## Abstract

The concentric fringe patterns created by features in holograms may be associated with a complex-valued orientational order field. Convolution with an orientational alignment operator then identifies centers of symmetry that correspond to the two-dimensional positions of the features. Feature identification through orientational alignment is reminiscent of voting algorithms such as Hough transforms, but may be implemented with fast convolution methods, and so can be orders of magnitude faster.

© 2014 Optical Society of America

Holographic microscopy records information about the spatial distribution of illuminated objects through their influence on the phase and intensity distribution of the light they scatter. This information can be retrieved from a hologram, at least approximately, by reconstructing the three-dimensional light field responsible for the recorded intensity distribution [1, 2]. Alternatively, features of interest in a hologram can be interpreted with predictions of the theory of light scattering to obtain exceedingly precise measurements of a scattering object’s three-dimensional position, size and refractive index [3]. The availability of so much high-quality information about the properties and motions of individual colloidal particles has proved a boon for applications as varied as product quality assessment [4], microrheology [5, 6], porosimetry [7], microrefractometry [8], and flow velocimetry [9, 10], as well as for molecular binding assays [9], and as a research tool for statistical physics [11–13] and materials science [14].

Fitting measured holograms to theoretical predictions requires initial estimates for the individual scatterers’ positions. This can pose challenges for conventional image analysis algorithms because the hologram of a small object consists of alternating bright and dark fringes covering a substantial area in the field of view [9]. Here, we introduce a fast, robust and accurate feature-identification algorithm that not only meets the needs of holographic particle tracking, but also should be useful in other image analysis applications.

Figure 1(a) shows a typical hologram of a colloidal polystyrene sphere in water. This hologram was recorded with an in-line holographic video microscope [1, 2] using a collimated linearly polarized laser for illumination (Coherent Cube, vacuum wavelength *λ* = 447 nm). Light scattered by the sphere interferes with the rest of the beam in the focal plane of a microscope objective (Nikon Plan Apo *λ*, 100× oil immersion, numerical aperture 1.45). The objective, in combination with a tube lens, relays the interference pattern to a video camera (NEC TI-324A II) with an effective magnification of 135 nm/pixel. The intensity distribution recorded by the video camera is normalized by a background image [3, 9] to suppress spurious interference fringes. Figure 1(a) shows a 480 × 480 pixel region of the normalized intensity, *b*(**r**).

The sphere’s hologram features bright and dark circular fringes all centered on a point in the focal plane that coincides with the sphere’s center. This point could be identified by performing a circular Hough transform, which additionally would identify the radii of all the rings [15]. Hough transforms, however, have a computational complexity of *𝒪*{*N*^{4}} in the number *N* of pixels on the side of an *N* × *N* image [15]. Variants of Hough transforms that identify centers but not radii can achieve a computational complexity of *𝒪*{*N*^{3} log*N*} [16].

More efficient searches for centers of rotational symmetry take advantage of the observation that gradients in the intensity of images such as Fig. 1(a) either point toward or away from the centers. Figure 1(b) shows the magnitude, |∇*b*(**r**)|, of the image’s gradient. Each pixel in the gradient image, ∇*b*(**r**), is associated with a direction,

*x̂*axis. Figure 1(c) shows

*ϕ*(

**r**) for the image in Fig. 1(a). Each pixel therefore offers information that the center of a feature might lie somewhere along direction

*ϕ*(

**r**) relative to its position

**r**. Voting algorithms [9] make use of this information by allowing each pixel to cast votes for pixels along its preferred direction, the votes of all pixels being tallied in an accumulator array. Hough transforms operate on a similar principle, but also incorporate distance information. Pixels in the transformed image that accumulate the most votes then are candidates for center positions, and may be located with sub-pixel accuracy using standard algorithms [17]. Alternatively, the intersections between pixels’ votes can be obtained as solutions of a set of simultaneous equations [18]. Voting algorithms typically identify the centers of features such as the example in Fig. 1(a) to within 1/10 pixel. Efficient implementations [9, 18] have a computational complexity of

