Ghost Imaging with the Optimal Binary Sampling
aa r X i v : . [ phy s i c s . d a t a - a n ] M a r Ghost Imaging with the Optimal BinarySampling D ONGYUE Y ANG , G UOHUA W U , B IN L UO AND L ONGFEI Y IN School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, 100876,China State Key Laboratory of Information Photonics and Optical Communications, Beijing University ofPosts and Telecommunications, Beijing, 100876, China [email protected]
Abstract:
To extract the maximum information about the object from a series of binary samplesin ghost imaging applications, we propose and demonstrate a framework for optimizing theperformance of ghost imaging with binary sampling to approach the results without binarization.The method is based on maximizing the information content of the signal arm detection, byformulating and solving the appropriate parameter estimation problem - finding the binarizationthreshold that would yield the reconstructed image with optimal Fisher information properties.Applying the 1-bit quantized Poisson statistics to a ghost-imaging model with pseudo-thermallight, we derive the fundamental limit, i.e., the Cramér-Rao lower bound, as the benchmark forthe evaluation of the accuracy of the estimator. Our theoertical model and experimental resultssuggest that, with the optimal binarization threshold, coincident with the statistical mean of allbucket samples, and large number of measurements, the performance of binary sampling GI canapproach that of the ordinary one without binarization. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ghost imaging (GI), which used to be considered as a counter-intuitive phenomenon from thefirst time it was demonstrated [1], allows an unknown object to be obtained by measuring thespatial-temporal properties of a light beam that never interacts with it. This indirect imagingmethod relies on the intensity or fluctuation correlation between a point-like bucket detectionand a spatial-resolved non-contact reference profile, either a light field detection [2, 3], or a pat-tern modulation [4]. Compromisingly, a large number of repeated measurements are requiredfor reconstructing a high quality image [5–7], which has become a major drawback preventingGI from practical applications, especially real-time tasks, even with the help of compressivesensing technique [8]. Considering this, reducing the dynamic range of detectors or recordingmeasurements with less bits, even 1-bit, would speed up GI process significantly when there areless data to be sampled, transported, stored, and calculated [9]. In fact, it is even more suitablefor computational GI [10], where the reference camera is replaced by a spatial light modulator,thus the sampling process of reference camera is equivalent to the pattern modulation of spatiallight modulator. Binary modulation would bring a much higher modulation rate, especially forthe digital micro-mirror device (DMD) [11], which is in nature a two-level device and has toconduct period multiplexing to accomplish gray scale modulation.For the GI applications under extreme weak echo response conditions [12–14], i.e., detectorshardly register more than one photon for each measurement, the detection noise would imposea significantly negative impact on the image reconstruction of GI [15]. Binary sampling hasprovided an effective way to suppress the background noise at low-light-level [16], also capableof high sensitivity, which is the inherent binary nature of single-photon avalanche detectors(SPAD), one of the most sensitive device to measure the light intensity. This natural combina-tion of binary sampling and SPAD could benefit GI in both sensitivity and robustness againstoise.However, binary sampling have its drawbacks in the loss of information. When we digitalizethe signal into 1-bit, we create at each output a quantization error: the difference between theoriginal signal and the binarization threshold. This quantization error does harm the imagequality of GI [9]. Here comes the question - given a binary sampling GI scenario with certaincharacteristics (e.g. laser light intensity, number of camera pixels, measurement number) - howgood can this binary sampling GI perform? Or given a goal of image quality - how can weoptimize the characteristics of our GI system? Is there an optimal binary detection that wouldyield maximal physical information about the object? To answer these questions, we treat theGI procedure as an estimation of the coincidence measurements for decoding the brightnessdistribution of the object. A powerful measure of the effectiveness of this procedure is based onFisher information [17, 18], a concept from statistical information theory. Fisher information isa mathematical measure of the sensitivity of an observable quantity (e.g. image quality assess-ments) to changes in its underlying parameters (e.g. binarization threshold). Using the Fisherinformation function, one may compute the Cramér-Rao lower bound (CRLB), which providesa theoretical lower bound on the variance of an unbiased estimator. With the right estimator, theperformance of GI, represented by the CRLB, can be evaluated quantitively.The purpose of this work is to fundamentally investigate the binary sampling in GI. We pro-pose a framework for optimizing the performance of binary sampling ghost imaging. Our imageestimation model based on the measure of Fisher information reveals an informative-optimalbinarization threshold for the samples of the signal arm, which is the statistical mean of allbucket samples. With the optimal binarization threshold and large measurement number, theperformance of binary sampling GI can approach the ordinary one without binarization. Theresults of the designed experiments demonstrate highly agreement with the predictions of ourmodel.
