## Abstract

Conventional and digital holographies are proving to be increasingly important for studies of marine zooplankton and other underwater biological applications. This paper reports on the use of a subsea digital holographic camera (eHoloCam) for the analysis and identification of marine organisms and other subsea particles. Unlike recording on a photographic film, a digital hologram (e-hologram) is recorded on an electronic sensor and reconstructed numerically in a computer by simulating the propagation of the optical field in space. By comparison with other imaging techniques, an e-hologram has several advantages such as three-dimensional spatial reconstruction, non-intrusive and non-destructive interrogation of the recording sampling volume and the ability to record holographic videos. The basis of much work in optics lies in Maxwell's electromagnetic theory and holography is no exception: we report here on two of the numerical reconstruction algorithms we have used to reconstruct holograms obtained using eHoloCam and how their starting point lies in Maxwell's equations. Derivation of the angular spectrum algorithm for plane waves is provided as an exact method for the in-line numerical reconstruction of digital holograms. The Fresnel numerical reconstruction algorithm is derived from the angular spectrum method. In-line holograms are numerically processed before and after reconstruction to remove periodic noise from captured images and to increase image contrast. The ability of the Fresnel integration reconstruction algorithm to extend the reconstructed volume beyond the recording sensor dimensions is also shown with a 50% extension of the reconstruction area. Finally, we present some images obtained from recent deployments of eHoloCam in the North Sea and Faeroes Channel.

## 1. Introduction

Maxwell's contribution to optical science is well known and documented. His work on colour photography (e.g. Evans 1961) paved the way for much of today's photographic methodology. His electromagnetic equations are of course the cornerstone of all our work in wave propagation, diffraction, interferometry and suchlike, and led eventually to the development of holography. Many authors (e.g. Bleaney & Bleaney 1976) show how the wave equation has its origin in Maxwell's equations and how this leads to the concepts of image formation in holography (Hariharan 1996).

The roots of subsea optics are more difficult to trace. However, one of the first recorded scientific uses of underwater photography is that of Boutan in the late nineteenth century (described by Vine 1975). The use of subsea optics accelerated rapidly with the development of sophisticated stills and video cameras and other optical devices, which were used extensively to study the distribution and interrelationships of micro- and macroscopic organisms such as phyto- and zooplankton. A comprehensive treatment of optical methods in subsea biological studies is given by Solan *et al*. (2003). However, it is with the development and exploitation of digital holography that we are concerned in this paper.

The ability to record, non-intrusively and non-destructively, high-resolution holograms of micro-objects in three dimensions in their natural environment gave marine scientists the opportunity to study the subsea environment in a way never possible before. Holography was seen to be particularly useful for the analysis and precision measurement of the species and spatial distribution of marine plankton and particles within the subsea water column (Knox 1966). This first laboratory use of underwater holography was quickly followed by the development of a series of subsea holographic cameras (e.g. Heflinger *et al*. 1978). These cameras employed the in-line holographic geometry owing to its simplicity of implementation and ability to record particles of a few micrometres in dimension. More recently, groups led by Katz *et al*. (1999), Craig *et al*. (2000) and Watson *et al*. (2001) have further developed and refined subsea ‘holocameras’. Katz's initial camera was based on a ruby laser and designed to drift with the water current. Watson's camera (HoloMar system) was built around a frequency-doubled Nd-YAG laser and designed to record simultaneous in-line and off-axis holograms. Although yielding poorer resolution than in-line holograms, off-axis holography can record larger organisms and at higher population densities. Analysis is generally performed manually but recent work (Malkiel *et al*. 2003) has led to automation of the task of data extraction.

All of these cameras suffered from being bulky, heavy and difficult to deploy at sea or on observation platforms such as remotely operated vehicles (ROVs) and autonomously operated vehicles (AUVs) and required wet-chemical processing of the holographic emulsion. Furthermore, and crucially, in recent years many photographic emulsion manufacturers have withdrawn their holographic materials from the market place. Owing to the above difficulties, the use of digital holography for subsea holography, following its introduction by Schnars & Jüptner (1994) for interferometric applications, has grown rapidly. The first reported use of digital holography for plankton studies was by Owen & Zozulya (2000). Since then, a number of workers such as Xu *et al*. (2001), Malkiel *et al*. (2003), Sun *et al*. (2002, 2005) and Jericho *et al*. (2006) have used the technique for a wide range of biological applications.

