1. Introduction

In 1995, a notable work in optical imaging initiated a new line of research and introduced an intriguing technique: two-photon ghost imaging [1]. Owing to the fact that the only optical source used in the earlier experiments consisted of entangled photon pairs, ghost imaging was considered, for several years, a remarkable effect of quantum non-locality. In 2002, a classical apparatus [2], based on the use of spatially-correlated needle-like beams, produced ghost images resembling those of entangled photon pairs. About two years after the publication of Ref. [2], the debate around the nature of ghost imaging (quantum vs classical) became more intense; this occurred when the needle-like beam pair was replaced by a correlated pair of speckle beams generated by a pseudothermal light source [3–5]. For completeness it is fair to recall that Abourrady et al [6] had previously clarified that “a quantum two-particle system in an entangled state exhibits effects that cannot be attained with any classically correlated system…” and that entangled-photon ghost imaging has assumed an important role in the conceptual comprehension of Quantum Mechanics. Today, the nature of ghost imaging (“quantum” vs “classical”) is still debated [7]; to simplify the description it is convenient to consider two distinct classes of experiments, i.e. “two-photon” quantum experiments (those relying on the quantum correlation of photon pairs) and “two-beam” classical experiments (those using needle-like or pseudothermal light beam). We also inform the reader that the present work (as all works classified under the heading of thermal light ghost imaging) deals only with ghost imaging produced by a specific classical source invented by Martienssen and Spiller [8] (variously named in the literature as thermal, pseudothermal, incoherent…); we refer to this technique as Pseudothermal Light Ghost Imaging (PLGI). Reference [9] simply describes the role of correlated fluctuations in the object and reference beams and also emphasizes that both classical and quantum entanglement correlations that meet this requirements will support “ghost” imaging. In this work we look at PLGI from a novel point of view (by considering only one beam impinging on the CCD detector) and demonstrate that the most distinguishing mathematical features of two-beam experiments [10] already emerge in single-beam experiments. More specifically, we show that by applying common signal processing to a single beam we obtain an equation that is identical to the fundamental imaging equation of conventional two-beam PLGI. Our analysis allows to shed new light on the conceptual roots of classical ghost imaging, which appears to be an ingenious product of signal processing devoid of any “ghost” content. To better clarify the role of the present contribution we first summarize the experimental setup and the mathematical procedure conventionally used in PLGI.

2. A summary of incoherent light ghost imaging obtained using a bucket detector behind the mask

An essential drawing of the setup commonly used in PLGI is depicted in Fig. 1. The light source consists of a laser beam hitting a slowly rotating ground glass disk (GGD); the waves scattered by the rough surface interfere randomly with one another and create, on any plane placed at some distance L from the source, a time-dependent speckle pattern whose average size depends both on L and on the diameter of the beam impinging on the disk.

 

Fig. 1. Schematic of the two-beam setup. GGD: ground glass disk; BS: 50% beam splitter; B1: reference beam; CCD: ccd camera; B2: object beam passing through the mask M; BD: bucket detector for measuring the total intensity crossing the mask. The two-beam correlation is done by first multiplying (⊗) the outputs and then cumulating (∑) the results

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The primary beam divides (BS) into a pair of identical speckle beams B1 and B2. B1 is sent directly to a CCD camera where it creates a time-varying speckle pattern that is sampled at regular intervals. The other beam goes through the transparent mask M (the object to be reconstructed) before falling onto the adjacent “bucket” detector (BD); the latter collects, in coincidence with the sampling performed by the CCD camera, the total energy of the light that goes across the mask (hence the name “bucket”). The image of the mask is subsequently retrieved (Fig. 1) by two-beam correlation [11]. To this end, the s^th speckle pattern recorded by the CCD camera is multiplied by the s^th output of the bucket detector and the result is cumulated with the previous (s-1) multiplications. Two-beam PLGI has been considered an intriguing effect because the mask image becomes visible by correlating two objects (the speckle pattern and the bucket detector output) none of which conveys information about the shape of the mask; it is just this fact that deserved to this technique the appellation of ghost imaging. As we will see in the next Section, an identical result occurs when each speckle pattern is correlated with the sum of the intensities belonging to a given subset (Fig. 2).

