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We use compressive in–line holography to image air bubbles in water and investigate the effect of bubble concentration on reconstruction performance by simulation. Our forward model treats bubbles as finite spheres and uses Mie scattering to compute the scattered field in a physically rigorous manner. Although no simple analytical bounds on maximum concentration can be derived within the classical compressed sensing framework due to the complexity of the forward model, the receiver operating characteristic (ROC) curves in our simulation provide an empirical concentration bound for accurate bubble detection by compressive holography at different noise levels, resulting in a maximum tolerable concentration much higher than the traditional back-propagation method.
© 2015 Optical Society of America
1. Introduction
In–line digital holography (DH) has found wide applications in imaging particle flows, such as holographic particle imaging velocimetry (PIV) [1], two–phase flows visualization [2], flow cytometry [3, 4], and marine biological imaging [5]. In these applications, the object space typically consists of many point–like scatterers, such as particles, blood cells, and air bubbles, sparsely distributed in a volume. The benefit of using in–line DH is the ability to record the shapes and 3D positions of the scatterers in a single shot with simple optics.
An in–line hologram records the interference between the scattered field from the object and the reference wave. The task of hologram reconstruction is to recover the in–focus information, e.g. locations of each plane that intersect the center of each particle, and the size and shape of each particle. A traditional algorithm often consists of two steps. First, a focal stack is generated by applying the back–propagation method (BPM) [6], in which the hologram is convolved at each plane with a depth–dependent free–space propagation point spread function (PSF). Each reconstructed slice contains both in–focus features and diffraction fringes from out–of–focus objects. The goal in the second step is to isolate in–focus objects by image segmentation techniques. A common method is to apply an experimentally determined threshold to a certain focus metric, such as edge sharpness [5], amplitude [2, 7] and correlation coefficient [8]. The advantage of this method is that the algorithm can be implemented very efficiently [2]. In practice, however, focus metrics are sensitive to noise and cross–talk between scatterers, and thus the segmentation results often contain spurious objects. Furthermore, localization accuracy is limited by the Nyquist sampling rate induced by the finite pixel size of the detector [9, 10].
The hologram is related to the scattered field by a nonlinear equation. Direct inversion of this equation does not produce satisfactory results because the equation is under–determined, since we are attempting to reconstruct scattered field in 3D from a 2D measurement. The solution uniqueness problem can be alleviated by linearization of the equation and adding constraints on the solutions based on prior knowledge about the objects e.g. least energy (Tikhonov) regularization and its variants [11]. Using focus metrics on reconstructions at different depths, as described in the previous paragraph, is also a prior, and in fact is assuming that the particles are sparse.
Sparsity can be leveraged in a way more systematic than the focus metrics of earlier work– namely, by converting the hologram reconstruction problem to a regularized nonlinear optimization [12–16]. Recent studies on the inversion of under–determined linear systems, known as compressed sensing, show that it is possible to obtain a highly accurate solution to such a problem as long as the expected solution is known to be sparse in some pre–determined basis; the optimal solution can be found by solving a standard convex optimization problem [17–19]. It has been shown that in digital holography the solution to the compressive model is robust to noise and ghost terms [15, 16]. In addition, a recent study on applying compressive holography to object localization has shown orders of magnitude improvement on lateral localization accuracy as long as the solution is sparse in its derivative [20,21]. Improvements in axial localization compared to BPM have also been reported [22–24].
For imaging particle flows, an important question is how reconstruction quality is affected by particle concentration [25, 26]. In [27], the influence of the shadow density on particle field extraction is studied for the back-propagation based method. In [28], exact guarantees for accurate 3D object reconstruction from its 2D diffracted field by compressed sensing are formulated theoretically, and a loose lower bound is provided for the required sparsity level of the object. The theoretical result is based on a simplified point source model and phase shift holography configuration. The authors provide a bound on the maximum concentration, although they concede it is pessimistic and hence lower than the actual bound. Furthermore, for two-phase flow applications, the point source model is not applicable since particles are often much larger than the wavelength, and rapid movement of particles precludes the use of multiple exposure methods such as phase shift holography.
In this paper, we model the particles as spheres of finite size and use rigorous Mie scattering as the forward model to investigate the performance of a compressive holography method (CHM) for bubble imaging in water at various concentrations. Due to the complexity of the forward model, no simple analytical bounds on concentration can be derived using classical compressed sensing theory. However, we do establish an empirical bound on concentration for accurate reconstruction via numerical simulations, and we show that CHM enables more accurate reconstruction at bubble concentrations higher than BPM. While empirical, this bound does provide guidelines for practical application of CHM to imaging particle flow. To our knowledge, this is the first quantitative analysis of compressive hologram reconstruction to particle detection application based on a physically rigorous particle model.