In the remainder of this paper, we describe the development of a pulsed underwater digital holographic camera (known as eHoloCam) and the algorithms employed in numerical reconstruction of subsea holograms. We also present results from some of the subsea deployments of this camera.

## 2. Electronic holographic camera (eHoloCam)

In digital holography, a hologram is recorded electronically on an imaging sensor such as a charge coupled device (CCD) or complementary metal oxide semiconductor (CMOS) array. Hologram reconstruction is then simulated by a computer using a numerical reconstruction algorithm, such as the angular spectrum method (e.g. Champeney 1973), or various others usually based on the Fresnel–Kirchhoff integral (Cuche *et al*. 1999; Poon & Kim 2000; Dong *et al*. 2004). Holocameras based on this approach have usually employed the in-line mode of recording owing to the lower resolution capabilities of electronic sensors compared with photographic films. A distinct advantage of electronic recording, apart from speed, convenience and freedom from wet-chemical processing, is the ability to record holographic videos (holovideo) which not only allow a three-dimensional interrogation but introduce the fourth dimension, i.e. time.

Since reconstruction is carried out numerically, without the need for a dedicated optical reconstruction facility, the reconstruction algorithms can readily include specialist techniques such as dark field, phase reconstruction, and pre- and post-processing. As with photographic holography, no lenses are necessary in recording, and stability and accuracy of the sensor geometry provide in-built calibration and accuracy of imaging. Digital holovideos either record the movement of organisms within a particular scene and track the path of individuals from the reconstructed holograms, or capture a large number of frames of nearby volumes, thus providing an enlarged sampling volume, with improved statistical description of marine populations.

Our new digital holographic camera (eHoloCam) is designed for *in situ* studies of marine plankton and other marine or freshwater organisms (Sun *et al*. 2007). Figure 1 shows the basic layout of the system. The laser, single board computer, storage hard drives and beam forming optics are contained in the primary housing, while the CMOS camera is located in the secondary housing. Both housings are designed to an operational pressure of 300 bar (3 km depth) and have been pressure-tested and certified to 180 bar (1.8 km). The primary housing is 724 mm length by 330 mm diameter and the secondary housing is 170 mm length by 100 mm diameter. Sapphire windows (*λ*/4 flatness, 75 mm diameter, 18 mm thickness, uncoated) are located in both the housings to allow passage of light from laser through the water to the sensor. The light path in water is 453 mm. Three connecting rods between the front face of the primary housing and the back plate of the secondary housing are used to maintain the system on a rigid baseline. The eHoloCam uses a pulsed frequency-doubled Nd-YAG laser (532 nm, 1 mJ per pulse, 4 ns pulse duration, maximum pulse repetition rate of 25 Hz) to freeze any motion in fast-moving organisms, and a high-resolution CMOS sensor (IBIS4-6600, 10 bit) which is integrated into a PixeLINK camera (PL-A781). The maximum pixel count of the sensor is 2208×3000 (sensor area of 10.50×7.73 mm^{2}) with a square pixel pitch of 3.5×3.5 μm^{2}. The system is capable of recording volumes of 36.5 cm^{3} of the water column (10.5×7.7 mm^{2} sensor area by 450 mm in-water path length) in a single video frame at frame rates up to 25 Hz. The combination of resolution, image size and video frame rate and laser repetition rate chosen depends on the operational needs of the system.

We used two negative lenses to expand the laser beam to approximately 90 mm diameter and then collimated to achieve a uniform (0.4*λ* wavefront flatness) beam of 40 mm diameter. A narrow bandpass filter for 532 nm wavelength is mounted in front of the CMOS sensor to avoid ambient light. Fiducial wires (50 μm diameter) were located at the rear surface of the collimator in the primary housing and in front of the sensor in the secondary housing, to give transverse and longitudinal references for reconstruction calibration.

