WO2008020083A2

WO2008020083A2 - High resolution digital holograms

Info

Publication number: WO2008020083A2
Application number: PCT/EP2007/058595
Authority: WO
Inventors: Bryan Hennelly; Thomas Naughton; John Macdonald
Original assignee: National University Of Ireland Maynooth
Priority date: 2006-08-17
Filing date: 2007-08-17
Publication date: 2008-02-21
Also published as: WO2008020083A3

Abstract

A method of digital holography comprises illuminating an object S, rotating the object wavefield relative to a digital holographic sensor CCD so that the object wavefield is shifted across the recording surface of the sensor, and recording a plurality of digital holograms ('input holograms') at respective angular positions of the object wavefield. The input holograms are combined to create an output digital hologram having a higher resolution that any of the input holograms. The object wavefield may be rotated by rotating the object (in the case of a 2-D object) or by rotating a mirror interposed between the object and the sensor (in the case of a 3-D object).

Description

High Resolution Digital Holograms

1. FIELD OF THE INVENTION

The present invention relates to a method for the recording of digital holograms, in particular the creation of a high resolution (superresolution) digital hologram from a number of lower resolution digital holograms.

2. PRIOR ART

Holography is the science of recording and reconstructing a complex electromagnetic wavefield, see T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods, (Wiley, 2004). The principle of holography is to record the interference pattern generated by the light emanating from an object and a reference beam, which are coherent with respect to one another. The interference pattern is recorded on a photosensitive material and to reconstruct an image of the object at the same distance away from the material, the material is illuminated with the same reference beam as was used in the recording process. If the observer moves, the object becomes visible from many different perspectives. During recording/reconstruction four terms are generated - the intensities of the object and reference waves and the real and conjugate images. Gabor invented electron holography in 1948 and this method suffered because the desired real image was degraded by the overlapping presence of the other three terms. With the onset of the laser Leith and Upatnieks appended the holographic principle with the introduction of the offset reference wave. This enabled the separation of the object wavefield from the other components that are generated in the optical reconstruction process.

Digital holography (DH) refers to the use of an array of individual electronic sensor devices to record the hologram in place of a photosensitive material. Such an array is referred to herein as a digital holographic sensor and a typical example is a CCD array, often referred to simply as a CCD. As in continuous (i.e. photographic) holography, DH comprises two parts - (i) recording and (ii) reconstruction. (i) DH Recording:

'In-line' or 'on-axis' DH refers to an architecture in which the reference wavefield travels in the same direction as the object wavefield, see Fig. l(i). In Fig. l(i) light from a laser L is split into coherent illumination and reference beams I and R respectively by a first beam splitter BSl . The illumination beam I is directed onto the object 10 via plane mirrors Ml, M2, and the reflected object beam O passes through a second beam splitter BS2 to fall on the recording (i.e. light sensitive) surface of a holographic sensor here in the form of a CCD. The reference beam R likewise falls on the recording surface of the CCD after reflection by a plane mirror M3 and the beam splitter BS2. The CCD records the interference pattern generated by the object wavefield and the reference wavefield at the recording surface of the sensor. Such systems are extremely well known, and no further explanation is necessary.

As in continuous holography this method suffers from poor reconstructed image quality, due to the presence of the intensity terms and the conjugate image that contaminates the reconstructed object image. While it is possible to remove the intensity terms with efficient numerical techniques, it remains difficult to remove the conjugate image in the reconstruction process, which is discussed shortly. This may be achieved using an off- axis recording setup equivalent to that used by Leith and Upatnieks, see Fig. l(ii). However, this reduces the range of angles/resolution of the reconstruction significantly as will be explained below. An alternative approach known as phase-shifting interferometry, Fig. l(iii), has been introduced allowing for an in-line set-up to be used with at least two successive captures (i.e. two successive holographic recordings) and enables separation of the object wavefield from all of the other terms. Between captures a constant phase change is introduced in the reference beams. Fig. 2 shows the same three architectures, this time with 2-D objects S as input to the systems. Such 2-D models are often used to describe biological specimen pressed between two plates. In addition a microscope objective is often placed between such objects and the CCD to magnify the object wavefield. In each of the three set-ups the object and the CCD array are separated by free space. In general there may be any bulk optical system in this free space (any system of lenses, sections of free space, gratings, prisms, Spatial Light Modulators, diffusers etc.), e.g. an optical Fourier Transform or a magnification system. Furthermore, there may be many variations of reference beam, e.g. spherical wave, plane wave, random phase.