3. Mathematical analysis of the single beam experiment

We now derive a result for an experiment that is not aimed at reconstructing the image of a real mask but can be generalized to model real experiments in which the light source is classical. The apparatus is described in Fig. 2; unlike setups used in ghost imaging, the light directly propagates from the source to the detector (the CCD camera).

In the following, we work with functions defined on a discrete domain, in line with the fact that images are universally captured by CCD cameras endowed with a discrete array of pixels; time is also assumed to be discrete, because the speckle patterns are “instantaneously” recorded at regular intervals.

 

Fig. 2. Schematic of the single-beam setup. Only one beam (upper part of figure) impinges on the CCD camera, whose front side is shown enlarged in the lower part. F stands for the matrix whose elements are the speckle intensities recorded by the whole ccd array A. W is the numerical sum of the speckle intensities belonging to the subset M

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We now formally derive the main result of our contribution whose experimental verification we report in Sect. 5. A front view of the CCD camera entrance is shown in the lower part of Fig. 2; it consists of N_xN_y pixels distributed on a rectangular array A, where i=1,2..,N_x and j=1,2…,N_y are, respectively, the horizontal and the vertical pixel position indices. Let a time-dependent random function F _S (i,j) represent the intensity pattern recorded at pixel (i,j) and time t_s=sτ, where τ is the sampling period (s=1,2,…S is the time index). Let’s also select a subset M of A, whose elements are pixels belonging to a certain (possibly unconnected) area M⊂A; the characteristic function χM (i,j) of M is defined to be identically one on M and zero elsewhere. Suppose that a large number S of samples {F _S (i,j)} of the function F, (for s=1,2,…S), have been recorded and are available for processing. From the numerical point of view, each sample F _S (i,j) is a large N_y×N_x matrix whose elements are spatially-correlated random functions of time; they are also spatially and temporally stationary in the sense that their correlation functions Γ(·) depend only on coordinate differences:

where the angular brackets denote an ensemble average.

Let’s now define and compute a vector W(s), whose s^th component is the sum of the values of F _S (i,j) for all pixels (i,j) belonging to M ; using the characteristic funtion of M, W(s) can be calculated as follows:

After normalization, W(s) will play the role of statistical weight for the s^th speckle pattern FS (i,j).

We now perform the zero-delay crosscorrelation between the speckle patterns F and their statistical weights W. To this end, all values F _S (i,j) are multiplied by W(s) to obtain, at each “time” s, the weighted pattern Φ_S (i,j), i.e.:

The pixel-by-pixel sum of the weighted patterns Φ_S (i,j) is then executed to obtain the final result, R (i,j):

When the number S of samples becomes large, the summation within square brackets in Eq. (4) becomes the discrete spatial autocorrelation of the stationary speckle pattern sampled by the CCD camera:

We thus obtain the final result R (i,j) for the weighted sum:

where “*” denotes the convolution operation.

Equation (5) can be stated as follows:

When a long sequence of stationary speckle patterns F(A) is correlated (at zero delay) with the sums of the values assumed by F(A) in a certain subset M∈A, the result is equal to the convolution of the autocorrelation function of the speckle patterns with the characteristic function of the subset M.

An immediate consequence of Eq. (5) is that when C_F is δ-like:

the final result R (i,j) is simply χM (i,j), namely the shape of the subset M.

Let’s now discuss why Eq. (5), which according to the above derivation holds for a single-beam setup, can be logically extended to cover the case of two-beam PLGI. This can be easily done by turning the single-beam setup of Fig. 2 into the two-beam setup of Fig. 1 as follows:

… insert a beam splitter in the path of the single beam so as to create a twin beam that propagates along a side arm;

… place in the path of the twin beam a real mask whose transparency function is identical to the characteristic function χM of the subset M that was used in the single-beam experiment;

… place a real “bucket” detector directly behind the real mask and define its output to be the weight of the speckle pattern falling on the camera.

By this procedure we create an exact replica of the two-beam ghost imaging apparatus shown in Fig. 1; and it is immediate to see that identical results will be obtained by the setups of Fig. 1 and Fig. 2. We finally note that Eq. (5) is the fundamental equation of PLGI.