The eHoloCam system was designed to operate on a towed sampler, Auto-Recording Instrumented Environmental Sampler (ARIES—developed by Fisheries Research Services FRS Marine Laboratory, Aberdeen), from the research vessel *Scotia*. Water flows through the ‘jaws’ of eHoloCam laterally and perpendicular to the beam path when it is towed at various water depths at up to 4 knots (approx. 2 m s^{−1}) speed. eHoloCam is self-contained in power and in data storage and contains embedded control software since no power or communication exists between the *Scotia* and ARIES. On-deck data transfer and system control are through a gigabit ethernet port which is activated prior to a dive to allow initialization of the system, and after a dive for transfer of data via a ‘topside’ PC. Custom in-house system control software, written in Visual C++, controls the laser firing, camera–laser synchronization and data acquisition and storage from the embedded single-board computer, which has two serial advance technology attachment drives permitting local storage of large amounts of holovideo data. The software is directed by a command script that permits specification of the required sequence of actions by an end-user of eHoloCam. The holographic camera is described in more detail by Sun *et al*. (2007).

## 3. Numerical reconstruction algorithms for digital holography

The roots of holographic recording and replay/reconstruction lie in the formulation of Maxwell's (1865) equations and the subsequent derivation of the wave equation. The theory of forming and reconstructing a hologram is based on wave superposition between a coherent light source and the scattered light from the object illuminated by the same coherent light source. When a monochromatic plane light wave ** E**(

*x*,

*y*,

*z*,

*t*) propagates in free space it can be expressed as(3.1)where

**is the position vector in (**

*r**x*,

*y*,

*z*) space;

**is the wavevector; ,**

*k**λ*is its wavelength; and

*E*_{0}is its amplitude. Equation (3.1) contains both the spatial and the temporal dependence of the electric wave field. The temporal term can be omitted in many cases when only spatial information is needed. The spatial term in equation (3.1) represents the complex amplitude of the wave, denoted by

*ψ*(

**) if we consider a linearly polarized wave, thus the solution can be expressed as(3.2)where is its amplitude.**

*r*### (a) Hologram recording

A hologram records the interference pattern formed by the waves scattered or reflected from an object (the object wave), and those arriving directly (the reference wave) at the recording plane (*ξ*, *η*, 0; figure 2). Two geometric layouts are commonly used in holographic recording: in-line (figure 2*a*) where the reference and object waves are both on the optical axis and collinear, and off-axis (figure 2*b*) where the reference wave is angularly separated from the object wave as they meet at the recording plane. As shown, a back-illuminating reference wave is used in in-line recording geometry and passes through the, largely transparent, recording volume. In contrast, in off-axis recording, large opaque objects can easily be captured and front or back illumination can be used.

A conventional hologram is recorded on a photographic plate/film which is then chemically processed to obtain the fringe pattern produced by the reference and object waves. During reconstruction, the hologram is illuminated by the reference beam (figure 2*c*) which propagates through the hologram disturbance and forms the reconstruction wave field after the hologram. In digital holography, an electronic photo-sensor directly records the interfering waves to form an electronic hologram (e-hologram) which is then reconstructed in a computer by numerically simulating the wave propagation. This is based on the theory of coherent diffraction through the hologram which acts as an aperture. The diffracted wave can be described by the Fresnel–Kirchhoff integral or the angular spectrum method (e.g. Champeney 1973). We discuss these two common methods of reconstruction to show their suitability for in-line and off-axis hologram reconstructions.

### (b) Angular spectrum reconstruction algorithm

Consider a plane reference wave propagating in the *z*-direction and approaching the hologram aperture plane (*ξ*, *η*) which is located at *z*=0 (figure 2*c*). After the hologram aperture, the wave will continue to propagate in the *z*-direction (*z*>0) to form the diffracted wave field with a modified phase and amplitude caused by the hologram disturbance. An arbitrary point (P_{1}) in the hologram plane and a point (P_{2}) in the diffraction field at *z* are determined using the position vectors ** ρ**,

**and**

*R***′, where**

*r***′ is the projection of**

*r***at the diffraction field at**

*R**z*. The vector

**represents the position difference between P**

*r*_{1}and P

_{2};

**(**

*k**k*

_{x},

*k*

_{y},

*k*

_{z}) represents the wavevector, and

**the projection of**

*q***on the reconstruction plane;**

*k***′ and**

*r***are both two-dimensional vectors which are conjugate variables in Fourier transformation.**