To provide an insight into the DH recording set-ups and their limitations a Wigner chart may be employed, such as those shown in Fig. 3. These charts represent 1-D signals and their extension to the 2-D case is trivial. They are plan views of the Wigner Distribution Function (WDF) enabling us to view a signal's energy as a function of space, x, and spatial frequency, k. To illustrate the usefulness of the Wigner Chart in the analysis of first order optical phenomena the case of free space propagation of the wavefield emanating from a 2-D sample, such as those shown in Fig. 2, is examined. In Fig. 3(i) a 2-D object is illuminated by a coherent source. The wavefield emanating from the plane of the object (i.e. the object wavefield) may be described by Fourier theory as being composed of an infinite sum of weighted plane waves (perfect laser beams of different strength) at different angles. For the purpose of illustration four such plane waves A, B, C, and D are shown. The Fresnel or paraxial approximation predicts that the object wavefield at an output plane some distance d away is composed of the same plane waves except they are now located in different regions in space. Fig. 3(ii) is a Wigner Chart for the input signal at the plane of the object. The plane waves are represented as horizontal lines over the region in x occupied by the object. The k coordinate represents the angle of the plane wave. Fig. 3(iii) is the Wigner Chart of the propagated signal at the output plane. The plane waves will retain their angle and therefore they will not move along the k axis in the Wigner Chart. They will however move in x by an amount proportional to the angle of the plane wave and the distance of propagation. Fig.4 further illustrates the concept of free space propagation, this time for a greater distance. Fig. 4(i), (ii) and (iii) correspond to the same diagrams in Fig. 3. It can be seen that the plane waves spread out further in space than in the previous case.

Each coordinate in the Wigner Chart represents a ray of light at some point in space x and traveling at some angle proportional to the k value. The dimension out of the page indicates the amount of energy of the ray. The object signal has a finite width in x and every point of the 1-D object in x has rays emanating at all angles in k. No upper bound is shown to the shaded regions in Fig. 3 and Fig. 4 since it extends to infinity in the positive and negative of the k axis indicating a full range of angles. If this object wavefield is allowed to propagate some distance away from the object plane (say to the CCD plane, i.e. the plane containing the recording surface of the CCD) it will be found that each of these rays maintains its angle of orientation (k → k) but the position has obviously changed depending on angle and the distance traveled (x → x + λdk) where λ is the wavelength of the light and d is the distance propagated). Therefore each coordinate in the Wigner Chart has undergone a positional transformation as shown in Fig. 3; if the wavefield is observed at a farther distance it will be seen that the WDF has undergone a greater positional transformation as shown in Fig. 4. This particular transformation is given by the paraxial approximation and is an interpretation of the Fresnel Transform in Wigner space.

Fig. 5(i) and (ii) illustrate the recording of the propagated wavefields shown in Figs. 3(i) and (ii) by a CCD having its recording surface in the output plane (the recording is implemented using a reference beam as shown in Fig. 1; however, inclusion of the reference beam in Fig. 5 is superfluous to the discussion at hand). The Wigner Charts at the output planes of Figs. 3(i) and (ii) are shown in Figs. 5 (iii) and (iv). These two figures show a second, rectangular, Wigner Chart centered in the x-k coordinate system. This is the chart of the CCD recording surface, which has some width (physical size) in space and has a finite bandwidth associated in k, which is determined by the resolution of the CCD, i.e. the spatial width of the box in x is equal to the physical extent of the CCD recording surface in space and the spatial frequency bandwidth of the box in k is determined by the pixel pitch or the sampling rate of the CCD. In this case there clearly exist plane waves incident on the face of the CCD (i.e. inside the spatial extent of the CCD recording surface). In order to record a signal satisfactorily it is important that none, or as little as possible, of the signal energy lies inside the width of the CCD recording surface inx and outside of the bandwidth of the CCD in k. Such a situation, illustrated by Fig. 5(iii) results in aliasing, i.e. this energy is a large source of noise since the camera cannot record the interference pattern that is generated, i.e. this interference pattern will be at too high a spatial frequency to be recorded. Using these charts it is possible to calculate the optimum distance to place an object from the CCD so that all angles that cannot be recorded are at x coordinates outside the region of the CCD, see Fig. 5(iv). Clearly such a calculation must take into account the object width, the CCD recording surface width and CCD bandwidth. From this discussion it may be concluded that it is only possible to record a small range of angles - i.e. a small bandwidth of the signal energy. Note that similar Wigner Charts may be used to analyse any DH system that has a bulk optical system between the object and CCD.

