AU2004201109A1

AU2004201109A1 - Method and apparatus for producing a phase image of an object

Info

Publication number: AU2004201109A1
Application number: AU2004201109A
Authority: AU
Original assignee: University of Melbourne
Current assignee: University of Melbourne
Priority date: 2003-03-26
Filing date: 2004-03-16
Publication date: 2004-10-14

Description

AUSTRALIA

Patents Act 1990 COMPLETE SPECIFICATION STANDARD PATENT Applicant(s): THE UNIVERSITY OF MELBOURNE Invention Title: METHOD AND APPARATUS FOR PRODUCING A PHASE IMAGE OF AN OBJECT The following statement is a full description of this invention, including the best method of performing it known to me/us: 2 METHOD AND APPARATUS FOR PRODUCING A PHASE IMAGE OF AN

OBJECT

Field of the Invention This invention relates to a method and apparatus for producing a phase image of an object.

Background of the Invention The production of phase images of an object has been of considerable interest for many years. Phase images are particularly useful for the examination of transparent or semi-transparent samples. Conventional contrast or intensity images of such samples often provides very little detail of the internal structure of the samples because of the transparent nature of the samples. Phase images are able to reveal more internal structure and therefore provide a better tool for examination and analysis of such samples. However, with a few exceptions, none of the conventional techniques are able to yield quantitative phase data. Quantitative phase information has typically called for interferometric techniques utilizing coherent illumination. Optimal resolution in microscopes calls for partially or completely incoherent illumination, and this requirement is inconsistent with interferometric techniques.

More recently, a technique for quantitative noninterferometric phase imaging has been developed and which can completely decouple the phase from the absorption in the sample. This technique is able to produce quantitative phase data relating to an object. The technique is described in International Patent Application No. PCT/AU99/00949 and International Patent Application No. PCT/AU02/01398. The contents of these two International applications is incorporated into this specification by this reference.

\\melb files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 3 The technique according to the above two International applications enables good quality phase images of an object to be produced. However, the effects of noise on the imaging technique can reduce the quality of the phase image which is produced.

Summary of the Invention The object of this invention is to provide method and apparatus for producing phase images which can address the issue of noise and produce better quality images in noisy environments.

The invention, in a first aspect, may be said to reside in a method of producing a phase image of an object, comprising: determining an optimal range of defocused distances; producing at least two differently focused images of the object with at least one of the images being captured at a defocus distance within said range; and processing the images to create a phase image of the object.

By choosing an image within the optimal range for at least one of the images, noise in the phase image is reduced and the quality of the phase image is improved.

Preferably the step of determining the optimal range comprises determining a function which relates the optimum defocused distance to both the amount of noise in an image and the intensity data contained within the image, the optimal range being defined as a range of defocused distances about the optimum defocused distance where the function is substantially flat.

Preferably the defocused distance 8z of at least one of the defocused images meets the requirement \\melbfiles\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 4 z VI _kr))R s k( 3ideal (1) where k is the wave number C is the system noise, Io is the in-focus intensity, V is a gradient operator, is the phase, r is the plane perpendicular to the propagation direction Z, Iideal is the noise free intensity which is the true image (ie. signal captured noise, that is Io Iideal and 8z is the defocus distance.

Preferably the method includes determining whether an optimum defocused distance 8z does exist by determining if the left and right hand sides of the above equations enable the inequality to hold, that is to say that if ko 6IoV2()) k a3ideal \aZ RMS Preferably the optimum defocused distance exists if the noise level a satisfies 3I (VI(ri))3 a2 3 RM S o k3 aIideal o z

/RMS

The invention, in a first aspect, may also be said to reside in an apparatus for producing a phase image of an object, comprising: \\melbfiles\homeS\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 5 processing means for determining an optimal range of defocused distances; a sensor for producing at least two differently focused images of the object with at least one of the images being captured at a defocus distance within said range; and processing means for processing the images to create a phase image of the object.

Preferably the processing means for determining the optimal range comprises processing means for determining a function which relates the optimum defocused distance in terms of the amount of noise in an image and the intensity data contained within the image and producing an intensity derivative, the optimal range being defined about the intensity derivative on either side of the intensity derivative where the function is substantially flat.

Preferably the defocused distance 8z of at least one of the defocused images meets the requirement 5z 61 -2Io (V L ))Rs k 3ideal az 3

RMS

Preferably the processing means determines whether an optimum defocused distance 8z does exist by determining if the left and right hand sides of the above equations enable the inequality to hold, that is to say that if ko 6Io<< (VI_ )9s I RM k aZ3

MS

\\melbfiles\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 6 Preferably the optimum defocused distance exists if the noise level a satisfies 2 3 k3 a3Iideal (ri)) a Z

RMS

The invention may also be said to reside in a method of producing a phase image of an object, comprising: producing at least two differently focused images of the object having a relatively large defocused distance; producing at least two differently focused images of the object at a relatively small defocused distance; and constructing a composite image for the differently focused images to produce a phase image of the object.