*q*The diffraction field at P_{2} may be obtained by superposing the plane monochromatic reference wave disturbances with various amplitudes and propagating in various directions after the hologram aperture (represented by ). According to equations (3.1) and (3.2), when a scalar plane wave approaches the hologram aperture, it has a simple monochromatic form. The hologram modulates this to give a field that may be described as a superposition of plane waves,(3.3)where Re represents the real part of the complex wave and represents the angular spectrum of the complex plane waves. The integral in equation (3.3) gives the spatial term of the diffracted wave and so, dropping the harmonic time dependence, the complex amplitude can be written as(3.4)where(3.5)

When *z*=0, ** r**′ becomes

**, and equation (3.4) is written as(3.6)Equations (3.4) and (3.6) represent Fourier transforms (FTs) of the waves at their respective planes and, therefore, an inverse transform can be obtained. If we let be the FT of hologram function , the diffraction equation for the angular spectrum method is obtained (considering equation (3.6)) as(3.7)Thus, the FT of the diffraction/reconstructed field at any**

*ρ**z*after the hologram aperture may be described by knowledge of the hologram aperture. To implement equation (3.7), we need to obtain from the recorded hologram matrix and the aperture properties given by the pixel pitch and pixel numbers or imaging area of the electronic sensor. Equation (3.7) can be implemented in either an ‘exact’ or an ‘approximate’ (Fresnel) form, as described below.

To obtain the FT function of , a two-dimensional fast Fourier transform (2DFFT) is applied to , and after pointwise multiplication by the ‘propagator’ matrix an inverse fast Fourier transform (IFFT) is applied to extract the reconstruction field . The exact form of angular spectrum reconstruction is therefore as follows:(3.8)The exponential propagator term inside the IFFT operation in equation (3.8) is implemented as a matrix over the whole sensor area at the *ξη* plane, where Δ*ξ* and Δ*η* are the sensor pixel pitch in *ξ* (horizontal) and *η* (vertical) directions; *m* and *n* are integers and represent the pixel indices counting along *ξ* and *η*; *M* and *N* are the total pixel numbers along *ξ* and *η*, respectively.

Now, considering the Fresnel region where , we may alternatively use a Taylor series expansion of to obtain(3.9)

Inserting equation (3.9) in (3.7) and retaining only the first two terms, the FT of in the Fresnel region becomes(3.10)where can be removed from the FT operation; therefore, the reconstruction field of is given approximately as(3.11)The intensity distribution for the reconstructed field is obtained by(3.12)The reconstruction is computed by providing any *z* value. At the recording distance *d* (figure 2), the in-focus image will be formed in the reconstruction space. Therefore, numerically extracting the object information of interest in the whole recorded volume becomes possible by scanning through *z* with a step of Δ*z* (e.g. 1 mm) in the reconstruction region corresponding to the recording volume. Equations (3.8) and (3.11) are the fundamental relationships for the exact and approximate computations of object reconstruction with the same number of sample elements in the object, hologram and reconstruction domains. Therefore, the reconstructed image dimension is independent of *z* if a collimated reference beam is used. Both equations have been used by us for reconstructing in-line digital holograms and holovideos captured in the laboratory and in eHoloCam cruises. Figure 3 shows the geometry used for laboratory and *in situ* recording of hologram with back illumination of the object volume.

### (c) Fresnel integration reconstruction algorithm

An alternative form of representation of the diffraction field is in terms of Fresnel integration of the wave (e.g. Champeney 1973). Equation (3.11) gives the Fourier domain representation of the wave field at the reconstruction plane; applying the convolution theorem of Fourier analysis, it can be expressed in the spatial domain as the convolution of the hologram aperture function and a ‘chirp’ function as follows:

In the Fresnel region, this becomes(3.13)This may also be obtained directly from the Fresnel–Kirchhoff integral. To implement equation (3.13), we replace *x*′, *y*′ by *u*Δ*x*′, *v*Δ*y*′; *ξ*, *η* by *m*Δ*x*, *n*Δ*y*, respectively, where *u* and *v* are integers and are in the range of 1<*u*<*M* and 1<*v*<*N*, respectively. Unlike the angular spectrum method, Fresnel integration has different scaling factors in the hologram and reconstruction domains. The image size in the reconstruction domain changes with the distance *z*, wavelength *λ* and sensor parameters *M*, *N*, Δ*ξ* and Δ*η* but can be defined as *M*Δ*x*′ and *N*Δ*y*′, where Δ*x*′ and Δ*y*′ are the pixel pitch in the reconstruction plane along *x*′ and *y*′, respectively, and given (Kreis 2005) by

## 4. Laboratory results and *in situ* performance

We now present some of the results from the laboratory investigations carried out in the Laser Laboratory at the University of Aberdeen and from the *in situ* deployment of the eHoloCam in the North Sea and Faeroes Channel to show how the camera and reconstruction algorithms performed.