The description so far does is incomplete. The Wigner Charts thus far have illustrated the WDF of the complex object wavefϊeld O_z(x) and the CCD. In fact there are four separate signals recorded in the interference pattern on the CCD. This is explained as follows:

The complex signal incident on the CCD recording surface is given as the sum of the two wavefϊelds,

h(x) = O_z (x) + R(x)

(1)

The CCD records the real-valued intensity |/z(x)|² which may be written as below in Equation (2),

|A(X)Γ = |O, (X)|² +|Λ(X)|² +O, (X)Λ^* (X)+Λ(X)O; (X) . (2)

Given that in DH it is possible to easily remove the first two signals - the intensities of the object and reference wavefields - Equation (2) may be re- written as follows:

h(xf = O_z (χ)R^* (χ) ₊ R(χ)o: (χ) . (3) Thus the two intensity terms may be dropped to leave the real and conjugate images. In the case of phase shifting DH the real image may be separated out and the analysis of DH recording given above using Wigner Charts is entirely valid. In the case of in-line and off-axis DH the two terms in Equation (3) are present in the final hologram. Fig. 6(i) shows the Wigner Chart of the final hologram for in-line DH. The conjugate image is present as a second signal (a mirror of the real image in k). The two signals cannot be separated in the region of overlap in reconstruction. In off-axis DH, the reference wave R(x) is chosen to move the real image WDF up the k axis and the conjugate image WDF down the k axis, see Fig. 6(ii). Choosing the appropriate off-axis setup ensures that the two WDFs do not overlap in the region of the CCD recording surface and can therefore be separated in the reconstruction process. However, to ensure that no aliasing occurs (i.e. that there is no energy from either of the signals within the x width and outside the k bandwidth of the CCD) the distance between the object and CCD must be increased further still. This imposes a much more severe limit on the range of angles that may be recorded and therefore the resolution of the reconstructed image.

(ii) DH Reconstruction:

While the recording process is similar to the continuous case, given that a photosensitive material can be replaced with a CCD array, the reconstruction process is quite different and is non-optical.

In DH, reconstruction is performed on a computer by simulating the propagation of the hologram back to the plane of the object some distance from the hologram/CCD plane. It is possible to simulate the propagation of the wavefield any distance so as to focus on different parts of a 3-D object at different depths on the axis of propagation. To view the object from a particular perspective one selects an appropriate section of the recorded hologram on the CCD to be used in the numerical reconstruction process. The line joining the center of this section of the recorded hologram on the CCD and the center of the object in the object plane determines the direction of perspective. Furthermore, the physical area of the section used determines the resolution of the reconstructed object. In off axis the reconstruction has at best only half the minimum resolution of the in-line architecture.

The numerical algorithms that may be used to simulate the propagation of the recorded complex hologram back to the object plane are based on analytical models that describe light propagation. For example, the Fresnel-Kirchoff integral and the Fresnel Integral/Transform may be used to analytically describe the propagation of light through free space. The linear canonical transform describes the propagation of light through any bulk optical system. Numerous efficient algorithms exist to implement these transformations on a computer. Note that all of these algorithms require a number of calculations in the order of MogiV where N is the number of samples (pixels).

One major advantage of DH over material holography is the ability to use discrete signal processing techniques to the recorded signals. In recent years DH has been demonstrated to be a useful method in many areas of optics such as microscopy, deformation analysis, object contouring, particles sizing and position measurement.

The main disadvantage of DH is the low resolution/range of angles of perspective of the reconstruction.