According to this aspect of the invention, the at least two differently focused images at large defocused distance will have low spatial frequencies well resolved but the high spatial frequencies will be damped or reduced, and the differently defocused images at relatively small defocus will have high spatial frequencies well resolved, but the low spatial frequencies will be damped or reduced.

Thus, by combining the two sets of data into a composite image, the image can be reconstructed with both high and low frequencies well resolved to produce a good quality image of the object, thereby reducing the impact of noise on the final image of the object.

Preferably the differently focused images taken at large defocused distance are used to create a first phase image of the object and the differently focused images at the \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 7 small defocused distance are used to create a second phase image of the object, and the two phase images are combined to construct a composite image by low pass filtering the phase image obtained using the large defocused distance, and high pass filtering the phase image obtained using the small defocused distance, and then adding the two filtered images together.

Preferably the high and low pass filters sum to unity.

In the preferred embodiment of the invention, the filter pair have the form Ti(U)= T, (J =exp(-Ui 2 loge 2 2

T

2

T

2 1-exp(-iU 1 2 log 2/ 2 for the low and high pass filters respectively.

The crossover frequency 4 is by definition equal to the radial spatial frequency UI for which the high-pass and low-pass filters both have the value of In practice, may be selected by trial and error to achieve the cleanest background for the resulting image.

Empirically, we find that a wide range of cutoff frequencies 4 give similar results for the composite image; this is a consequence of the fact that there is band of spatial frequencies in the recovered phase maps which is reliably reconstructed for both defocus distances.

In the preferred embodiment of the invention, each phase image is constructed from three differently focused images of the object, one image being an in focus image, one image being a positively defocused image, and the other \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 8 being a negatively defocused image, the positively and negatively defocused images being taken respectively at the large and small defocused distances.

In the preferred embodiment, two phase images are used to create the composite image. However, in other embodiments, more than two phase images can be used to create the composite image by creating further phase images at various defocused distances.

The defocused distance at which one of the differently focused images having relatively large defocus, and one of the differently focused images having relatively small defocus distance are captured, may be determined in accordance with the first aspect of the invention described above.

Whilst in the preferred embodiment of the invention, the high-pass and low-pass filtering takes place by a pair of filters, the high-pass and low-pass filtering may include more than two filters so that a low range filtered image, an intermediate range filter image and a high range filter image are produced, and which are added together to create the composite image.

The invention may also be said to reside in an apparatus for producing a phase image of an object, comprising: a sensor for producing at least two differently focused images of the object having a relatively large defocused distance and for producing at least two differently focused images of the object at a relatively small defocused distance; and processing means for constructing a composite image from the differently focused images to produce a phase image of the object.

Preferably the processing means is for creating from the \\melb_files\home\Luisa\Keep\Speci\atia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 9 differently focused images taken at large defocused distance a first phase image of the object, and from the differently focused images at the small defocused distance a second phase image of the object, and the processing means is for combining the two phase images to construct a composite image by low pass filtering the phase image obtained using the large defocused distance, and high pass filtering the phase image obtained using the small defocused distance, and then adding the two filtered images together.

Preferably the high and low pass filters sum to unity.

In the preferred embodiment of the invention, the filter pair have the form Tii() T(iiJ) exp(- UJ 2 log, 2/42)

T

2 1

=T

2 (iu 1-exp(-U_ 1 2 loge 2/2) for the low and high pass filters respectively.

The crossover frequency is by definition equal to the radial spatial frequency UjI for which the high-pass and low-pass filters both have the value of In practice, 4 may be selected by trial and error to achieve the cleanest background for the resulting image.

Empirically, a wide range of cutoff frequencies 4 give similar results for the composite image; this is a consequence of the fact that there is a band of spatial frequencies in the recovered phase maps which is reliably reconstructed for both defocus distances.

In the preferred embodiment of the invention, each phase image is constructed by the processing means from three \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 10 differently focused images of the object, one image being an in focus image, one image being a positively defocused image, and the other being a negatively defocused image, the positively and negatively defocused images being taken respectively at the large and small defocused distances.

The sensor may comprise a CCD array, photodiode, photographic film, or the like, depending on the nature of the radiation which is used to create the images. The radiation used to create the images may be electromagnetic radiation and, in particular, visible light, ultraviolet light or x-rays, particle radiation such as electrons or neutrons, and mechanical radiation such as acoustic waves.

In the preferred embodiment of the invention, the sensor comprises a single sensor for capturing all images, and the sensor is moveable relative to a lens system to capture the images at the required defocused distances.

However, in other embodiments, the sensor may comprise a plurality of separate sensor elements which are either moveable relative to the lens system or which are fixed and arranged so that a different optical path length exists thereby providing the different defocused distances.