### (a) Capturing fast-moving plankton with a pulsed laser

The pulsed laser described earlier was used for imaging fast-moving planktonic animals in the laboratory. Living plankton samples were collected from off the coast at Stonehaven, Scotland. The samples were placed in natural seawater in a tank with high-quality flat windows at both the sides (figure 3). The electronic sensor chosen was a Sony ICX-082AL-6 CCD incorporated in a Hitachi HP-E/KM1-S10 interlaced camera associated with an image grabber (ViewCast Corporation Osprey 210). The sensor dimensions were 8.7 mm (H) by 6.5 mm (V) with corresponding pixel numbers of 756×581 and 11.6 μm by 11.2 μm pitch. An eholovideo and its corresponding reconstruction (at *z*=37 mm) shows the rapid motion of a jumping copepod. The exact form of the angular spectrum (equation (3.8)) was used for the reconstruction. The corresponding digital holovideo and reconstruction are available as videos 1 and 2, respectively, in the electronic supplementary material.

To ensure laser–camera synchronization, the Hitachi interlaced camera must be set to ‘field-on-demand’ to allow the output of odd horizontal lines while leaving even lines blank (grey level of 0, shown in the electronic supplementary material, video 1). The holograms must be deinterlaced before reconstruction using the ‘line doubler’ method, where the captured odd fields can be processed similarly, filling the adjacent empty blank with even lines. Deinterlacing the fields using the ‘weave’ method (combines the most recent odd and even fields) is not effective, since the hologram would consist of two holograms captured at different time instances, while the line doubler method maintains the captured aspect ratio and improves the numerical reconstruction vibrational stability of the system. Some extracted images (frame 1, 22, 24, 26, 31 and 32) from video 2 in the electronic supplementary material are shown in figure 4*a*–*f*.

### (b) Pre- and post-processing of in-line e-holograms captured by eHoloCam

The eHoloCam has been deployed in the North Sea from the RV *Scotia* in four cruises covering all seasons (December 2005, April, July and December in 2006). The camera was towed (at speeds from 1.5 to 2.0 m s^{−1}) through the water column at various water depths down to 430 m and was automatically operated to capture from 10 m downward. Several hundreds of eholovideos of *in situ* plankton were recorded and data analysis is ongoing.

Digital holograms recorded using eHoloCam inevitably contain periodic background noise caused by interference in the cover glass of the CMOS sensor or other optical components in the system. This noise can be removed from the e-hologram with spatial frequency-domain filters to improve the quality of the reconstructed image. Frequency-domain filtering is carried out either as a pre-processing or a post-processing step, but appears to give maximum benefit when used in pre-processing. Periodic noise in the holograms appears as two peaks on either side of the central zero order (DC term) in the frequency domain (indicated in figure 5*a*). The line joining these two peaks is perpendicular to the periodic fringes (enlarged square in figure 5*b*) in the hologram, while their distance from the DC term is proportional to the inverse spatial wavelength of the noise in the hologram. Removal of this noise involves generating a mask and multiplying this with the frequency-domain representation of the e-hologram to force these unwanted peaks to 0 while leaving other frequency components untouched. Figure 5*b* shows a hologram with some periodic noise and figure 5*c* shows the same hologram with the periodic noise removed.

It is also easy to construct a filter mask in the frequency domain to remove the DC component (zero-order diffraction) of an image, thereby implementing a dark-field reconstruction. By performing this procedure as a processing operation after reconstruction, the image visibility can be increased, potentially providing better image contrast for auto-recognition system and certainly improving visibility to the human eye. The zero-order filter (high-pass filter) operates in exactly the same manner as the periodic noise filter described above, except that it masks out the term in the frequency domain which represents the DC (average brightness) level in the image as opposed to those representing periodic noise. For instance, figure 6*a* shows a reconstruction of the hologram in figure 5*b* without post-processing, where the bright background indicates the presence of the DC term. The reconstructed image is then applied with the high-pass filter, which showed that the DC term is visibly removed from the background and the copepod image turns to bright (figure 6*b*) thereby increasing the image visibility.