2. Theoretical Analysis
In classical imaging, the image sought could be considered as a parametric approximation of λ ( x ) [19], the (normalized) object brightness distribution, written as λ ( x ) = N Õ n = s n h ( N x − n ) , (1)where s n represents the reflective or transmissive function of the object, with n being the numberof subfields on the object plane (1 ≤ n < N ), h ( x ) is the point spread function (PSF) on theimage plane. Due to the finite size of the lens, the impulse response cannot be a Dirac delta,which builds a "point-to-spot" correspondence from a geometrical light point on the object planeto a unique geometrical light spot on the image plane, inducing a resolution limit of the imagingsystem. Similarly, the physics of second-order-measuring GI diagram can also be understoodvia Klyshko’s "unfolded picture" [20, 21], which also build a "point-to-spot" correspondence tomake the "ghost" image of the object aperture possible [22].In the equivalent imaging model of GI, as shown in Fig. 1, s n , j refers to a subfield of the dif-fuser located on spatial unit n at j th measurement, corresponding to the spatial unit x filtered bythe object O ( x ) , and the light emitted from the source is then divided by a beam splitter (BS) intothe reference and the signal arm, where GI requires the same light intensity distribution λ j ( x ) onboth the object plane and reference plane in each j th measurement. To simplify the presentation,we base our discussions on a 1D sensor array, but all the results can be easily extended to 2D case. .2. Reference detection s n,j / (cid:307) m I m,j I Bj Photon Counting y m,j BucketDetection Photon Counting BinarizationCameraObject (cid:540) j C o i n c i d e n ce M ea s u r e m e n t y j b j Light Exposure Binary Sampling
Fig. 1. Signal processing diagram of the equivalent imaging model of binary sampling GI.
On the reference plane, a spatial-resolved image sensor, i.e., a camera works as a samplingdevice of λ j ( x ) . Suppose that the sensor consists of M pixels on the area of interest (AOI),assumed to be the same scale of O ( x ) , the m th pixel covers an area between [ m / M , m + / M ) ,for m ∈ [ , M ) . Besides, GI requires J independent coincidence measurements to reconstruct animage. For the j th ( ∈ [ , J ] ) frame or modulation pattern, within an exposure time τ , we denotethe light intensity accumulated on the m th pixel by I m , j , I m , j def = ∫ τ ∫ ( m + )/ Mm / M λ j ( x ) dxdt = τ h λ j ( x ) , β ( M x − m )i , (2)where β ( x ) is a box function, reads, β ( x ) def = , if 0 ≤ x ≤ , otherwise. (3)Substituting Eqs. (1) and (3) into (2), we have I m , j = τ N Õ n s n , j (cid:28) h ( x ) , β (cid:18) M ( x + n ) N − m (cid:19) (cid:29) , (4) I m , j denote the exposure values accumulated by the m th sensor pixel in j th measurement,which has a stochastic relation to y m , j , the number of photons impinging on the surface of the m th pixel during the exposure time τ in j th measurement. More specifically, according to thesemi-classical theory of photoelectric detection [23], y m , j can be modeled as realizations of aPoisson random variable Y m , j [24], with intensity parameter I m , j , i.e., P (cid:0) Y m , j = y m , j ; I m , j (cid:1) = I y m , j m , j e − I m , j y m , j ! , for y m , j ∈ Z + ∪ { } (5)The expectation (statistical mean) of this Poisson process is E [ Y m , j ] = I m , j , since the averagenumber of photons captured by a given pixel is equal to the local light exposure I m , j . Con-sidering the random nature of the s n , j , both in spatial ( n ) and time ( j ) domain, caused by thepseudo-thermal property [23] and independent repeated measurement in GI, we simplify themarginal distribution of s n = Í j s n , j , λ ( x ) = Í j λ j ( x ) , and I m = Í j I m , j . .3. Bucket detection & Binary sampling For modeling of the "point-like" bucket detection in the signal arm, the bucket detector canbe considered as a collection of pixels filtered by the object O ( x ) with an intensity summationoutput. In what follows, and without loss of generality, we assume the O ( x ) is box function witha length of M , which can be imagined to be a M length "single-slit" object. Recall that the lightfield λ ( x ) on the object plane, and the number of photons impinging on each pixel of the spatialunit subject to the Possion process with a same parameter I m , according to the central limittheorem (CLT) when the number M of pixels is large, the bucket signals are { I B j = M I m , j } ,and the photon counts subject to a Gaussian random variable (cid:8) Y B j (cid:9) ∼ N ( M I m , M I m ) in time ( j ) domain. Using the approximation of Gaussian to Poisson process with a large M I m , we denote P ( Y B j = y j ; M I m ) = p ( M I m ) , i.e., P (cid:16) Y B j = y j ; M I m (cid:17) = ( M I m ) y j e − MI m y j ! . for y j ∈ Z + ∪ { } (6)In this contribution, we apply binarization only on the bucket signals. However, this binariza-tion optimization framework can be easily extended to the samples of reference detectors, butnot included here for simplicity. The binary output b j are drawn from the mapping of randomvariables Q : Y B j B j , such that Q ( y ) = , if y ≥ q ;0 , otherwise. (7)where q > P ( B j = b j ; M I m ) = p b j ( M I m ) , b j ∈ { , } , i.e., p ( M I m ) def = q Õ k = ( M I m ) k k ! e − MI m , (8) p ( M I m ) def = − p ( M I m ) = − q Õ k = ( M I m ) k k ! e − MI m , (9) The previous discussions reveal the relation between the subfield s n , j and the double armcoincident measurements at j th frame, and GI relies J consecutive repeated measurements toreconstruct the image T ( m ) of object by the means of second-order correlation function, T ( m ) = (cid:10) Y m , j × B j (cid:11) j h Y m , j i j h B j i j , (10)where h·i j denotes average overall the measurements, and Y m , j , B j (or Y B j , without binarization)represent the intensity distribution at the reference and binarized bucket detection, respectively.After modeling the detection process, reconstructing the image of object boils down to estimatingthe unknown deterministic parameters { s n } . Input of our estimation problem is two coincidencesequences of binary samples of bucket detector { B j } and ideal samples of the camera { Y m , j } .The PDF of { B j } depends on the averaged light intensity { I m } over all measurements, which arelinked to the subfield parameter { s n } of the light source. In our analysis, we assume that the GIsystem is piecewise stable, i.e., M , N are constant and the PSF h ( x ) in Eq. (1) has a permanentdistribution, the spatial sampling factor 1 / ϕ m is a constant supported within [ , M / N ] [25].oting that convolution can be considered as a linear operator, the mapping between s n and I m can now be simplified as I m = s n / ϕ m , f or m ∈ (cid:20) M ( n − ) N + , MnN (cid:21) (11)Besides, due to the random modulation of R.G.G. and pseudo-thermal light property, thecoincidence measurements are assumed to be independent and identical distributed (iid.) onthe time domain, i.e., the temporal sampling factor 1 / ω j could also be considered as a constant,which simplifies the mapping between s n , j and I m , j , reads, I m , j = s n , j / ϕ m ω j , f or m ∈ (cid:20) M ( n − ) N + , MnN (cid:21) , j ∈ [ , J ] (12)It is apparent that the parameters { s n } have disjoint spatial regions of influence, thus wecan do the estimation one-by-one, independently of each other. Without loss of generality,we focus on the estimation of parameter s from a sequence b j = [ b , . . . , b J ] T and I m , j = (cid:2) I , , . . . , I M / N , J (cid:3) T . For simplicity, we drop the subscript of s and use s instead. The likelihoodfunction L b ( s ) of observing the binary sampling coincidence measurements is written as, L b ( s ) def = P (cid:0) B j = b j , j ∈ [ , J ] ; s (cid:1) P (cid:0) Y m , j = y m , j , m ∈ [ , M / N ) , j ∈ [ , J ] ; s (cid:1) = J Ö j = p b j ( Ms / ϕ m ) J Ö j = M / N Ö m = p ( s / ϕ m ω j ) . (13)Defining J ( ∈ [ , J ] ) to be the number of "1"s in the binary sequence, we can simplify Eq.