of blurred and noisy data was generated using this approach for use in PCID reconstructions. Pixel-by-pixel saple variances were produced fro support-constrained PCID restorations using this dataset with M = 10. In this paper, perfect object support where the support boundary confors perfectly to the ocnr object, and the blur2 and circular object support shown in Fig. 1 were eployed in PCID restorations. CRBs for this iaging situation were obtained fro previous work. 5 CRB, PCID, and PCID with positivity enforced results for the read noise only case are presented in Figure 2. These results are depicted as pixel-by-pixel variance aps, noralized to the highest valued pixel of the associated CRB variance ap. PCID saple variances and CRBs, for the case of perfect object support, tight but non-perfect blur2 object support, and circular support are shown. The CRB and PCID results were generated fro the sae iaging scenario, so that the CRBs can be eployed as a etric to evaluate the PCID results. The siilarities in orphology show that the CRBs and the PCID saple variances are in qualitative agreeent for all three support constraints sizes. Also, the nuerical value of the sued CRB arrays and PCID saple variance arrays shows

3 that PCID is approaching the theoretical liits to noise reduction for this iaging scenario, as given in Table 1. The ean-square error (MSE), i.e., the total variance sued with the total squared bias (these results were largely unbiased), of the PCID reconstructions nearly eet the CRBs. 6 In order to achieve the CRBs, the PCID algorith ust drive to the global iniu solution of the given noise corrupted, blind deconvolution proble. In iaging situations where positivity was not eployed, the blurred and noisy data could be used to seed the PCID algorith. In challenging iaging situations where the object support is significantly larger than the object and where positivity is eployed, the PCID algorith can get stuck in local inia. In this case, the initial object guess eployed to seed the algorith consisted of the ocnr object with noise that aounted to three or ore ties its associated CRB variance. The sued saple variances of the positivity constrained case are the sae or lower than the unconstrained case. Although the positivity constrained CRBs are not provided, it sees reasonable to conclude that the variances in the positivity constrained case are lower because negative intensity values are eliinated thereby reducing the range of possible variance fro the ean. Also, the squared bias of the reconstructions rose above the unconstrained PCID saple variance levels as ight be expected fro the application of the positivity constraint. Also, positivity appears to operate like an object support constraint when the applied object support constraint is loose, as shown in the positivity enforced, circular support constrained results in Figure 2. (a) (b) (c) Fig. 1. (a) Coputer-siulated satellite odel, (b) the blur2 support constraint and (c) circular support used for the PCID and CRB calculation. Totals within support constraint: true support blur2 support circular support CRBs 4.8 x x x 10 6 PCID 4.9 x x x 10 6 PCID positvity enforced 4.9 x x x 10 5 Table 1. Read noise case. The total CRBs within a support constraint consist of the suation of all the iniu possible pixel-by-pixel variances that any unbiased estiate could achieve. For each PCID result presented in this table, the sued pixel-by-pixel saple variances were drawn fro a set of fifty reconstructions. Each reconstruction was post-processed fro easureents fored fro the ocnr object, the ten PSFs, and independent read noise realizations.

CRBs ax CRB PCID PCID positivity enforced 0 true blur2 circle Fig. 2. Read noise only case. The first row consistts of pixel-by-pixel CRB lower bounds to variance aps.

Support constrained cases are organized by colun: perfect or true support, blur2 non-perfect support, and circular support.

CRB, PCID, and PCID with positivity enforced results for the photon and read noise are presented in Figure 3.

PCID saple variances and CRBs, for the case of perfect object support, tight but non-perfect blur2 object support, and circular support are shown.