The invention may also be said to reside in a computer program for producing a phase image of an object, comprising: code for determining an optimal range of defocused distances; code for producing at least two differently \\melbfiles\home$\Lui a\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 11 focused images of the object, with at least one of the images being captured at a defocused distance within said range; and code for processing the images to create a phase image of the object.

The invention may also be said to reside in a computer program for producing a phase image of an object including: code for producing at least two differently focused images of the object having a relatively large defocus distance and for producing at least two differently focused images of the object at a relatively small defocused distance; and code for constructing a composite image from the differently focused images to produce a phase image of the object.

Brief Description of the Drawings Preferred embodiments of the invention will be described, by way of example, with reference to the accompanying drawings, in which: Figure 1 is a schematic diagram of the preferred embodiment of the invention; Figure 2 is a set of images created to illustrate the problem addressed by the preferred embodiment of the invention; Figure 3 is a set of experimental images created also illustrating the problem addressed by the preferred embodiment of the invention; Figure 3A is a graph illustrating the preferred embodiment of the invention; Figures 4A, 4B and 4C are images also created according to the preferred embodiment of the invention; and Figure 5 is a graph illustrating a filter set used in the embodiment of Figures 4A, 4B and 4C.

\\melb_file\home\Luisa\Keep\Speci\atia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 12 Description of the Preferred Embodiments With reference to Figure 1, a schematic illustration of the preferred embodiment is described which utilizes a source of radiation 10 which, for the purpose of the preferred embodiment, will be described with reference to visible light, a condenser lens 12 and a focusing lens 14.

An object 16 to be imaged is located between the lenses 12 and 14. A sensor 20 such as a CCD array is used to capture differently focused images of the object 16 at planes A, B and C shown in Figure 1. For ease of illustration, the sensor 20 is simply shown moved out of the path of the radiation passing through the object, but in use, the sensor is located at the planes A, B and C.

The image at plane B will be an in focus image and the images at planes A and C will be respectively a positively defocused image and a negatively defocused image of the object. The defocused images are obtained by either moving the sensor 20 relative to the object or moving the lens systems 12 and 14 so as to focus at the plane B a negatively focused image or a positively focused image of the object 16.

The sensor 20 is connected to a processor 40 which creates the phase image from the data captured by the sensor 20 in accordance with the algorithm disclosed in the aforementioned International applications. The processor also determines the optimal defocused distance in the manner which will be described below, and also combines the at least two images relating to large and small defocused distances to produce a combined phase image in the manner described below. Thus, in other words, the processor 40 performs all of the processing of the data captured by the sensor 20 to determine optimal distances and create the phase images in accordance with the following description.

\\melbfiles\home$\Luisa\Keep\Speci\atia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 13 Whilst it is well known that the intensity and phase structure over a given surface uniquely determines the intensity distribution in three-dimensional free space it is less well known that, in the absence of phase discontinuities, the converse is true that the intensity distribution uniquely determines the phase.

The method is established on the basis that energy conservation must be satisfied and this requires that the time-averaged probability flow vector or Poynting vector must obey the equation: (1) where V* is the divergence operator in three-dimensional space and r is.a position vector in that space. For the case of a coherent wave, this may be written as: V 0 (2) where I(i) and 4(r) denote the intensity and phase of the wave, respectively. This equation has a unique solution for the phase if the intensity is known everywhere and the phase is continuous.

If we make the paraxial approximation, then equation (2) becomes the so-called transport-of-intensity equation

(TIE):

-k aI(O,0) V. (3) az where k is the average wave number of the radiation, V± and fL respectively denote the gradient operator and position vector over the in-focus image plane, and z denotes the optic axis (nominal direction of radiation propagation). If the intensity and \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 14 intensity derivative aI(iL,O)/az are known then this elliptic partial differential equation may be uniquely solved for the phase provided that the phase is continuous. Experimentally, the intensity is easy to obtain. The intensity derivative may be estimated by acquiring the intensity in two planes, separated by a distance 6z either side of the plane of interest, and forming: I(i 8z)- I(,-8z) (4) az In the case of optical microscopy, the derivative along the optical axis in equation may be obtained by taking images at positive and negative differential defocus about the plane of interest. In this way, the amplitude and phase of the object may be determined by substituting the intensity and intensity derivative data into the transport-of-intensity equation and then solving for the phase numerically. Previous work has shown that, using this algorithm, a quantitative phase measurement may be acquired reliably and quickly, using radiation of insufficient coherence for interferometric phase determination.

This technique clearly depends on the phase information being rendered visible through the redistribution of intensity on propagation, a phenomenon well known to both optical and electron microscopists, who often use defocus to visualise transparent objects using a conventional microscope. However, it is apparent that phase gradients with a small magnitude may not produce a clear signal in the presence of noise. The aim of this paper is to explore the nature of the noise sensitivity of the technique and to propose a simple and direct method for improving the image obtained from noisy data.

\\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 15 The structure of the recovered phase image 1 is determined (up to an irrelevant additive constant) solely by the intensity distribution I(fl,0) and longitudinal intensity derivative aI(iL,O)/8z in the measurement plane.