3. SUMMARY OF THE INVENTION

According to the invention there is provided a method of digital holography comprising illuminating an object, rotating the object wavefield relative to a digital holographic sensor so that the object wavefield is shifted across the recording surface of the sensor, recording a plurality of digital holograms ("input holograms") at respective angular positions of the object wavefield by recording the interference pattern generated by the object wavefield and a reference wavefield, and combining the input holograms to create an output digital hologram having a higher resolution that any of the input holograms. In certain embodiments the object is substantially two dimensional and the object wavefield is rotated by rotating the object.. In other embodiments the object is three dimensional and the object wavefield is rotated by rotating a mirror interposed between the object and the sensor.

The invention is applicable to holograms recorded using an in-line holographic system, an off-axis holographic system, or a phase shifting holographic system.

There is also provided, as an independent invention, a method of digital holography comprising illuminating an object, recording a digital hologram ("input hologram") by recording the interference pattern generated by the object wavefield and a reference wavefield, the input hologram having both real and conjugate images, and processing the input hologram by removing one of the real and conjugate images in its plane of focus and then numerically propagating back to the plane of the other of the real and conjugate images to create an output digital hologram in which the said one image is substantially removed.

4. BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of example, with reference to the accompanying drawings, in which:

Fig. 1 shows prior art examples of in-line, off-axis and phase shifting digital holography systems for 3-D objects. Fig. 2 shows the same three prior art digital holography systems for transmissive 2-D objects.

Fig. 3 shows Wigner Charts illustrating free space propagation. Fig. 4 shows Wigner Charts illustrating free space propagation for a greater distance than in Fig. 3. Fig. 5 illustrate the recording of an object wavefϊeld at different distances from the object.

Fig. 6. shows the Wigner charts for in-line DH and off axis DH. Fig. 7 demonstration of the effect of rotation of the object wavefϊeld. Fig. 8 illustrates capturing a hologram, then rotating the object and capturing a second hologram.

Fig. 9 shows embodiments of the invention implemented using the same three DH architectures as Figs. 1 and 2, the object being a transmissive 2-D sample that may be rotated in x and y. Fig. 10 shows embodiments of the invention implemented using the same three DH architectures as Figs. 1 and 2, the object being a 3-D object whose wavefϊeld is reflected onto the CCD by a rotatable plane mirror.

Figs. 11 to 14 illustrate a method of combining individual low resolution holograms to form a higher resolution superresolution hologram.

5. DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

5.1 2-D OBJECTS - ROTATING THE OBJECT As stated, the main disadvantage of DH is the low resolution/range of angles of perspective of the reconstruction. To increase this quantity a number of digital holograms are captured where a 2-D object is rotated about at least one axis between captures, and preferably about orthogonal x and y axes, so that the object wavefϊeld is shifted across the face (i.e. recording surface) of the CCD in at least one direction between captures. Rotation may be intermittent, i.e. the object is halted temporarily while each recording is taken, or continuous with the recordings being taken "on the fly". The following discussion refers only to a rotation about a single x axis, but similar considerations apply to rotation about the y axis. The idea is based on rotation of the object being mathematically equivalent to multiplying the input object signal by a linear phase term, i.e. if the object is rotated then the input object signal changes:

θ{x) → 0{x)exp(j2πxa/λ) . (4)

The propagated version of O(x) is recorded, i.e. the Fresnel Transform of O(x) which is denoted as O_z(x) where z denotes the distance propagated. Its Fourier Transform is

denoted as O_z(k) . It is a property of the Fresnel Transform that if the input field rotates as in Equation (4), then this has the following effect on the recorded field:

This effect is best demonstrated using the Wigner Distribution Function. Fig. 7 illustrates the effect of rotation of the input wavefield. Fig. 7(i) shows the rotation of the object and the resulting shift in angle of the plane waves (mathematically equivalent to multiplying by a linear phase factor). This diagram should be compared with Fig. 5 (ii). While at the input plane the plane waves do not move in space and only change in angle, in the output plane the plane waves have moved in space and also changed in angle. Fig. 7(ii) shows the Wigner Chart of the input object wavefield. As the input object is rotated (multiplied by a linear phase factor) the input WDF translates along the k axis (shifts in angle). In Fig. 7 (iii) the Wigner Chart of the signal in the output plane of the CCD is shown. The input rotation has caused this Fresnel transformed signal to translate in both x and k along the dashed line. The chart of the CCD is also shown and one can see that the Wigner Chart of the signal passes through that of the CCD without ever causing aliasing. It is this effect that is exploited as a method for superresolution.