### (c) In situ recording of plankton using eHoloCam

It was observed that the types of plankton, their population density and behavioural characteristics in different seasons and locations varied significantly. It is not the intention in this paper to report and reach conclusions on the biology of plankton. However, we present here some examples of the recorded images (figure 7) to demonstrate the capability and viability of eHoloCam for imaging plankton. The selected images show copepods (e.g. the middle image in figure 7*e*) and Larvacea (figure 7*a*) at different development stages. To capture edible particles, these Larvacea secrete a complex mucous ‘house’ (Todd *et al*. 1996), e.g. the circled structure in figure 7*a*, which is extremely fragile and always damaged in netted plankton samples. eHoloCam is capable of recording these characteristics with no damage to the animal. In addition to copepods and Larvacea types, decapods (e.g. figure 7*d*) and worms (not shown) were found to be highly abundant in April and July but not in December.

Since in-line holography has the ability to record phase objects in general, apart from being able to record opaque objects (e.g. figure 7*d*,*e*), eHoloCam is also capable of recording transparent (figure 7*f*) and semi-transparent organisms (figure 7*b*,*h*). Both large organisms, e.g. up to 10 mm long, large zooplankton (arrow worms in the electronic supplementary material, figure 1*a*,*b*), and small organisms, e.g. single marine diatom cells (dimension size of 20–100 μm), were recorded and reconstructed. Figure 7*g* is an example of a single phytoplankton cell (*Chaetoceros peruvianus*) in a side view. The size of the valve face of the individual cell is close to 90 μm. Both organisms in figure 7*f*,*g* were observed in the open sea (Faeroes Channel) in December 2006. Figure 7*c* shows the behavioural characteristics of an animal (Ctenophora) folding its long (greater than 5 mm) tentacles for capturing food (Todd *et al*. 1996); this behaviour will not be observed satisfactorily with other imaging devices because a large focusing depth (at least several millimetres) is essential along with resolution at the micrometre level. A phytoplankton chain such as *Thalassiosira* (figure 7*e*), consisting of more than 26 individual cells, was imaged with the connecting structure (5 μm) not visible owing to the limitation of the sensor resolution. An interesting phenomenon was observed only in April's cruise, i.e. the existence of cast-off carapaces (Exuvia; e.g. Figure 7*h*) from zooplankton. The shape and size of the carapaces indicate the development stage of plankton and are a useful clue for biologists in identifying the behavioural characteristics of species in these different stages. Again, the advantages of holography are apparent since such highly transparent materials can be easily damaged or altered in shape during sampling.

The image resolution for reconstructions in the eHoloCam system, using the IBIS4-6600 sensor, was measured in air to be 125 lp mm^{−1} (line-pairs per mm) or 8 μm, by reconstructing a hologram of a 5 bar resolution target placed in air at 100 mm from the sensor. This is in accordance with sampling theorems which indicate that the sampling frequency (the inverse of pixel pitch) should be at least twice that of the spatial frequency of the fringes formed on the hologram. For larger object distances, the resolution becomes diffraction limited and is determined by the hologram (sensor) aperture in the usual way. The presented images show that eHoloCam is capable of resolving most zooplankton and marine diatoms (phytoplankton) in a range of 20 μm to 10 mm.

## 5. Effect of overfilling illumination

In normal application of the angular spectrum method, only the collinear (in-line) volume delineated by the sensor dimensions is reconstructed. However, when using the Fresnel reconstruction algorithms it is possible to extend this volume beyond the sensor area if particles outside the sensor area receive illumination. In eHoloCam, the diameter of the illuminating beam (40 mm) is larger than the sensor dimensions (10.5×7.73 mm^{2}) and this ‘over-illumination’ can be used to increase the reconstruction volume. This is demonstrated in figure 8, where two organisms are reconstructed using the in-line angular spectrum (figure 8*a*,*b*) and off-axis Fresnel integration (figure 8*c*,*d*) reconstruction algorithms. Using the in-line method, a jellyfish larva at the top of the image is reconstructed (figure 8*a*). There is a segment missing from this image which appears as a weak ‘ghost’ image at the bottom of the frame. Similarly, in figure 8*b*, a Larvacea is reconstructed at the r.h.s. of the image with its missing segment also appearing as a ghost image at the l.h.s. For the same e-hologram, the off-axis reconstruction algorithm (Fresnel integration) allows the missing segments of these organisms to be reconstructed in their true locations (figure 8*c*,*d*, respectively).