The working principle on which two-beam PLGI is based emerges clearly by comparing the basic mechanisms on which single-beam and two-beam experiments are based. In the single-beam case, a single speckle pattern illuminates both the whole camera and the subset M; in the two-beam case, a real mask is located in a separate arm but this requires the creation of a twin copy of the speckle pattern passing through the mask. The final results output by the two setups are identical because, in both cases, the same mathematical operations are performed on the same sequences of numbers.

4. Statistical results from the single beam experiment: variance and visibility of the weighted sum

In this Section we derive some results (considered of primary importance in ghost imaging) which can be readily generalized from one-beam to two-beam experiments.

When the average speckle size is smaller than the pixel dimension, we can work in the δ- like approximation (the intensities at neighbouring pixels are uncorrelated). Let’s consider the following simplified situation: The photodiode array is 1-dimensional and consists of P pixels. Let’s chose a subset M covering the leftmost T pixels, i.e. pixels corresponding to indices k={1…T}. On the s^th acquisition, W(s) is the sum of the intensities recorded in the subset M:

We further suppose that the intensity recorded at any pixel is distributed according to the negative-exponential function (of mean µ) typical of speckle patterns.

It is convenient to preliminarily determine the mean, the variance and the standard deviation, calculated over N independent trials, of the variables listed below:

Is (k) (k=1..... P) intensity recorded at the k^th pixel on the s^th trial W(s) has been defined above

The relevant results are summarized in Table I; the most important fact is that the average level of R for 1<k≤T [denoted by (R_(in) in Table I] is higher than for k>T [denoted by R_(out)]. Incidentally, it is this difference that determines the visibility of the subset M and lies at the root of any ghost-imaging reconstruction. Working with only one beam, the explanation of this crucial fact is extremely simple and intuitive: the sum of the intensities lying within a subset M is correlated only with the intensity recorded at pixels located inside the same subset; moreover, the correlation degree decreases as the extent of the subset M increases (if we throw N dice, the correlation between the output of one die and the sum of all dice decreases as N increases). Matlab simulations have accurately confirmed these theoretical considerations in a wide variety of different cases.

Table I. Summary of statistical variables

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The values listed in Table I are obtained on the hypothesis that the intensity at each pixel is distributed according to a negative exponential; the average and the variance, respectively, of two correlated variables (X,Y) are calculated using the formulas:

The most interesting results emerging from Table I are the following.

First, the weighted sum R at pixels inside the subset M is higher than outside by the quantity Nµ². This difference is the signal to be detected.

Another result stems from comparing signal with noise. As a practical rule for an acceptable perception of the subset “image”, we might impose that the signal be larger than three times the noise or: $N{\mu }^2>3{\mu }^2\sqrt{\mathrm{NT}\left(T+4\right)}$. Considering that usually T≫1, this inequality is equivalent, as an order of magnitude, to N>10 T². This result is interesting because it shows that the required ensemble size rapidly increases with the size of the target; the same expression allows to assess the minimum number of independent patterns that must be summed in order to obtain a distinct perception of the “image”. To see whether this request is reasonable, let’s try a subset M consisting of, say, 30 pixels; the order of magnitude of N must be then of the order of 104, an estimate that agrees both with experiments and simulations.

Finally it is important to note that, as long as we work in the δ-like speckle approximation, the 1-dimensional length T of the subset M simply becomes, in two dimensions, the area of the subset M; this prediction, too, has been successfully tested by supplementary experiments described in Sect. 5.2.

For larger speckles, the effective parameter becomes the ratio between the mask area and the average correlation area of speckles. This is the natural generalization of the δ-like condition considered above, when the effective area of each speckle coincided with one pixel and so the ratio of the mask area to the speckle effective size was equal to T.

5. Experimental results

5.1 Reconstruction of a two-hole mask

Our work is based on two experiments. In the first, we employ the conventional two-beam ghost-imaging apparatus (Fig. 1) to retrieve the shape of a mask located in the object plane. In the second, we use the single beam setup to show that the shape of a subset M can be retrieved by correlating each speckle pattern F with the total intensity W of the same speckle falling on M (lower part of Fig. 2).