*𝒪*{

*N*

^{3}}. Achieving this efficiency involves first identifying pixels with the strongest gradients, typically by imposing a threshold on |

*b*(

**r**)|.

Here, we introduce an alternative to discrete voting algorithms that is based on a continuous transform of the local orientation field. This approach eliminates the need for threshold selection and further reduces the computational burden of localizing circular features in an image. The spatially varying orientation of gradients in *b*(**r**) may be described with the two-fold orientational order parameter [19, 20]

*b*(

**r**)|

^{2}emphasizes contributions from regions with stronger gradients.

To identify symmetry-ordained coincidences in the orientation field, we convolve *ψ*(**r**) with the two-fold symmetric transformation kernel,

*K*(

**r**) complements the phase of

*ψ*(

**r**), as can be seen in the inset to Fig. 1(c). The integrand of Eq. (4) therefore is real-valued and non-negative along the line

**r′**−

**r**that is oriented along

*θ*=

*ϕ*(

**r′**), and is complex-valued along other directions. Real-valued contributions directed along gradients of

*b*(

**r**) accumulate at points

**r**in Ψ(

**r**) that are centers of symmetry of the gradient field, as illustrated schematically in the inset to Fig. 1(d). Complex-valued contributions, by contrast, tend to cancel out. Centers of symmetry in

*b*(

**r**) therefore are transformed into centers of brightness in

*B*(

**r**) = |Ψ(

**r**)|

^{2}, as can be seen in Fig. 1(d). The centroid of the peak then can be identified and located [17].

Circular features at larger radii from centers of symmetry subtend more pixels in *b*(**r**) and thus would tend to have more influence over the position of centers of brightness in *B*(**r**). The factor of 1/*r* in Eq. (3) ensures that all of the fringes in a sphere’s hologram contribute with equal weighting to the estimate for its centroid.

The orientation alignment transform defined by Eqs. (2), (3) and (4) is related to the Fourier-Mellin transform, which is used to detect geometrically invariant features in images [21, 22]. It can be computed efficiently using the Fourier convolution theorem,

where*ψ̃*(

**k**) is the Fourier transform of

*ψ*(

**r**), and where is the Fourier transform of

*K*(

**r**). The orientation alignment transform therefore can be calculated by performing a fast Fourier transform (FFT) on

*ψ*(

**r**), multiplying by a precomputed kernel,

*K̃*(

**k**), and then performing an inverse FFT. Computing the gradient image by convolution with a Savitzky-Golay filter [23] reduces sensitivity to noise in

*b*(

**r**) and can be performed in

*𝒪*{

*N*

^{2}} operations. The transform’s overall computational complexity is set by the

*𝒪*{

*N*

^{2}log

*N*} cost of the forward and inverse FFT, and so is more efficient than voting algorithms. Rather than requiring sequential analysis of above-threshold pixels, moreover, the orientation alignment transform lends itself to implementation on parallel processors. Our implementation in the IDL programming language achieves real-time performance (30 frames/s) on a 2 Gflop/s processor for holograms such as the example in Fig. 2.

Figure 2 illustrates the orientation alignment transform’s performance for identifying and locating multiple particles in a single image simultaneously. This hologram records twelve 3 μm-diameter colloidal silica spheres that were arranged in four different planes using holographic optical tweezers [24]. Despite interference between the spheres’ scattering patterns and uncorrected motion artifacts in the hologram, the spheres’ contributions to *b*(**r**) are transformed into peaks in *B*(**r**) whose locations are identified by crosses superimposed on the original hologram.

The widths and heights of the transformed peaks depend on the particles’ axial positions, as can be seen in Fig. 2. This dependence can be calibrated on a particle-by-particle basis to facilitate real-time three-dimensional tracking with minimal additional computational burden. Two-dimensional tracking requires no separate calibration.