Fig. 8 demonstrates the superresolution method. A digital hologram is captured, then the object is rotated and a second digital hologram is captured. In the first hologram recording in Fig. 8 (i) the range of plane waves that are recorded are between A and B. In Fig. 8 (ii) the rotation shifts the wavefield in x and k. That part of the hologram field that contained light at angles too great for their interference to be recorded by the CCD resolution (and was therefore arranged to be at an x position adjacent to and not on the camera face) is shifted in x onto the face of the camera and the angles are shifted down in k to angles that may be recorded. In the second capture the range of plane waves that can be recorded are between C and D. It is clear from the diagram that B < C < A. This allows for some region of overlap to exist between successive captures, i.e. successive holograms correspond to overlapping regions of the shifted object wavefϊeld, and facilitates correlation algorithms that calculate precise shifts in x and k. This may be repeated again and again for successive angular positions of the object in both x and y directions, a respective digital hologram being recorded at each rotation. The next step in the process is to join, or "stitch", these separate recordings together, analogous to a patchwork quilt, in both spatial position x and spatial frequency position k. This will be discussed later.

The method described above is based on the equivalence between rotation of the object and mathematical multiplication of the object signal by a linear phase factor. This equivalence is only valid in the case of rotation of a planar object. However, in many practical cases the object of interest will approximate a 2-D planar object. For example, a biological specimen pressed between two plates meets such an approximation.

The above method may be implemented in various ways, as shown in Fig. 9 which illustrates embodiments of the invention in which the DH architectures of Figure 1 are modified to implement this method, i.e. rotating a substantially 2-D object and recording multiple digital holograms. In each case the optical distance between the object and the sensor is selected so that when recording each hologram substantially no energy falls on the recording surface of the CCD at unrecordable angles.

5.2 3-D OBJECTS - ROTATING A MIRROR

In order to deal satisfactorily with 3-D objects an imaging system and a mirror are introduced into the system. As shown in Fig. 10, which shows modifications of the three architectures shown in Fig. 1, an imaging system 12 (represented as a one lens imaging systems) is placed between the 3-D object 10 and a mirror MR to image the object onto the mirror. The mirror is rotatable about x and y axes and reflects the object wavefϊeld onto the CCD where it is recorded using a digital holographic method. Since the mirror is a 2-D plane its rotation will have an identical effect as described in Section 5.1 for the case of 2-D objects. Again, in each case the optical distance between the object and the sensor is selected so that when recording each hologram substantially no energy falls on the recording surface of the CCD at unrecordable angles.

Rotation of the 3-D object itself will not result in the desired motion of the propagated field in x and k. This is only valid in the case of rotation of a planar object. The equivalence between rotation of the object and mathematical multiplication of the object signal by a linear phase factor is ensured by rotating a mirror onto which the object has been imaged.

Care must be taken to shift the object wavefield, and hence the hologram field, by some fraction of the spatial extent of the CCD when the mirror is rotated. If this is implemented some portion of the signal energy will be common to both holograms, i.e. successive holograms will correspond to overlapping regions of the shifted object wavefield, as described for the 2-D object. This region of the holograms that is common to both is used in the next part of the method, which uses correlation to find the shifts in x and k that have taken place. As before, shifting preferably takes place in both x and y directions.

Note that the above embodiments may be modified to include any bulk optical system placed between the rotating 2-D object and the CCD, or between the 3-D object and the rotating mirror, or between the rotating mirror and the CCD. All such cases can be analysed and developed using the approach outline herein.

The case of the imaging system placed between the object and the reflecting mirror, as described above, is the simplest to deal with conceptually. However there exist any number of possible bulk optical systems that can be placed between (i) object and mirror and (ii) mirror and CCD. For example, the imaging system between the object and mirror could be replaced with a magnification system for recording digital holograms of smaller objects or indeed a demagnification system for recording large real world objects. In either of these cases the above analysis would be identical. It is to be noted that in the case of imaging, magnification and demagnification systems the limit to the resolution/angle of the digital holograms that can be recorded is related to the f number of the imaging system, i.e. the range of angles of light (plane waves) that pass through the imaging system.