4 CRBs ax CRB PCID PCID positivity enforced 0 true blur2 circle Fig. 2. Read noise only case. The first row consistts of pixel-by-pixel CRB lower bounds to variance aps. The second and third row consist of PCID saple variance aps and for unconstrained and positivity constrained cases. Support constrained cases are organized by colun: perfect or true support, blur2 non-perfect support, and circular support. All PCID variance aps are noralized to the highest value within the associated CRB ap located at the top of the colun. CRB, PCID, and PCID with positivity enforced results for the photon and read noise are presented in Figure 3. As in the read noise only case, these results are depicted as pixel-by-pixel variance aps, noralized to the highest valued pixel of the associated CRB variance ap. PCID saple variances and CRBs, for the case of perfect object support, tight but non-perfect blur2 object support, and circular support are shown. The CRB and PCID results were generated fro the sae iaging scenario, so that the CRBs can be eployed as a etric to evaluate the PCID results. Again, the siilarities in orphology between the two aps show that the CRBs and the PCID saple variances are in qualitative agreeent. Also, the nuerical value of the sued CRB arrays and PCID saple variance arrays shows that PCID is approaching the theoretical liits to noise reduction for this iaging scenario, as given in Table 2. The MSE of the PCID reconstructions nearly eet the CRBs. 6 It is worth noting that the MSE was alost entirely coprised of variance; these results were largely unbiased. In order to achieve the CRBs, the PCID algorith ust drive to the global iniu solution of the given noise corrupted, blind deconvolution proble. In iaging situations where positivity was not eployed, the blurred and noisy easureent data could be used to seed the PCID algorith. In challenging iaging situations where the object support is significantly larger than the object and where positivity is eployed, the PCID algorith can get stuck in local inia, e.g., the positivity constrained case with circular support, shown at the botto right in Figure 3, did not converge to the global iniu. In this case, the initial object guess eployed to seed the algorith consisted of the ocnr object with noise that aounted to three or ore ties its associated CRB variance. Totals within support constraint: true support blur2 support circular support CRBs 2.0 x x x PCID 2.3 x x x PCID positvity enforced 2.2 x x Not available Table 2. Photon and read noise case. The total CRBs within a support constraint consist of the suation of all the iniu possible pixel-bypixel variances that any unbiased estiate could achieve. For each PCID result presented in this table, the sued pixel-by-pixel saple variances were drawn fro a set of fifty reconstructions. Each reconstruction was post-processed fro easureents fored fro the ocnr object, the ten PSFs, and independent photon noise and read noise realizations.

crbs ax CRB pcid 0 pcid positivity enforced true blur2 circle Fig. 3. Photon and read noise case. The first row consistts of pixel-by-pixel CRB lower bounds to variance aps.

5 crbs ax CRB pcid 0 pcid positivity enforced true blur2 circle Fig. 3. Photon and read noise case. The first row consistts of pixel-by-pixel CRB lower bounds to variance aps. The second and third row consist of PCID saple variance aps and for unconstrained and positivity constrained cases. Support constrained cases are organized by colun: perfect or true support, blur2 non-perfect support, and circular support. All PCID variance aps are noralized to the highest value within the associated CRB ap located at the top of the colun. 4. Conclusions and Future Work We have eployed CRBs as a etric to easure the perforance of the PCID algorith for the cases of a satellite object, ten blurring PSFs, and photon and read noise. Pixel-by-pixel saple variance aps were copared to CRB aps for the case of perfect, tight, but non-perfect, and circular support constraints on the ocnr object. When these support constraints are eployed, variances are in nuerical and orphological agreeent with CRB theory. The PCID saple variances are roughly equal to or slightly higher than the CRBs, however, because PCID reconstructions achieved the global iniu of the associated PCID conjugate gradient search algorith. PCID saple variances for the positivity enforced case were copared to the CRBs for the unconstrained case. In this case, sued PCID variances dropped below unconstrained CRBs and biases increased, as ight be expected. Also, the positivity constraint acts as an object support constraint when loose object supports are eployed. In future work, factors that ake the PCID algorith get stuck in local inia will be investigated. Also, the noise reduction benefit of additional fraes of data to MFBD algoriths will be investigated. CRB theory indicates that the relative aount of noise reduction per iteration is greater than 1/ for the first few fraes, where is frae count, and then shallows out to approxiately 1/ relative noise reduction per iteration. Saple variances fro PCID will be tested for agreeent with CRB theory. The authors wish to thank the Air Force Office of Scientific Research for their financial support of this research.

6 5. References 1 C.L. Matson and A. Haji, Craér-Rao lower bounds for support-constrained and pixel-based ulti-frae blind deconvolution, Proceedings of the SPIE, vol (2006). 2 M.C. Roggeann and B. Welsh, Iaging Through Turbulence (CRC Press, Boca Raton, FL, 1996). 3 S.M. Kay, Fundaentals of Statistical Signal Processing (Prentice Hall, Upper Saddle River, NJ, 1993). 4 C.L. Matson, C. C. Beckner, Jr., and K.J. Schulze, Fundaental liits to noise reduction in iages using support - benefits fro deconvolution, Iage Reconstruction fro Incoplete Data III, SPIE, vol (2004) 5 C.L. Matson and A. Haji, Noise reduction in support-constrained ulti-frae blind-deconvolution restorations as a function of the nuber of data fraes and the support constraint sizes, Proceedings of the OSA Topical Workshop on Signal Recovery and Synthesis (2007) 6 C. L. Matson and C. C. Beckner, Jr., Regularization, support constraints, and noise reduction in iages a Craér-Rao bound analysis, Proceedings of the OSA Topical Workshop on Signal Recovery and Synthesis (2005)

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