2. Theory and method
The in–line hologram is a record of the interference between reference light Er and the scattered light E. A schematic diagram of the experimental geometry for in–line holography is shown in Fig. 1. Assuming that the wavefront of the illuminating plane wave with unit amplitude is not significantly disturbed by object, the intensity g recorded on the camera located at the z = 0 plane is
Here, the object of interest consists of micron–size dielectric air bubbles sparsely distributed in water. If we further assume that the effect of the multiple scattering between bubbles is small such that it can be ignored, the total scattered field at the detector plane is where f is the object function, and h is the depth–dependent PSF of free–space propagation. Under the paraxial approximation, Together with the expression for the plane reference wave Er(x′,y′, z′) = exp(i2πz′/λ), the total scattered field is Equation (4) is the Born approximation: the integration inside the square brackets with respect to du and dv denotes field contribution from the object at a fixed depth slice z′; the integration with respect to dz′ calculates the superposition of fields from all depth slices.Next, let us consider the discretization model for Eq. (4). Assume that the object is discretized into N × N × Nz voxels (the number of samples in both x and y have assumed to be the same without loss of generality) with lateral spacing Δ, and axial spacing Δz. The element in the object matrix f is determined by fm1m2l = f (m1Δ, m2Δ, lΔz). We define a 2D matrix f(l) denoting the lth 2D axial slice from f, whose element is . The term enclosed by the square bracket in Eq. (4) can be written as
where E(l) is a vector with length of N2 denoting the field contributed from the lth axial object slice, f(l) is the rastered form of f(l) defined as with length of N2, and is the free–space propagation operator for the lth object slice with dimensions of N2 × N2, which takes 2D Fourier transform ℱ2D of f (l), multiplies a depth dependent quadratic phase function , and then inverse Fourier transform back to the spatial domain. The spatial frequency sampling pitch along the lateral dimension is Δu = 1/(NΔ), p and q denote the indices. The total scattered field is the linear superposition of E(l) from all axial slices: where E is the rastered form of E with length of N2, defined as E(N−1)m1+m2 = Em1m2, and H denotes the scattering operator for all object slices with dimensions of N2 × (NzN2). The discretization model for the hologram recording equation in Eq. (1) is where g is the rastered form of the matrix g for hologram with elements determined according to g(N−1)n1+n2 = gn1n2.Similarly to most prior works in digital holography and compressive holography, we linearize Eq. (8) by dropping the halo term e = |Hf|2. In section 3, however, we will treat this term as background noise so as to account for its effect on our reconstruction results. After linearization, consider an unknown vector x with length of 2NzN2consisting of the real and imaginary parts of f, Eq. (8) is rewritten as
where y = g − 1 with length of N2 and n denotes additive noise. Equation (9) is under– determined because the number of unknowns in x is 2Nz times as the number of measurements in y. As a result, this is an under-determined problem and the solution is not unique unless prior information is utilized to constrain the solution space. Compressed sensing tries to find a unique solution by enforcing sparsity constraints on the signal represented in a proper set of basis function.Successful implementation of compressed sensing is based on two requirements: incoherence of measurements and sparsity [29]. The first requirement does not mean “incoherence” in the typical sense we assign in statistical optics; rather, it means that the information of the unknown vector must be evenly spread over the set of basis vectors that describe it at the detector. Here, we utilize Fresnel diffraction to generate incoherence at the sensing plane. The mixing produced by the Fresnel propagator is not provably optimal, but it is extremely easy to attain by simple free–space propagation [30]. The second requirement is fulfilled by finding a set of basis function such that the original unknown vector can be expressed by a few non–zero coefficients under this basis. Although optimal sparsity basis may further improve the reconstruction [31], here we choose total variation (TV) as our sparsity basis as used in [15]. TV regularization prefers solutions with sharp boundaries since it minimizes the derivatives of images; this is consistent with our expectation of the object being small particles with sharp edges in an otherwise empty space.
The sparse solution under the TV basis is obtained by solving the following minimization,
where α is the regularization parameter and ‖x‖TV is defined as a function of the gradient of the object function according to [32]3. Simulation and analysis
3.1. Single bubble analysis
We first consider the case where only a single air bubble is immersed in water. The bubble is modeled as a perfect sphere with 10μm diameter located at 2.5 mm away from the camera plane consisting of 256 × 256 pixels of 2 μm pixel pitch. Under plane wave illumination of 632 nm wavelength, the scattered field at the camera plane is calculated based on the Lorentz– Mie theory [33]. The hologram is then calculated as the interference between the scattered field and the plane wave. Fig. 2 shows the reconstructed intensity distribution by BPM and CHM.