Whilst in theory these two quantities can be measured to arbitrary precision, the sensor 20, such as a CCD camera or scanning pinhole, is subject to the influences of experimental noise. As indicated before, the quantity aI(i 1 ,0)/8z is usually estimated via the intensity difference between two closely spaced planes, and as such can be sensitive to noise in the detection system. One would therefore expect the recovered phase to be highly sensitive to the effects of noise. However, as is shown below, the inversion process is surprisingly robust in the presence of significant amounts of experimental noise.

Denote the intensity distribution in two planes separated by 28z as If(f)=I( 1 ,5z) and and let Io(i 1 be the intensity distribution over the plane located half way between and I The measurements in each of these planes will, in any real experiment, be affected by measurement noise. If we assume the noise distribution in all of the measurement planes to be statistically identical and independent of the measured intensity then we can express the intensity distribution in each of the measurement planes as being the sum of the ideal, noise-free intensity distribution Iideal(rI) plus a term G(L) representing system noise: I+ lii( I(L) 'ieal(') We now perform a cubic-order Maclaurin series expansion of about the zero defocus, subtract the resulting pair of simultaneous linear equations, and divide by 2 8z, to give: \\melbfiles\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 16 I+ I a) Iideal 6Z 2 I ideal (6) 26z az 3! z 3 /256z where the noise terms have been assumed to be statistically identical and have therefore been combined in quadrature. This immediately tells us that equation will be an accurate measure of the intensity derivative lIideal(r,O)/aZ provided that the second and third terms on the right-hand side of equation are each much smaller in modulus than the first term.

In order to relate the results directly to the phase distribution, we assume that the intensity is sufficiently slowly varying that the intensity derivative is dominated by the effects of phase curvature, so that equation (3) becomes: aI(f,0) -k =I V 1 Io(r)* Io(f)Vi( 0) (7) dz Io This approximation, while too crude for our quantitative phase recovery algorithm, is sufficiently good for our present purpose of investigating the effects of the noise on the phase recovery.

Using equation we arrive at conditions for equation to be an accurate estimate for the intensity derivative aI(? ,O)/az: ka 28zlo RMS(8) and k 3 ideCal 8 2 61\S\ M a a a a \\melbfiles\homeS\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 17 where )RMS is used to denote the root-mean-square (RMS) spatial average of the bracketed quantity over the field over the field of view, and a RMS Significantly these two inequalities relate the measurement parameters to the RMS curvature of the wavefield being investigated, It is therefore the magnitude of the wavefront curvature rather than the absolute magnitude of the phase shifts that determines the quality of the recovered phase image for a given noise level and defocus distance.

Inspection of equations and shows that they are two competing inequalities which have to be balanced in order to obtain an acceptable measure of the phase from two displaced intensity measurements. Equation is concerned with the signal-to-noise ratio of the data and states that the noisier the data the greater the value of a) the greater the separation 8z between measurements planes required in order to maintain the same level of noise-induced artefacts in the recovered phase; too small a defocus distance implies that the reconstructed phase will be dominated by noise-induced artefacts. The presence of the factor of on I

RMS

the right hand side of equation makes the physically intuitive statement that the stronger the phase curvature, the stronger the phase-induced signal in the intensity derivative data and, hence, the smaller the defocus 6z required to maintain a given noise-induced artefact level in the reconstructed phase obtained by solving equation It is tempting to think that the phase sensitivity can be arbitrarily increased by simply increasing the defocus distance 8z; this is only true up to a point because equation has to be balanced against equation in which 8z appears in the numerator rather than the \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 18 denominator. Equation says that as the defocus distance is increased the proportionality of the measured intensity difference to the true intensity derivative beings to break down because the second term on the righthand side of equation is no longer negligible compared to the first term on the right-hand side. This provides an upper limit to the acceptable defocus distance which may be used. To summarize, we have a intuitively reasonably trade-off: the defocus distance must be sufficiently small for the intensity difference to be proportional to the intensity derivative and so provide accurate input data for the equation (ii) the defocus distance must also be sufficiently large for the intensity difference signal to rise above the noise in the data.

To illustrate this trade-off, Figure 2 shows some simulated experiments. We took two uncorrelated pictures as the phase structure 4(KO0) and intensity I(i,O) of the object in the plane z=0. An image of a squirrel was used as the phase (maximum phase excursion 1.00 radians).

Both images were 256x256 pixels 1.00 cm square in size.