As noted above that there exist any number of possible bulk optical systems that can be placed between (i) object and mirror and (ii) mirror and CCD. These are not limited to imaging and magnification systems. Any bulk optical quadratic phase system may be used and may impose an entirely different limit to the invention than the 'f-number' in the case of the imaging system. For example, if the object is placed close to the mirror with only free space between them this corresponds to a Fresnel Transform. In this case all angles of plane waves will reach the mirror. However, it will be found that after some number of rotations of the mirror, the plane waves hitting the CCD will be at too high an angle to record. This limit can easily be determined by applying the shift invariant properties of whatever optical systems exist between the mirror and object and between the mirror and CCD.

5.3 IDENTIFYING THE RELATIVE POSITIONS - STORING AND RECONSTRUCTING

It is very difficult to know the angle of rotation of the mirror/planar object and the distance of propagation to the CCD to the level of accuracy required in order to accurately determine the precise movement of the object wave field inx and k across the face of the CCD between successive recordings. It is necessary to know these precise displacements in x and k between successive recordings in order to stitch the individual holograms together to create the larger, superresolution hologram. The following describes how to overcome this obstacle, i.e. to determine the shift in x and k of the object wavefϊeld, and hence the corresponding hologram wavefϊeld, caused by rotation of the mirror/planar object about the x axis between successive recordings (similar considerations apply to rotation in the orthogonal y direction).

Two overlapping complex holograms, denoted as O\{x) and 02<X), have been recorded. From Equation 5.1 |O₂(x)| = \O\(x - ξ)|. The absolute values of the two holograms are shifted versions of one another. Since it has been arranged for the shift to be less than the width of the CCD, some region of the two holograms is common to both. By correlating the absolute values of the two holograms, |Oi(x)| and |O₂(x)| a peak at the position x = ξ will be obtained.

Correlation may be implemented using a variety of methods on a computer. The correlation of two discrete images of size N will result is a discrete image of size 2N- 1. Given that the two holograms will appear as a type of random speckle noise the correlation procedure is optimized by subtracting the mean values from |Oi(x)| and |O₂(x)| before they are correlated. It is important that the value of the shift ξ is known to a level of accuracy less than the size of the CCD pixel. The correlation function that is obtained is discrete with the 2N- 1 samples separated by a sampling interval equal to the pixel size. The position of the maximum valued sample with respect to the origin indicates a first approximation of the shift ξ. Since the discrete correlation function that has been obtained is well sampled in the Nyquist limit, the continuous correlation function may be obtained from these samples using a variety of interpolation procedures. Thus one may obtain the position of the correlation peak and therefore the value of the shift ξ to a level of accuracy much less than the pixel size. Fig. 11 shows the correlation peak obtained using the superresolution technique. Two holograms were captured using phase shift digital holography. The mirror was rotated between captures and the absolute values of the two holograms had their means subtracted and were correlated. The discrete correlation function obtained was interpolated so that each pixel shown has a size equal to one fifth the CCD pixel size. The next step is to determine the value of the shift in spatial frequency, denoted K. To do this the common spatial regions of the holograms are isolated using knowledge of the precise ξ value, i.e. by taking the section from each hologram that has the same absolute value. Given that ξ has some sub-pixel value, in order for the two sections (which are discrete images) to match up precisely the second function will have to be interpolated. Fig. 12 shows the Wigner Charts of the two isolated sections. It can be seen that the only difference between these two holograms is a shift in the spatial frequency domain.

In order to find the value of the shift in spatial frequency, K., the same sub-pixel correlation is applied as before. This time the complex Fourier Transforms of the two isolated regions are correlated. There is no need to subtract the mean in this case. The peak obtained when the Fourier Transforms (obtained using the FFT algorithm) of the two isolated regions are correlated is shown in Fig. 13. The same sub-pixel correlation technique as before is applied. The distance between the samples above is one fifth that afforded by standard correlation techniques.

This completes that part of the method that concerns itself with position identification. Now all that is left is to stitch the two holograms together - in fact , to stitch the first hologram and that part of the second that is not contained in the first. The first step is to align the two parts in space. In general, since ξ is known to a level of accuracy much less than the pixel size it will be necessary to apply interpolation to the new section from the second hologram in order to obtain perfect alignment between the samples. Fig. 14(i) shows the Wigner Charts of the two aligned signals.