The images shown in figure 9 demonstrate that the procedure outlined above is effective in increasing the reconstruction area of the in-line hologram by recovering the missing parts of the animal in figure 8*c*,*d*. Figure 9*a* shows the reconstruction of a 1101×1365 pixel hologram using the approximate angular spectrum (in-line) method (equation (3.11)). The small organism, *Appendicularian*, was found in focus at *z*=210 mm. The same hologram was then cropped to 1101×1101 pixels so that the organism (located 2 mm from the upper cropped edge) is no longer within the directly illuminated area. Figure 9*b* shows that the small animal is clearly reconstructed using the Fresnel integration algorithm at *z*=−210 mm. The phytoplankton chain (also seen in figure 9*b*) was also out of the direct illumination but was satisfactorily reconstructed (not shown) at *z*=−185 mm. The ‘minus’ sign in *z* represents a virtual image of an off-axis reconstruction with the organism having the same orientation as in figure 9*a* for the in-line reconstruction. The pseudoscopic real image (not shown here) can be reconstructed at *z*=185 mm. When compared with the in-line reconstruction an extra area of 40 mm^{2}, or approximately 50% of the sensor area, can be achieved.

It is worth pointing out that to achieve this enhanced reconstruction area, the off-axis method described above has limitations. For instance, (i) the reconstruction will be invalid when *z* is small and the paraxial approximation is no longer satisfied and (ii) the reconstruction will not be formed when the interfering fringes from particles outside the directly illuminated area are not sufficiently recorded in the hologram. This can occur when, for example, a one-dimensional object is perpendicular to the sensor edges and lies outside the sensor area. In this case, the interference fringes have infinite curvature, are perpendicular to the sensor edges and do not intercept the sensor.

## 6. Conclusion

Holography has clear advantages over other optical and measurement techniques for the analysis and identification of marine organisms and particles. The ability to record in three dimensions over a large field of view makes the technique invaluable to marine biologists for the studies of species populations and dynamics. By digital recording of the holograms the benefit of the time dimension is added. Our subsea holographic camera (eHoloCam) is based on a pulsed laser so that holograms and holographic videos of fast-moving particles can be recorded. A range of algorithms for numerical reconstruction of the holograms have been developed and compared for application in eHoloCam. We have shown how the algorithms can be subjected to pre- and post-processing methods for improving quality and reducing noise levels of the holograms. The camera has been deployed on four occasions in the North Sea and Faeroes Channel and the results of some of the recordings are shown. The angular spectrum algorithm is well suited for the reconstruction of in-line digital holograms. The exact and approximate formulations of the angular spectrum method have been implemented for the underwater digital hologram reconstruction of plankton. The Fresnel algorithm is well suited to off-axis reconstruction since its scaling properties enable large object distance to be reconstructed with the limited sensor size. Preliminary results show that data extraction outside the in-line volume was possible but the effectiveness of this method is largely dependent on the shape and size of the object. The method was demonstrated to be effective in extending the reconstruction area up to 50% of the sensor size.

## Acknowledgments

The authors wish to thank their industrial partners, CDLtd and Elforlight Ltd, for their significant contribution to this work, especially Dr Gary Craig (CDL) who contributed to the housing design and Mr Keith Oakes (Elforlight) who designed the pulsed laser used in the eHoloCam. Many thanks are given to all at Fisheries Research Services Marine Laboratory, Aberdeen, UK, for their valuable support in the deployment of eHoloCam, especially to Steve Hay for his help in species identification, Mr John Dunn and Mr Paul Fernandes for their support in the *Scotia* cruises. We would like to thank the funding bodies UK DTI (Department of Trade and Industry) in conjunction with EPSRC (UK Engineering and Physical Sciences Research Council) under the OSDA (Optical System for the Digital Age) Link Programme for the financial support they have given to the eHoloCam project.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rsta.2007.2187 or via http://journals.royalsociety.org.

One contribution of 20 to a Theme Issue ‘James Clerk Maxwell 150 years on’.

- © 2008 The Royal Society