The radiation wavelength was taken as X=632.8 nm (HeNe laser), with the in-focus phase and intensity structure numerically propagated the distance ±8z indicated in Figure 1 before recovering the phase using our phase recovery algorithm described in the above International applications. Pseudo-random Gaussian noise of zero mean was added independently to the in-focus and defocused intensity distributions to simulate the effect of experimental noise on the recovered phase. Note from Figure 2 how at low noise levels small defocus distances produce a clear image of the phase, but that as more noise is added to the image it is necessary to increase the defocus distance in order to see the image through the noise, as predicted in equation and the subsequent discussion. However, note also that as the defocus distance is increased the recovered phase image \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04

I

19 loses fine detail. This is the result of equation (9) which says that the smallest possible defocus distance should be used in order to obtain the sharpest possible image of the object.

For a given large defocus distance, there exists a certain critical length scale which is defined as follows: both the transport-of-intensity equation and the finitedifference approximation are good approximations for Fourier components in the phase which do not change significantly over distances larger the critical length scale. Thus suitably gross features of the phase are well retrieved by solving the transport-of-intensity equation with intensity data taken using the given defocus distance. For details finer than the critical length scale, equation breaks down, together with the approximation Since working with will greatly overestimate the contribution of high spatial frequencies in the phase to the defocused image (an argument which can be made rigorous using the theory of phase-contrast transfer functions), solution of for the phase will greatly suppress the high spatial frequencies of the phase relative to their true level. Thus, as the defocus distance is increased, the recovered phase image loses detail over length scales finer than the defocus-dependent critical length scale introduced above.

Figure 3 shows some experimentally obtained results. To produce this figure we imaged an unstained cheek cell in saline solution using a 40x UPlanApo objective on an Olympus BX60 microscope operating in transmission mode.

This sample was chosen because of its ready availability as a test object with complex structure, and because it exhibits detail on many length scales. Images were collected using a Kodak KAF1400 CCD with 6.8pm square pixels in a 12-bit Photometrics SenSys CCD camera attached to the 0.5x video port on the microscope, whilst precise \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 20 defocus settings were obtained using a stepper motor attached directly to the microscope focus knob. To achieve different levels of noise the illumination level was changed by inserting neutral density filters into the illuminating beam path and/or adjusting the intensity of the light source so as to change the number of counts detected by the CCD. Assuming that the number of counts measured by any CCD pixel follows a normal distribution, and that this is the primary source of noise in our data, one standard deviation uncertainty in counts for any pixel is simply ±N counts where N is the number of counts registered in a given pixel. For 12-bit digitisation the maximum value for any given pixel is 4096 counts, but to guard against inadvertent CCD saturation in the defocused images we choose to operate at or below a maximum of 2500 counts; thus the minimum noise level obtainable in one image is Bearing in mind that the intensity derivative is estimated by subtracting two defocused images the minimum error in the intensity derivative is actually the error in each image having been added in quadrature. To obtain higher noise levels the illumination is simply reduced without altering the exposure time as the number of counts in each pixel decreases the noise level increases, enabling us to investigate the effects of higher noise levels on the quality of phase recovery. The effect of varying defocus on the quality of the resultant image data sets was done by simply taking data sets at different defocus distances for each illumination setting.

A summary of the results is presented in Figure 3, which mirrors the layout of Figure 2; going down the table we increase defocus distance whilst going across the table we increase the noise level. Inspection of this table clearly shows how higher noise levels require a greater defocus distance in order to recover an acceptable phase image, but that this comes at the expense of losing fine \\melbfiles\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 21 detail in the sample.

We now turn our attention to computing the optimum defocus distance for a given object which will maximum spatial resolution whilst at the same time minimising the level of noise-induced artefacts in the reconstruction. To do this we rearrange the inequalities in equations and in terms of defocus distance: ko o z I ')/RMS As can be seen from this expression an optimum defocus distance 6z can exist only if the left and right sides of equation (10) enable the inequality to hold, that is to say if: kcy -510o I

))RMS

(11) Thus an optimum defocus distance will only exist if the noise level a satisfies: C 2 (12RMS 2 (12) which places an upper bound on the acceptable level of \\melb files\homeS\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 22 noise that may be present. If the noise level is low enough equation (12) will always hold. However, if the curvature of the wave is too weak and/or the noise too strong equation (12) will be violated and no choice of defocus will yield an accurate and acceptable estimate of the intensity derivative.

To find the optimal defocus distance, the existence of which is subject to the validity condition we minimise the sum of the squares of the error terms in equation to obtain the following expression for the optimal defocus distance 8zoptimm: 1/3 a3( I Thus provided equation (12) is satisfied, the defocus distance 8zoptium given by equation (13) will provide the best measure of the intensity derivative and, hence, the phase. Note that the defocus distance increases as the cube root of the noise level a: 8Zoptimum oC 1 3 (14) As can be seen by Figure 3A, the optimum defocused distance is given by the lowest point in the curve Sz 2 However, on either side of this point, the curve is basically flat, for example between 6zi and 6z 3 This flat region between the sloping section A and sloping section B of the curve therefore defines an optical range for the defocused distance 6z. A number of images can be taken within this range and generally a smooth transition from one image to another will result. Thus, the defocused distance z can be anywhere within this range, thereby any two images taken with the range 6z 1 to 6z 3 will provide good quality noise-free results with the image at 8z 2 being the \\melb files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 23 optimum defocused distance.