Fig. 2 The intensity at the focal plane reconstructed by BPM (a) and CHM (b), and the longitudinal cross section from BPM (c) and CHM (d), and the corresponding line profiles (e,f,g,h) across the center of the bubble.
First, from the reconstructed focal plane intensity distribution and line profile across the bubble centre by BPM and CHM, we can find the BPM reconstruction xy-BPM contains small fluctuation around the bubble due to the interference effects from the ghost terms, as shown in Fig. 2(a) and Fig. 2(e). However, the fluctuation is completely eliminated in the CHM result xy-CHM as shown in Fig. 2(b) and 2(f).
Next, we compare the reconstructed intensity in the axial (z) direction. In in-line DH, localization accuracy in the axial direction is much worse than in the lateral direction because the depth of focus (DoF) is inversely proportional to the square of the numerical aperture (NA) of the imaging system. In this simulation, the bubble diameter determines the effective aperture size [15] and the corresponding DoF is 632 μm, which agrees well with the full width at half maximum (FWHM) measured in the BPM reconstructed intensity cross section though the center of the bubble along z in Fig. 2(g). The CHM improves the localization accuracy by 3 times in this case, as shown by the narrower FWHM in Fig. 2(h).
3.2. Multiple bubbles analysis
3.2.1. Signal-to-background-ratio analysis
In the single bubble detection case, it is relatively easy to localize the bubble using either BPM or CHM based on the symmetry of the reconstructed amplitude. However, the localization is more difficult in the multi–bubble case, where the cross–talk between bubbles greatly affects the reconstruction quality. First, let us briefly analyze the difference between the BPM and CHM in terms of the sources of signal and background. In BPM, at any reconstructed focal plane, the total field consists of all the propagated fields from all the bubbles in the entire volume and the propagated twin-image and halo terms. If one tries to isolate a particular particle, only the scattered field from this particle is the useful signal, whereas all the rest terms contribute to the background. As a result, the signal-to-background ratio (SBR) (ignoring other source of random noise) for the BPM model can be defined as
and SBRBPM degrades quickly as the concentration increases. However, this is not the case in the CHM model. In CHM, the entire 3D distribution of particles are taken as an unknown to solve for in the linear model in Eq. (9). In other words, the optimization in Eq. (10) tries to find all the particle locations that best fit the hologram simultaneously, rather than isolating a single particle locally. As a result, all the scattered fields from the entire volume contribute to the signal term. Furthermore, the twin-image terms also contribute to the signal based on the construction in Eq. (9). The background in the CHM model entirely comes from the halo term and the SBR can be defined as The SBRCHM decreases almost linearly as the concentration increases (both in log–scale), as shown in Fig. 3. Here, the concentration is measured by a dimensionless variable termed the two–dimensional geometrical occlusion factor Rg, defined as where Np is the number of bubbles, r̄ is the mean radius, and the hologram contains N pixels on each side with pixel size Δ. Rg can be interpreted as the maximum fractional area covered by all the bubbles projected on the hologram plane.Fig. 3 The SBR in the CHM model decreases as the concentration Rg increases. Insets: sample holograms at the corresponding concentration.
3.2.2. Qualitative comparison
To compare the performance of BPM and CHM in the multi–bubble system, holograms containing various numbers and sizes of bubbles were simulated and reconstructed. In this simulation, the diameters of bubbles were chosen randomly from a uniform distribution ranging from 10 μm to 20 μm. They were located at random positions within a volume of 512 × 512 × 6000 μm3. The center of the volume was 6 mm away from the hologram plane. Assuming that the effect of multiple scattering between bubbles is negligible, the total scattered field is the linear superposition of the scattered fields from each bubbles. The hologram was then calculated as the interference between the total scattered field and the plane reference wave.
Holograms containing 8, 16, 32, 64, 128, 256, 512, 1024 bubbles were considered. For each data set containing the same number of bubbles Np, 20 holograms with different size and position distributions of bubbles were simulated. In the reconstruction, the estimated object function is expressed as a 256 × 256 × 25 matrix corresponding to discretization of the actual volume with lateral spacing Δ = 2 μm and axial spacing Δz = 250 μm. Since the diameter of a bubble is much smaller than Δz, each bubble is represented as a single disk located at some depth slice.