Once alignment/interpolation has been performed the two sections are again separated. Fig. 14(ii) and (iii) shows the charts of these two sections centered in x. Both hologram parts are independently reconstructed using discrete Fresnel Transforms and the new Wigner Charts are shown in Fig. 14(iv) and (v). Finally using knowledge of K the two reconstructions must be stitched together in the Fourier domain. This is illustrated in Fig. 14(vi). Firstly, the two holograms are transformed to the spatial frequency domain. This is done using FFT algorithm. These two Fourier Transforms must now be positioned correctly relative to one another. In order for the first Fourier Transform to be shifted along the negative k axis and for the second to be shifted along the positive k axis there must be room for the two to move. Zero values are placed along the negative and positive k axes and then the signals are moved into these regions. The movement of the two signals is again determined by the a priori knowledge of the value of K. Since K is a sub-pixel value again it will be required to apply interpolation to obtain perfect alignment. Carrying out an inverse FFT to the new stitched Fourier Transform, the superresolved reconstructed object is obtained.

The reconstruction process above should be contrasted with that described in the prior art. The increase in the number of samples/storage required is kept to a minimum as is the time taken to perform the numerical reconstruction. As stated, all of the above analysis may be applied to the y dimension also where rotation of the object/mirror takes place in two orthogonal directions between captures. However, in certain applications it may be satisfactory to increase the resolution in one direction only, in which case rotation will take place only about one axis.

Note that the method (capture and reconstruction) can be derived for the case when any bulk optical system (system of lenses, sections of free space, prism, gratings etc.) occupies the space between the mirror and the CCD, or object and mirror and using any other numerical reconstruction algorithm other than the Fresnel Transform.

When two holograms are joined there is an inherent increase in the number of samples.

This is primarily due to the increase in bandwidth of the newly constructed hologram - i.e. the increase in the width of k as a number of holograms are stitched together in k.

The inverse of the total extent in k is equal to the resolution of the reconstructed image.

This is coupled with the fact that there is an increase in the extent in x with every stitched hologram. If a large number of holograms are to be stitched, the number of samples, which grows exponentially with the number of holograms, would increase beyond any practical amount that could be stored by a computer. Also, the FFT algorithm in the joining process and the proceeding numerical Fresnel Transform algorithm in the reconstruction process are both order NlogN. The time taken to implement these procedures would increase beyond any practical amount as N increases exponentially. Further computation time arises when one considers that one must take into account the number of samples in the y dimension in the same way as above for the x. The number of samples which affects the storage and time of the algorithms is an exponential function..

As stated above, it is very difficult to know the angle of rotation of the mirror/planar object and the distance of propagation to the CCD to the level of accuracy required in order to accurately determine the precise movement of the object wavefield in x and k across the face of the CCD between successive recordings. For this reason, overlapping holograms are recorded and matched in spatial frequency position and spatial position using correlation techniques, as described. However, the invention does not rule out measuring the precise displacements in x and k between successive recordings in order to stitch the individual holograms together to create the larger, superresolution hologram, in which case non-overlapping recordings may be taken. If direct measurements of the displacements in x and k are able to be made, the individual holograms may be stitched by using the same procedure outlined above. With a precise knowledge of a (the angle of rotation), λ and z, we can determine (without the need for correlation) the displacement in k to be sin(α) and the displacement in x to be λza. Thus we can now continue from the fourth paragraph in this section 5.3 to construct the superresolved digital hologram.

5.4 A METHOD FOR THE REMOVAL OF THE VIRTUAL IMAGE FROM SINGLE CAPTURE IN-LINE HOLOGRAMS

The following is a separate invention, which can be used to facilitate the previously described invention. In other words, it can be applied to the individual input holograms prior to their being stitched together to form the output, superresolved hologram. However, this second invention is of general applicability to digital holography. Single capture, in-line holography, see Fig. l(i), offers superior spatial and temporal resolution when compared with other architectures. Off-axis holography, Fig. l(ii), has the same temporal resolution but greatly inferior spatial resolution while phase shifting holography, Fig l(iii), has the same spatial resolution but has a significantly worse temporal resolution (requires multiple captures). The great disadvantage of single capture, in-line holography is the presence of the virtual (i.e. conjugate) image which appears as a speckle noise in the real image reconstruction and greatly impairs the visual quality of the reconstruction. This invention is a digital signal processing technique that allows removal of the virtual image.