(13) as L b ( s ) = ( p ( Ms / ϕ m )) J ( p ( Ms / ϕ m )) J − J J Ö j = M / N Ö m = p ( s / ϕ m ω j ) , (14)In order to measure the sensitivity of a measurement (in our case, the coincidence measure-ment) to the parameters being estimated (source field s ), we introduce the Fisher informationmatrix and the CRLB [18], since the accuracy of parameter s , its true value, is at best equal tothe square root of the CRLB [26]. For the Fisher information matrix in our case, which can besimply written as I ( s ) = E h − ∂ ∂ s log L b ( s ) i . By substituting Eq.(14) and after some straightmanipulations, I b ( s ) can be simplified as I b ( s ) = E − ∂ ∂ s © « J log p ( Ms / ϕ m ) + ( J − J ) log p ( Ms / ϕ m ) + J Õ j = M / N Õ m = log p ( s / ϕ m ω j ) ª®¬ = J M ϕ m p ′ ( Ms / ϕ m ) p ( Ms / ϕ m ) p ( Ms / ϕ m ) + E J Õ j = M / N Õ m = (cid:0) y m , j (cid:1) / s , (15)Using the definition of p ( x ) in Eq. (8), the derivative p ′ ( x ) can be computed as p ′ ( x ) = − e − x x q ( q ) ! . (16)Since { y m , j } are drawn from Poisson distributions as in Eq. (5), we have E [ y m , j ] = I m , j = s / ϕ m ω j for all m . Then, E J Õ j = M / N Õ m = (cid:0) y m , j (cid:1) / s = J MN ( s ϕ m ω j ) s = J MN ϕ m ω j s , (17)ubstituting Eqs. (7), (15) and (16) into (14), using the definition of CRLB b = / I b ( s ) , andafter some straightforward manipulations, we have CRLB b = sJ M N ϕ m ω j Γ N ω j + Γ , w here Γ = q Õ j = ( q ) ! ( Ms / ϕ m ) − j ( q − j ) ! ∞ Õ j = ( q ) ! ( Ms / ϕ m ) j ( q + + j ) ! . (18)For comparison, the case without any binarization are also investigated, where bucket outputsare y def = [ y , y , . . . , y J ] T , i.e., the number of photons impinging on each pixel. The likelihoodfunction L y ( s ) of the ideal coincidence measurement in this ideal case is, L y ( s ) def = P (cid:0) Y j = y j , j ∈ [ , J ] ; s (cid:1) P (cid:0) Y m , j = y m , j , m ∈ [ , M / N ) , j ∈ [ , J ] ; s (cid:1) = J Ö j = p ( s / ϕ m ) J Ö j = M / N Ö m = p ( s / ϕ m ω j ) , (19)Using the Fisher information I i ( s ) = E h − ∂ ∂ s log L y ( s ) i and similar calculation process ofEq. (17), we get CRLB i = s E " J Í j = (cid:0) y j (cid:1) + E " J Í j = M / N Í m = (cid:0) y m , j (cid:1) = sJ M N ϕ m ω j N ω j + , (20)For the comparison between the performance of binary sampling GI under different q , we (a) (b) CR L B ( × - ) Binarization Threshold ( q ) (cid:3) Binary (cid:3)
Ideal CR L B D i ff e r e n ce ( (cid:3) × - ) Measurement Number ( J ) (cid:3)(cid:3) (cid:507)(cid:38)(cid:53)(cid:47)(cid:37) Fig. 2. Simulation results: (a) The CRLB of binary sampling GI vs. q , compared with theCRLB of ideal sampling GI. Source intensity s =
8, number of measurement J = , M =
5, subfields N = q = Ms / ϕ m =
40. Data fitted to an exponential decay function. do the numerical simulations to calculate the CRLB of different schemes with certain param-eters. For reducing the computational complexity, the spatial sampling factor are assumed tobe ϕ m =
1, refers to a PSF of box function when m ∈ [ , M / N ] , and the temporal samplingfactor ω j =
1, refers to a uniform sampling when j ∈ [ , J ] . The behavior of CRLB of differentsampling schemes against threshold q is shown in Fig.2 (a). With the optimal q = Ms / ϕ m ,RLB of binary sampling GI reaches the minimum, indicating the optimal estimation of s ,yields the highest quality of reconstructed image of binary GI. The physical meaning of thisoptimal threshold can be simply understood that the temporally-averaged light field s / ϕ m emit-ted from the subfield s is filtered by a M-pixles object and summed by a bucket, which is thestatistical mean of bucket outputs. Besides, the difference between the two CRLBs could alsoprovide a measure of performance degradations incurred by the binary sampling operation, asshown in Fig. 2(b). Under the optimal q , the best-precision gap between the binary samplingGI and the ordinary one, denoted by the difference ∆ CRLB = CRLB b − CRLB i , are subjectedto a negative exponential decay behavior and converging to 0 with the increasing measurementnumber J , which heralds the clue of the reconstructed image of binary sampling GI approachingthe ordinary one under large number of measurements.