As a qualitative comparison, reconstructions and localizations from sample holograms with Np = 16, Np = 128, and Np = 512 are visualised in Fig. 4. The corresponding ground truth of the bubble positions and sizes are shown in the left column. In each image, the bubbles are represented as circles with different colors which represent the axial positions. The BPM reconstruction shown in the middle column is obtained by minimum intensity projection along the axial direction after applying an optimal threshold to the back-propagated intensity focal stack matrix [2]. As can be seen, even in the low concentration case (Np = 16), the BPM result contains some artifacts due to the cross–talk between bubbles which are close to each other. As the concentration increases (Np = 128), it is practically impossible to find a single optimal threshold for BPM to isolate any bubble as the SBR is too low. The CHM reconstruction shown in the right column are obtained by first solving the optimization in Eq. (10) followed by a thresholding and minimum intensity projection along the axial direction. This method successfully localizes most of the in-focus bubbles and removes most of the unwanted artifacts in the low (Np = 16) concentration and moderate (Np = 128) concentration cases. At high concentration (Np = 512), neither BPM and CHM can reconstruct the bubbles accurately, since the cross– talk dominates the signal and object is no longer sparse. In next section, we will quantitatively analysis the concentration bound which allows accurate bubble reconstruction.
Fig. 4 Left: Ground truth of bubble distribution; middle: BPM reconstruction; right: CHM reconstruction. First row: Np = 16; second row: Np = 128; third row: Np = 512. Bubbles are represented as circles with colors representing their corresponding axial positions.
3.2.3. Receiver operating characteristic curve
To localize and identify bubbles in the volume, a thresholding step must be used to process the reconstructed intensity distribution from BPM or CHM and classify the pixels into air bubbles or water. From this point of view, the BPM and CHM can be both regarded as binary classifiers. Receiver operating characteristic (ROC) curve can be used to measure the performance of a binary classifier as its threshold is varied. The curve is created by plotting the true positive rate (TPR) against the false positive rate (FPR) at various threshold settings. In our application, the detection is true positive when a pixel classified as an object by the algorithm is occupied by the air bubble, and the detection is false positive when a pixel classified as a bubble by the algorithm is actually water.
Fig. 5 demonstrates the ROC curve for BPM and CHM at different bubble concentrations. At low concentration (Np = 8, 16), both methods show very good detection performance which means it has a high true positive rate and low false positive rate since there is not too much cross-talk between bubbles. At moderate concentration (Np = 32, 64, 128), CHM outperforms the BPM significantly. As can be seen from Fig. 5(c–e), while BPM degrades quickly due to the cross-talk between bubbles CHM still shows good detection ability. As expected, at high concentration (Np = 256, 512, 1024), both methods fail to detect the bubbles accurately, since the object no longer meets the sparsity criterion for CHM. As can be seen in Fig. 5-(h), when the concentration is too high, the ROC curve approaches the no-discrimination line which is equivalent to a random classifier.
Fig. 5 ROC curve of BPM (red curves)and CHM (blue curves) reconstruction of holograms without noise with number of bubbles Np = 8, 16, 32, 64, 128, 256, 512, 1024 as shown in (a)–(h).
To further investigate the robustness of CHM to noise, holograms with added Gaussian noise of different variance were reconstructed by BPM and CHM. The area under the ROC curve measures the accuracy of discrimination that is the ability of a classifier to correctly classify those pixels with and without the bubbles. The larger the area, the higher discrimination accuracy a classifier has. Fig. 6 illustrates the accuracy of discrimination for BPM and CHM at different concentration and noise levels with SNR = 0, 10 and infinite. As can be seen, at the same noise level, the accuracy of discrimination of both CHM and BPM degrades with the increase of bubble concentration but the CHM can tolerate higher bubble concentrations. We can also find that with the same bubble concentration, the CHM is more robust to the Gaussian noise. From simulation results, we can conclude that with noise level of SNR > 10, the maximum number of bubbles is about 128 corresponding to Rg of 10% which can be regarded as an empirical concentration bound for CHM to give good detection ability with accuracy of discrimination higher than 0.95.
Fig. 6 Accuracy of discrimination of BPM and CHM reconstruction of holograms with noise level of SNR = 0, 10, and infinity at various concentration.
4. Discussion and conclusion
Our simulation results show that in the particle flow application, compressive holography can recover bubble locations and sizes more accurately as compared to the back-propagation method, even at modestly increased concentration, and the maximum projected concentration allowed for accurate reconstruction is around 10%. Compressive holography is also found to be more robust to the additive Gaussian noise. Moreover, since our simulation is based on a physically rigorous forward model, this empirical bound should provide guidelines for practical application of compressive holography to imaging particle flows. Because our analysis is based on ROC curve, our results are applicable especially to the problem of particle detection and localization that often is the most interesting application for fluid mechanics studies. An improved reconstruction algorithm using the sizes and positions of particles as the optimization variables instead of intensity of voxels as in [36], combined with a rigorous forward model and experimental validation of our simulation are promising topics for future research.
Acknowledgments
Financial support was provided by the Singapore National Research Foundation (NRF) through the Singapore-MIT Alliance for Research and Technology (SMART) Centre for Environmental Sensing and Modeling (CENSAM), and the Chevron Energy Technology Company.
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