 

Fig. 3. (a). Result of two-beam correlation, measured according to the procedure explained in Sect. 2. Note the pair of prominent hills that represent the mask transparency function described in Sect. 4 are smoothed by the convolution operator. b) Result of single beam correlation, determined according to the procedure explained in Sects. 3–4. Also in this case, the image of the (software) mask is smoothed by the convolution operator.

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In the two-beam experiment, the mask consisted of a metal plate pierced by two 1mm holes separated by a centre-to-centre distance of about 4mm. The speckle size was on the order of one third the dimension of the holes. The two-beam “ghost” reconstruction of the image of the holes (using 3000 trials) is shown in Fig. 3(a).

The corresponding single-beam result is shown in Fig. 3(b). Each speckle pattern, i.e. each realization of the random function F _S (i,j) to which the single-beam processing is applied, was recorded when the ground disk stopped in the position it reached after a pulse of fixed amplitude and random duration (between 0.5s and 10s) was supplied to the driving motor. Moreover, in order to grant the maximum mutual independence of the stored speckle patterns, the whole set consisted of a large number of groups; each of these was obtained by imparting a random lateral shift to the disk’s position.

The transparency function of the “software mask” employed to obtain Fig. 3(b) was designed to be equal to that of the real mask used to obtain Fig. 3(a).

5.2. Experimental dependence of visibility and variance on the area of the subset M

The availability in the PC memory of a large number of independent speckle patterns obtained under the same experimental conditions, i.e. all with the same average speckle dimension, allows to test the efficiency and more generally the properties of a large number of possible procedures. In particular we used them to find the dependence of the image visibility and of the signal variance on the subset size. To this end, the single-beam procedure was applied to circular subsets (holes) whose diameters were progressively increased and whose centers were randomly selected at the start, always using 3000 independent patterns.

Figure 4 reports the experimental visibility plotted vs the ratio: (subset M area)/(coherence area) ranging from about 0.5 up to about 22. The expected behaviour is: Visibility=1/ν_s where ν_s is the average number of speckles contained within the subset M.

 

Fig. 4. The plotted values of visibility are the differences [R_(in)-R_(out)] properly normalized, i.e. divided by [R_(in)+R_(out)]

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Finally, Fig. 5 shows how the experimental variances of R_(in) (upper graph) and R_(out) (lower graph) depend on the same areal ratio used in Fig. 4. It is natural for R_(out) to give a more satisfactory agreement with theory because it is calculated using an area much more extended than the subset M. In both graphs, the theoretical fit is made using the variance expressions reported in Table I.

 

Fig. 5. Values of the experimental variances of: (a) R_(in) and: (b) R_(out) plotted vs the ratio: (subset M area)/(coherence area). The continuous lines represent the theoretical functions (variances) given in Table I

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6. Conclusion

In conventional PLGI a real mask of unknown geometry is placed in a side path and the mask shape is recovered by correlating the CCD random patterns with the bucket detector output. We can perform a conceptually equivalent single-beam experiment by storing a large number of different subsets M in the PC memory; then we can instruct the PC to randomly select one of them without giving us any information about the choice performed. At this point, the PC acquires and stores a long sequence of speckle patterns from an optical system and, for each pattern, calculates the sum of the intensities falling in the chosen subset. The unknown geometry of the subset M emerges from the correlations of two sets (the sequence of the speckle patterns and the sequence of their weights) neither of which carries information about the shape of M; this procedure is mathematically identical to PLGI. The latter, therefore, appears as a result of signal processing which is noteworthy and interesting, but does not contain any “ghost”. While this is the conceptual meaning of our single-beam experiment, we would like to point out that its purpose is not solely to recover the shape of a hidden subset using a contrived procedure. Our apparatus, in fact, may be also practically useful. If PLGI has a future in practical applications, systematic tests of its efficiency will require that a large number of experiments be carried out with different speckle features and many different masks. A nice feature of our setup is that we first record the whole sequence of speckle patterns and eventually calculate the results. This procedure is not real-time but it offers an extraordinary flexibility for mathematical analysis: individual speckle patterns may be filtered, rotated, translated, space-limited or averaged in various ways, statistically analyzed and compared; as regards the statistical weights, they may be raised to some power or discriminated by a threshold and the effects of these changes measured. All these facilities will allow an easier treatment of the acquired data and a more complete test of the efficiency of the imaging procedure.