3. Experimental Verification
Laser
R.G.G. BS Object Lens BucketDetectorCamera
CoincidenceMeasurement Q z z 2f 2f Fig. 3. Experimental setup. Red mark Q : binarization. R.G.G., rotating ground glass. BS,beam splitter. Experiments are designed to verify the above analysis of the performance of binary samplingGI. For the conventional GI setup [27, 28], as shown in Fig. 3. The pseudo-thermal light source,which mainly consists of a 532 nm CW laser and a rotating ground glass (R.G.G.), generatesrandom speckle patterns. The intensity of the laser is 5 mW, the rotating speed of R.G.G. is 0.32rad/s, z = 300 mm, f = 100 mm. The light emitted from the source is then divided by a beamsplitter (BS) into the reference and the signal arms. The signal arm penetrates a transmissiveobject aperture, a ’GI’ pattern of 500 µ m, be focused by lens and then to be registered by a point-like bucket detector (Thorlabs PDA100A2) as an intensity sequence. The spatial profile, I m , j ,comes from the reference arm, which never interact with the object, is recorded by a commercialCMOS camera (XiMEA MQ003CM) with an AOI of 140 ×
140 pixels, placed on the imageplane and synchronically triggered with the bucket detector. The image is reconstructed by thesecond-order correlation defined in Eq. (10).We introduce binarization on the outputs of the bucket detector to mimic the binary detection,as shown in Fig. 3. Binarization with different threshold q is applied to the recorded quasi-continuous signals of each bucket measurement, as the mapping Q defined in Eq. (7). Herewe want to mention that, although this can easily extend to the reference samples, it is notincluded in our case for simplicity and practical considerations, since the reference arm of someGI applications usually be compacted into the transmitting system with the light source, asdescribed in Ref. [29–31], so the sampling bits of reference detectors would not be compressedin our experiment.When we change the binarizaiton threshold q on the bucket signals, the best and worstreconstructed images under 20,000 measurements are shown in Fig. 4, row I (a) to (d). Theriginal ’GI’ aperture, and the reconstructed images from non-binarized samples, optimal-binarized samples and the first- q -binarized samples are listed in sequence. Considering thecapability of reconstructing gray object, we replace the binary object aperture ’GI’ by a grayobject pattern - the chinese letter ’north’. We mount a neutral density attenuation filter (DahengGCC-301021, transmission 50%) right after the right half of the etched transmissive ’north’letter, which produce a gray object pattern ’north’ with 3-level grayscale. The gray objectpattern and experimental reconstructions under the same 20,000 measurements with differentbinarization operations are listed in the same sequence, shown in Fig.4, row II (a) to (d). Ourexperimental results with the gray object also shows high agreement with the conclusions inRef. [32]. (a) (b) (c) (d) (cid:741)(cid:266) Fig. 4. Experimental verifications: (I) Binary object pattern ’GI’, (II) Gray object pattern ofchinese letter ’North’; (a) Object aperture; Reconstructed image of 20,000 measurementswith (b) no binarization, (c) optimal q , (d) the first q . B u c k e t V a l u e Measurement Number
Optimal Threshold Bucket Mean
Experimental Counts Gaussian Fit P r obab ilit y Bucket Value
Experimental Results Gaussian Fit S N R ( / d B ) (a) (b) Fig. 5. (a) Up: Reconstructed image SNR vs. q (step by 0.00069 = bucket sampling interval)of bucket values. Down: Gaussian PDF of bucket detector output values. (b) The optimalthreshold and statistical mean of bucket outputs against measurement number. We use signal-to-noise ratio (SNR) of reconstructed image to evaluate the performance ofimaging system quantitively, and the definition is,SNR =
20 lg Í x O ( x ) pÍ x ( O ( x ) − T ( x )) , (21)here the O ( x ) and T ( x ) represents the "0-1" object and reconstructed image, respectively. Thehigher SNR is, the better image one gets. And the SNR performance against binarization thresh-old q under the same measurement number 20,000 is shown in upper part of Fig. 5 (a), comparedwith the frequency distribution of bucket outputs, both fitted well to a Gaussian PDF as assumedin our theoretical model. Apparently, there exists an optimal q to approach the performanceof GI without binarization, and the optimal q is very close to the mean of the bucket outputs,according to the property of the Gaussian PDF. Furthermore, we make more efforts to verifyif the optimal q is the statistical mean of bucket outputs. So we compare the optimal q , whichcorresponds to the highest SNR of reconstructed image, and the mean of bucket outputs in Fig.5 (b). For different measurement numbers, the optimal q behaves coincidentally to the bucketmean. Here we have to mention