Results such as those in Fig. 2 confirm reliable detection of micrometer-scale spheres down to separations of two or three wavelengths. Beyond this, superposition of overlapping patterns can displace centers of symmetry and introduce spurious features. The symmetry considerations underlying the orientation alignment transform are most useful therefore for dilute samples.

Applying the same analysis to each snapshot in a holographic video sequence yields the in-plane trajectory for each sphere in the field of view. Figure 3(a) shows the trajectory of the sphere from Fig. 1 obtained in this way from 16,500 consecutive video frames. Each frame, moreover, yields two measurements of the particle’s position because the even and odd scan lines are recorded separately. Given the recording rate of 29.97 frames/s the time interval between interleaved video fields is Δ*t* = 16.68 ms. The camera’s exposure time, 0.1 ms, is fast enough to avoid artifacts due to the particle’s motion [10, 25, 26]. The 33,000 position measurements plotted in Fig. 3(a) record the particle’s Brownian motion over more than 9 min.

Assuming that the sphere diffuses freely without significant hydrodynamic coupling to surrounding surfaces, the mean-squared displacement,

should satisfy the Einstein-Smoluchowski equation where*r*(

_{j}*t*) is the sphere’s position along one of the Cartesian coordinates with

*r*

_{0}(

*t*) =

*x*(

*t*) and

*r*

_{1}(

*t*) =

*y*(

*t*), where

*D*is the diffusion coefficient along that direction, and where

_{j}*ε*is the error in the associated position measurement. Analyzing trajectories with Eq. (8) therefore provides a method to measure tracking errors [17, 25, 26].

_{j}The data in Fig. 3(b) show the mean-squared displacements along *x̂* and *ŷ* computed from the trajectories in Fig. 3(a) using Eq. (7). The error bars in Fig. 3(b) reflect statistical uncertainties. Although results along the two directions agree to within these uncertainties, least-squares fits to the Einstein-Smoluchowski prediction in Eq. (8) yield slightly different values for the particle’s diffusion coefficient: *D _{x}* = 0.292 ± 0.002 μm

^{2}/s and

*D*= 0.281 ± 0.002 μm

_{y}^{2}/s. This discrepancy may be attributed to blurring along the

*ŷ*direction that arises when the even and odd scan lines are extracted from each interlaced video frame. The resulting loss of spatial resolution along

*ŷ*tends to suppress the apparent diffusivity along that direction [25,26]. This artifact may be avoided by using a progressive scan camera. The larger of the measured diffusion coefficients is consistent with the Stokes-Einstein prediction

*D*=

*k*/(6

_{B}T*πηa*) = 0.296 ± 0.002 μm

_{p}^{2}/s for a sphere of radius

*a*= 0.805 ± 0.001 μm [27] diffusing through water with viscosity

_{p}*η*= 0.912 ± 0.005 mPa s at absolute temperature

*T*= 297.1 ± 0.2 K.

Fits to Eq. (8) also yield estimates for errors in the particle’s position of *ε _{x}* = 8 nm and

*ε*= 9 nm, or roughly 0.06 pixel in each direction. This performance is comparable to the precision obtained with voting algorithms [9, 18]. Because of its speed advantage, the orientation alignment transform should be immediately useful for in-plane particle tracking applications. Its results also can be used to bootstrap more detailed analyses [9] for applications that require greater precision or simultaneous tracking and characterization.

_{y}The orientation alignment transform performs well for identifying features composed of large numbers of closely spaced concentric fringes. It does not fare so well with simple disk-like features whose few alignment coincidences occur at comparatively large ranges. Such images are better analyzed with Hough transforms, voting algorithms, or related morphological methods. The orientation alignment transform, by contrast, is better suited to holographic images whose gradient-rich structure is computationally burdensome for conventional methods.

## Acknowledgments

An open-source implementation of the orientation alignment transform is available online at http://physics.nyu.edu/grierlab/software/. This work was supported primarily by a grant from Procter & Gamble and in part by the MRSEC program of the National Science Foundation through Grant Number DMR-0820341.

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