Physically, equation 14 shows that the noisier the data the greater the required image separation in order for the intensity difference signal to rise above the noise; however, if the noise is so strong that we are defeated at low separations by poor signal-to-noise ratio and at high separations by non-linear terms in the longitudinal intensity evolution then no appropriate defocus exists.

The noise-induced artefacts in the phase retrieval are of very low spatial frequency (c/f Figs. 2 and An estimate for the noise-induced artefact in the phase reconstruction can be obtained by looking at the contribution to this image which is made by the last term on the right-hand side of equation We will again work with the approximation to the TIE, to simplify the present analysis while still bringing the essential results into the light. If we make use of the left-hand side of equation to estimate aI(Y 1 ,0)/az in the TIE, and assume I 0 constant I 0 then we obtain the following Poisson-type differential equation for the artefact component artefact(fl) of the reconstructed phase which is due to the noise in the data: V2artefact ka( We perform a Fourier transform on both sides, and rearrange slightly, to give: rtefact(ii) (16) 4n2 2 05zIo 12 where f( denotes the Fourier transform of the function \\melbfiles\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 24 f(f) Ui, is the Fourier space coordinate which is conjugate to rL, and we have made use of the identity Vi =-472U1 We see that the detector noise spectrum is S-2 suppressed by a damping envelope of the form 1-2 in forming the Fourier transform of the phase artefact. In particular, if the noise distribution in the measured intensity data is spatially uncorrelated and follows a white noise distribution then is a random variable with a mean and standard deviation that is independent of U and so a(Ui)/U± 2 is a random variable whose mean is very strongly peaked at the origin Ui =0 of Fourier space.

This is consistent with the results shown in Figures 2 and 3, where the phase reconstruction is seen to be contaminated with increasingly significant random lowfrequency error as the noise level increases.

Having completed our mathematical derivation of the nature of the noise-induced artefacts in the phase reconstruction, we give a physical explanation of why lowfrequency artefacts are a result of (high frequency) noise. The notion of wavefront curvature contains the solution. Areas of the wavefront with very strong curvature will have a significant effect on the intensity derivative because they rapidly focus or defocus optical energy on free-space propagation, but areas of the wavefront with weak curvature will have a comparatively much smaller effect on the intensity derivative. More precisely, rapidly-varying areas of the wavefront will result in a strong and rapidly-varying signal in the intensity and intensity derivative, but gently-curved areas of the wavefront will result in a weak and gentlyvarying signal. Since we are solving the inverse problem, let us reverse the argument: in the phase-retrieval algorithm, the necessarily gently-varying signals in the intensity and intensity derivative which are generated by \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 25 low-frequency phase curvatures must be much more strongly amplified than their high-frequency counterparts.

Therefore the phase retrieval algorithm is very sensitive to errors in the low-frequency components of the intensity data. Next, recall that the power spectrum modulus of the Fourier transform) of white noise is flat. The high Fourier frequencies of the noise are strongly suppressed in the phase-retrieval procedure because of the great amplification of high-frequency phase structure in the direct problem of free-space propagation. The low Fourier frequencies of the noise are, however, suppressed much more weakly, and so it is the low-frequency components of the noise which result in the low-frequency errors of the recovered phase map.

We saw from our earlier discussion that, as the distance between planes increases, the range of spatial frequencies that may be accurately recovered reduces. To recover high spatial frequencies at the expense of low spatial frequencies, we required a small plane separation; to recover the lower spatial frequencies at the expense of the higher, we required a larger separation between the imaging planes.

If one chooses to work with a single defocus distance, we saw that a compromise was necessary; the optimal defocus distance in equation (13) was the best that could be done in the face of a tradeoff between phase information at high and low spatial frequencies. However, one could instead choose to work with two defocus distances. A phase image obtained using intensity data taken at large defocus will have the low spatial frequencies well resolved (at the price of a damping of the high spatial frequencies); a phase image obtained from intensity data taken at relatively small defocus will have the high spatial frequencies well resolved (at the price of significant errors in the lower spatial frequencies).

\\melbfiles\home\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 26 Using the pair of retrieved phase images, one can construct a composite image. This is done by low-pass filtering the phase image obtained using intensity data at large defocus, high-pass filtering the phase image obtained using intensity data at small defocus, and then adding the two filtered images. The high and low pass filters must sum to unity. The exact functional form of these high pass and low pass filters is not important, provided that they are sufficiently smooth in both real and Fourier space. We chose a filter pair of the form:

T

1 T(ii) exp(-i 1 2 loge 2/ 2) (17a) T2( ij)= l-exp(-il 210ge 2/( 2 (17b) for the low- and high-pass filters respectively. The crossover frequency 4 is by definition equal to the radial spatial frequency ui 1 for which the high-pass and low-pass filters both have the value of Figure 5 shows graphically the effect of the filter pair described above. As can be seen from Figure 5, curve C provides for low-pass filtering and curve D for high-pass filtering. That is, the filtering performed by curve C provides the lower frequencies in the image and the filtering performed by curve D provides the higher frequencies in the image. When both curves C and D are added together, the result is unity as explained above.