that, the varying threshold is the physical sampling intervalsof our bucket detector, which cannot be determined by the statistical properties of our bucketoutputs.Further experimental verifications also show the performance comparison of GI with differ-ent binarization strategies from under-sampling (100) to over-sampling (50,000) conditions, theSNR of optimal binarized GI is very close to the ordinary one, which is far more better than thefirst-threshold binarized GI, as shown in Fig. 6 (a). For comparison, the SNR of binarized GIwith the bucket mean threshold are also presented, which behaves almost the same to the optimalbinarization. What’s more, there still exist a gap of SNR between the convergence upper limitof optimal binarized GI and the ordinary one, and the gap has shown a clue to shrink with theincreasing measurement number.Here, it is noted that the behavior of the experimental results is qualitatively similar to the E rr o r F r e q u e n cy Error Value (/ Sampling Interval)
Detect Error Poisson Fit (a) (b) S N R ( / d B ) Measurement Number
No Binarization Optimal Threshold Bucket Mean Threshold First Threshold
Fig. 6. (a) Reconstructed image SNR of the ordinary, optimal binarized and the first-threshold binarized GI against measurement number. Reconstructions with bucket meanbinarization threshold are listed for comparison. (b) Detection error distribution of bucketsamples under 50,000 measurements. theoretical CRLB calculations, while we can also find some apparent discrepancies betweenthe theoretical and the experimental results. For instance, the mismatch between the optimal q and mean of bucket outputs. This stems from several possibilities, such as the mismatch ofnoise model, the quantization error, the non-constant of background fluctuations, polarizationeffects, and some unaccounted aberrations in our experiments. And we attribute this mainly tothe detect error in our experiment. For estimating the noise level, we pick the bucket outputsinduced by the same light fields, and record the error distribution in Fig. 6 (b), which can bemodeled as a Poisson noise with an intensity of 2.2 times the sampling intervals. Due to thespace constraints, we leave further discussions on this additional noise. . Conclusion & Discussion In this work we have proposed and demonstrated a general model for optimizing the performanceof binary sampling GI, enabling the informative-optimal image reconstructions subject to theexplicit GI system conditions. To be sure, the CRLB of binary sampling is always larger thanthe ordinary one. It is not surprising that binary sampling loses information, but our theoreticalmodel indicates that the binary sampling scheme have the potential to behave arbitrarily closeto the ordinary one with the optimal binarization threshold and large number of coincidentmeasurements, which is the surprise. The optimal binarization threshold, corresponding to theCRLB minimum, are coincident with the statistical mean of bucket samples. Furthermore, wemay ask the question, what would we do to optimize the performance if we can only observebinary samples? It should be noticed that the optimal binarization threshold also determinesan optimal distribution of the sequence of binary signals, which is an uniform distribution onthe "0-1" binary sequence, to extract the maximum information about the object. Thus, theoptimization problem of binary samples can be solved by adjusting the system design of GI orthe charteristics of detectors to meet the optimal distribution of binary sequence.For the experimental verifications based on pseudo-thermal light GI, we optimize the bina-rization threshold of bucket signals to maintain highest image quality assessed by SNR. Theperformance after optimization and the behavior of optimal binarization threshold both verifyour theoretical analysis and demonstrate the effectiveness of our optimization method. More-over, the similar behavior of the CRLB and image quality assessments, e.g., the image SNR,against the varying binarization threshold suggest that the CRLB is not only a mathematicallimit, but indeed a promising candidate for optimization criterion, which yields a measureableperformance benefit.For the future technical improvements, this optimized binary strategy is not only beneficialto the ordinary GI, since binary data can dramatically reduce sampling, storage, transfer, andcalculation cost, but also pave the way for fundamentally optimizing the system design of GI inmany challenging applications.
Fundings
National Natural Science Foundation of China (61631014, 61401036, 61471051); National Sci-ence Fund for Distinguished Young Scholars of China (61225003); the BUPT Excellent Ph.D.Students Foundation (CX2019224).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
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