This invention can be used with any DH architecture in which the virtual image has not been fully removed, and when applied to superresolved holograms is applicable to both object and mirror rotation, whether continuous or intermittent.

A single capture, in-line digital hologram is made up of four component terms as previously discussed - the real and virtual images and the intensity of the object and reference wavefields. The latter two terms are quite easily removed using DSP methods as outlined in T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods, (Wiley, 2004). Thus, the virtual and real images are left. Numerically propagating the hologram a distance +z to the real image plane allows reconstruction of the real image but the virtual image appears as a speckle like noise which is a result of a propagation a distance 2z from its own plane of focus. If the hologram is numerically propagated a distance -z the virtual image will be reconstructed and the real image will appear as a noise like speckle term.

The method for removal of the virtual image is as follows:

1. Remove the object and reference wavefield intensities from the hologram by low pass filtering the hologram's FFT. 2. Numerically propagate the filtered hologram a distance -z to the plane of focus of the virtual object. In this plane the virtual image is clearly in focus. 3. Apply any standard object recognition/segmentation software to create a mask of the virtual object in focus. This mask will have zeros in the region of the virtual object and ones elsewhere.

4. Occlude the virtual object by multiplying the complex wavefϊeld at the distance -z by SMASK, effectively occluding all of the energy in the hologram due to the virtual image. In removing the virtual image some energy of the real image will also be removed; i.e. that energy that is in the region of the virtual object.

5. Numerically propagate a distance of +2z to the real image plane where the real image will now be in focus without the noise due to the virtual image.

Care should be taken during the numerical propagation algorithms to allow for the spreading of the real image energy in the plane of focus of the virtual image. This may require zero padding the hologram or its FFT before step (2) above.

Note that the terms 'real' and 'virtual' can be swapped in the above method.

The invention is not limited to the embodiments described herein which may be modified or varied without departing from the scope of the invention.

Claims

1. A method of digital holography, comprising illuminating an object, rotating the object wavefield relative to a digital holographic sensor so that the object wavefϊeld is shifted across the recording surface of the sensor, recording a plurality of digital holograms ("input holograms") at respective angular positions of the object wavefield by recording the interference pattern generated by the object wavefϊeld and a reference wavefield, and combining the input holograms to create an output digital hologram having a higher resolution that any of the input holograms.

2. A method as claimed in claim 1, wherein input holograms are recorded at respective positions of the object wavefield displaced in two orthogonal directions relative to the sensor.

3. A method as claimed in claim 1 or 2, wherein the object is substantially two dimensional and the object wavefield is rotated by rotating the object.

4. A method as claimed in claim 1 or 2, wherein the object is three dimensional and the object wavefield is rotated by rotating a mirror interposed between the object and the sensor.

5. A method as claimed in claim 4, further including a system for imaging the object onto the mirror.

6. A method as claimed in any preceding claim, wherein the optical distance between the object and the sensor is selected so that when recording each input hologram substantially no energy falls on the recording surface of the sensor at unrecordable angles.

7. A method as claimed in any preceding claim, wherein at least one pair of input holograms correspond to overlapping regions of the object wavefϊeld, and the input holograms are matched as to spatial frequency and spatial position.

8. A method as claimed in claim 7, wherein the matching is performed using correlation techniques.

9. A method as claimed in any preceding claim, wherein the input holograms are recorded using an in-line holographic system.

10. A method as claimed in any one of claims 1 to 8, wherein the input holograms are recorded using an off-axis holographic system.

11. A method as claimed in any one of claims 1 to 8, wherein the input holograms are recorded using a phase shifting holographic system.

12. A method of digital holography comprising illuminating an object, recording a digital hologram ("input hologram") by recording the interference pattern generated by the object wavefϊeld and a reference wavefϊeld, the input hologram having both real and conjugate images, and processing the input hologram by removing one of the real and conjugate images in its plane of focus and then numerically propagating back to the plane of the other of the real and conjugate images to create an output digital hologram in which the said one image is substantially removed.