In practice, may be selected by trial and error to achieve the cleanest background for the resulting image.

\\melb_files\homeS\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 27 However, to gain some insight into the nature of this cutoff frequency, its derivation from basic principles will be described.

We work in one transverse dimension for simplicity, and begin with the convolution form of the Fresnel diffraction integral: z 2z y(x,z) i O)exp{ k(xz 1 dx' (18) where x denotes the dimension perpendicular to the optic axis z, X is the radiation wavelength, and y(x,z) is the propagated complex disturbance as a function of distance z along the optic axis. The unpropagated disturbance x(x,z=0) can always be written in the form: 0) exp(i4(x)- (19) where O(x) is the phase and J(x)-V lnI(x,z=0) In(x,z=0)| Assuming O(x) and p(x) to be 2 sufficiently small, we write: 0) exp(i((x) 1 j(x) To first order in O(x) and the Fresnel diffraction integral (18) implies that: I(x,z) 2 1- 2(x)*Fcos(hzu 2 24(x)*Fsin(7Xu 2 (21) where F denotes Fourier transformation and denotes convolution. The estimated intensity derivative azI(X) is therefore: \\melb_files\homeS\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 1 28 2 I(x, z) z)2 sin(7Czu 2 (22) az 2z z where u is the Fourier space coordinate which is conjugate to x. Interestingly, p(x) has dropped out of the expression for Iz(x). The Fourier representation of (22) is: Fa x 1 2z'$(u)sin(tzu 2 (23) 8z If z is sufficiently small, then the small argument expansion of the sine gives: F 2()7 (24) 8z This equation is the Fourier transform of the transportof-intensity equation for the case of uniform unit intensity. We conclude that phase retrieval based on the transport-of-intensity equation will correctly retrieve the Fourier coefficients of the phase for all spatial frequencies up to the cutoff frequency t in which is sensibly chosen to be the frequency beyond which the transfer function sin(7hzu 2 of equation (23) is no longer well approximated by the second-order Maclaurin expansion.

Thus we may write an equation for t: sin(XzX 2 12 1-(n^zS2) _1-8 nXkz42 3! where the tolerance e is some small positive number, say 0.05. Solving for the cutoff frequency t, we obtain: F, z) J6 /(7tz) (26) \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 29 We note that the ideas of this section are trivially generalised to the case of N>2 images at N different propagation distances: 0 Z1 z2 ZN (27) leading to N filters T(U which sum to unity. For the filters T(Ui±) have the passband: (8,X,Zj+l) <ii1 ,Z (28) while for j=l and j=N we have the passbands: U, Z2), j=l 1 (29a) i1 ZN) j N (29b) This generalisation to N>2 images may be of some benefit in the context of quantitative phase imaging using extremely noisy data.

Figures 4A, 4B and 4C show results obtained experimentally according to the preferred embodiment of the invention, where we have taken a subset of the data presented in Figure 3 and applied the multiple defocus strategy based on the filters defined in equations (17a) and (17b). Note that whilst the image taken at small defocus distance (Figure contains noticeably more fine detail, the low-frequency contamination significantly obscures the object itself. This is as expected. Conversely, note how the image taken at a larger defocus distance (Figure 4(b)) has far fewer low-frequency artefacts but also contains much less fine structural detail. The composite image (Figure combines these two images using the filter described above, with the crossover frequency t being selected by eye. A significant improvement in image quality is apparent, with both the low and high spatial \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 30 frequency information being accurately represented in the composite image.

\\melb_files\home$\Luisa\Keep\Speci\Iatia Prov method and Apparatus for Producing a Phase Irnage.doc 15/03/04

Claims

1. A method of producing a phase image of an object, comprising: determining an optimal range of defocused distances; producing at least two differently focused images of the object with at least one of the images being captured at a defocus distance within said range; and processing the images to create a phase image of the object.

2. The method of claim 1 wherein the step of determining the optimal range comprises determining a function which relates the optimum defocused distance to both the amount of noise in an image and the intensity data contained within the image, the optimal range being defined as a range of defocused distances about the optimum defocused distance where the function is substantially flat.

3. The method of claim 1 or 2 wherein the defocused distance 6z of at least one of the defocused images meets the requirement k_ 6IoK(is)) k 5z Io (I())RMs k 3ideal (r) 8Z3 where k is the wave number a is the system noise, Io is the in-focus intensity, V is a gradient operator, is the phase, r is the plane perpendicular to the propagation direction Z, Iideal is the noise free intensity which is the true image (ie. signal captured noise, that is Io Iideal and 8z is the defocus distance. \\melbfiles\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 32

4. The method of any one of claims 1 to 3 further comprising determining whether an optimum defocused distance 5z does exist by determining if the left and right hand sides of the above equations enable the inequality to hold, that is to say that if ko 6I )R I )R k aItdealfrI) z RMS The method of claim 4 wherein the optimum defocused distance exists if the noise level a satisfies 3I Vi(T) 3 2 3 0I R M S o<<2 k3 Iideal 1 Z3 /RMS

6. An apparatus for producing a phase image of an object, comprising: processing means for determining an optimal range of defocused distances; a sensor for producing at least two differently focused images of the object with at least one of the images being captured at a defocus distance within said range; and processing means for processing the images to create a phase image of the object.

7. The apparatus of claim 6 wherein the processing means for determining the optimal range comprises processing means for determining a function which relates the optimum defocused distance in terms of the amount of \\melb_files\homeS\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 33 noise in an image and the intensity data contained within the image and producing an intensity derivative, the optimal range being defined about the intensity derivative on either side of the intensity derivative where the function is substantially flat.

8. The apparatus of claim 6 or claim 7 wherein the defocused distance 8z of at least one of the defocused images meets the requirement k_ 6IoK i )R S 6z IV Riw s))S k aideal z

9. The apparatus of claim 8 wherein the processing means determines whether an optimum defocused distance does exist by determining if the left and right hand sides of the above equations enable the inequality to hold, that is to say that if ko 610(VI ))RM S 1 o V2())RMS k ideal Z o RMS The apparatus of claim 9 wherein the optimum defocused distance exists if the noise level a satisfies 3I I o<<2 a\ Zal3 RMS

11. A method of producing a phase image of an object, \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 34 comprising: producing at least two differently focused images of the object having a relatively large defocused distance; producing at least two differently focused images of the object at a relatively small defocused distance; and constructing a composite image for the differently focused images to produce a phase image of the object.

12. The apparatus of claim 11 wherein the differently focused images taken at large defocused distance are used to create a first phase image of the object and the differently focused images at the small defocused distance are used to create a second phase image of the object, and the two phase images are combined to construct a composite image by low pass filtering the phase image obtained using the large defocused distance, and high pass filtering the phase image obtained using the small defocused distance, and then adding the two filtered images together.

13. The method of claim 12 wherein the high and low pass filters sum to unity.

14. The method of claim 12 or 13 wherein the filter pair have the form T,(iu 1 )=exp(-U 1 2 log,2/2) T 2 1 T 2 i 1 1-exp(-ii 2 log 2/ 2) for the low and high pass filters respectively. The method of any one of claims 11 to 14 wherein \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 35 each phase image is constructed from three differently focused images of the object, one image being an in focus image, one image being a positively defocused image, and the other being a negatively defocused image, the positively and negatively defocused images being taken respectively at the large and small defocused distances.

16. The method of claim 15 wherein two phase images are used to create the composite image.

17. An apparatus for producing a phase image of an object, comprising: a sensor for producing at least two differently focused images of the object having a relatively large defocused distance and for producing at least two differently focused images of the object at a relatively small defocused distance; and processing means for constructing a composite image from the differently focused images to produce a phase image of the object.

18. The apparatus of claim 17 wherein the processing means is for creating from the differently focused images taken at large defocused distance a first phase image of the object, and from the differently focused images at the small defocused distance a second phase image of the object, and the processing means is for combining the two phase images to construct a composite image by low pass filtering the phase image obtained using the large defocused distance, and high pass filtering the phase image obtained using the small defocused distance, and then adding the two filtered images together.

19. The apparatus of claim 18 wherein the high and low pass filters sum to unity. The apparatus of claim 17 or 18 wherein the \\melb_files\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 36 filter pair have the form 1 exp(-iU 2g loge 2/4 2 T2 =T2( 1- exp(-U 2 loge 2/ 2 for the low and high pass filters respectively.

21. The apparatus of any one of claims 17 to wherein each phase image is constructed by the processing means from three differently focused images of the object, one image being an in focus image, one image being a positively defocused image, and the other being a negatively defocused image, the positively and negatively defocused images being taken respectively at the large and small defocused distances.

22. The apparatus of claim 21 wherein two phase images are used to create the composite image.

23. A computer program for producing a phase image of an object, comprising: code for determining an optimal range of defocused distances; code for producing at least two differently focused images of the object, with at least one of the images being captured at a defocused distance within said range; and code for processing the images to create a phase image of the object.

24. A computer program for producing a phase image of an object including: code for producing at least two differently focused images of the object having a relatively large defocus distance and for producing at least two \\melb_files\homeS\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04 37 differently focused images of the object at a relatively small defocused distance; and code for constructing a composite image from the differently focused images to produce a phase image of the object. \\melbfiles\home$\Luisa\Keep\Speci\Iatia Prov Method and Apparatus for Producing a Phase Image.doc 15/03/04