Survey of Medical Imaging Techniques for Detection of Breast Cancers

Transcription

1 Survey of Medical Imaging Techniques for Detection of Breast Cancers MENG JOO ER *, WEN HSIANG CHOW *, EUGENE SZE-YEN PHANG * AND NIKOS MASTORAKIS * Intelligent Machines Research Laboratory, School of Electrical and Electronic Engineering Nanyang Technological University, Singapore SINGAPORE Military Institutions of University Education Hellenic Naval Academy Terma Hatzikyriakou, 18539, Piraues, GREECE. Abstract: - This paper presents a survey of medical imaging techniques currently available and highlights their advantages and disadvantages. It focuses on the aspect of medical imaging which uses a diffracting source and some of the reconstruction techniques that are used in obtaining images of the object from the nearest neighbour interpolation technique, the bilinear interpolation technique and the filtered backpropagation technique. The limitations of the reconstruction techniques are also presented. Mathematical theories behind the reconstruction process are briefly mentioned for completeness. Key-Words: - Medical Imaging 1. Introduction Breast cancers are now one of the most common forms of cancers diagnosed in women. Neither the causes of breast cancers nor the means of preventing the cancer are well understood. Early detection of cancerous lesions in the breast is the key to successful treatment and reduction of mortality. Mammography is widely recognised as the most reliable method for early detection of breast cancers. The appearance of clustered microcalcifications in mammogram films is one of the important early signs of breast cancers. Clustered microcalcifications have a higher attenuation than normal breast tissue and appear as a group of small and localised granular bright spots in the mammograms. However, microcalcifications are difficult to detect because of variations in their shapes and size and because they are embedded in and camouflaged by varying densities of the overlapping tissue structures. Other abnormalities such as masses and architectural distortions are also probable signs of breast cancers. The greater the asymmetry, size, lack of circularity, edge unsharpened and radio density of the mass, the greater the suspicion of breast cancers. Since early detection of cancerous tissues is crucial in enhancing the success rate of curing breast cancers, high quality cross-sectional images, which clearly distinguish the various tissue structures, are imperative. The introduction of computed tomography (CT) in the early 1970s proved to be the solution. CT refers to the cross-sectional imaging of an object by computerised processing of either transmission or reflection data collected by illuminating the object from many different directions. As a technique, CT has been proven extremely useful in diagnostic medicine especially since it enables doctors to observe the internal structure of the human body with unprecedented precision that rivals the information obtained via exploratory surgery. Biomedical applications of CT techniques include use of radioisotopes (e.g. X-rays), ultrasound and Nuclear Magnetic Resonance (NMR), allowing the reconstruction 1

2 of more than one indices or parameters of the tissue and thus revealing more information.. Medical Imaging Modalities Radioisotopes, ultrasound and NMR are the most widely accepted applications that utilise CT techniques in present medical imaging systems. These various CT disciplines present different technical characteristics such as resolution, accuracy and fidelity. Resolution is the measure of fine details displayed in the image. It represents the ability to identify individual details in an image and distinguish them from other details. It also determines the smallest size of structure that can be depicted by an imaging system. Fidelity is the quantitative accuracy to which an individual parameter is imaged. Distortion indicates the geometric accuracy of the final display. In addition, side effects, limitations and applicability must be considered as well. This includes things like noise, detailed contrast and presence of artifacts. Noise is the variability of the image signal not due to an inherent property of the object imaged but due to random fluctuations following laws of probability. It could also be due to the random nature of emission and absorption by tissue or random nature of the detection process. Detailed contrast affects the clarity in differentiating between the various entities imaged and is largely affected by the difference in attenuation coefficient between the structure concerned and the surrounding background. Artifacts are distortions in the images as a result of errors introduced in the image reconstruction process leading to misrepresentation of attenuation values of the entities concerned. The field of medical imaging is concerned with interactions of all forms of radiation or entity with tissue and the development of appropriate technology to extract clinically useful information from observations of such interactions..1 Ultrasound Imaging Ultrasound is defined as an acoustic wave with frequencies ranging from about 0 khz to several hundreds of MHz and is beyond the audible range of the human ear. The frequency range of 1 MHz to 10 MHz of the ultrasound spectrum is typically used for medical instrumentation due to the combined needs of good resolution, fidelity and penetrating ability with minimal distortion. Ultrasound interacts with the tissue in several different ways so that images can be produced from the acoustic impedance, velocity, attenuation and frequency shift of the object. In recent years, there has been a growing interest in the use of ultrasound for noninvasive imaging of human bodies by virtue of its promising nature..1.1 Advantages Non-invasive. The sound waves generated are external to the body and no foreign substances need to be introduced into the body to interact with the waves. Safe at low-power levels. There are no documented health risks associated with ultrasound imaging. This apparent advantage of ultrasound has made its use more popular in potentially sensitive regions such as the pregnant abdomen and the eyes. High sensitivity to soft tissue types. As the ultrasonic wave propagates through the body mass and impacts on tissues of different characteristics (e.g. density and elasticity), distinct changes in properties of the incident wave such as acoustic sound speed enable clearer differentiation between tissue types. Wide variations in refractive indices of materials. It is possible for selective imaging of specific planes of the object using ultrasound..1. Disadvantages Low penetrating power. Unlike X-rays, ultrasonic waves attenuate rapidly within the body and thus it is unable to penetrate air or bone areas within the body. Thus, certain parts of the anatomy, primarily the lungs, cannot be studied by ultrasound. Low resolution. Currently, medical images obtained via ultrasound have poorer resolution as compared to X-ray images probably due to diffraction effects. Ultrasonic energy does not propagate along straight lines. When the object inhomogeneities are large compared to the wavelength of the ultrasonic waves, energy

3 propagation is characterised by refraction and multiple scattering.. Radioisotopes In this area of medical application, we subject the body to some form of radiation, either by transmitting the radiation through the body (Xray CT) or by introducing the emitter of the radiation into the body (Emission CT). We measure the radiation transmitted through or emitted from the body at a number of points in order to obtain the distribution of the physical parameters inside the body that have an effect on the measurements. In X-ray CT, the physical parameter to be reconstructed is the (X-ray) linear attenuation coefficient inside the human body. It involves determination of the distribution of the X-ray attenuation coefficient within the body by mathematically processing X-ray transmission measurements from many angles. Emission CT involves the determination of spatial distribution of radioactivity from radionuclides injected into the body, usually via circulation. This is accomplished by external detection of radioactive emissions from the radionuclides within the tissues of the body. Emission events are measured from many view angles and distribution and the magnitude of the radiation is computed...1 Advantages High penetrating power. X-rays and gamma radiation from radionuclides do not attenuate drastically as they enter tissues or bones. It is sufficiently powerful in penetrating any mass within the body. High resolution. X-rays images distinguish bone boundaries and organs well. They also discriminate different tissue types having considerably different linear attenuation coefficients well. Low distortion. X-rays and gamma radiation from radionuclides, being nondiffracting, travel in straight lines and thus they do not suffer from diffraction effects such as multiple scattering... Disadvantages Health risks. Although diagnostic X-ray levels have been considerably reduced over the years, there are considerable statistics to suggest that a small damaging effect, which can increase the probability of diseases such as eye cataracts, exists. Due to the associated radiation hazards, X- rays-based techniques are not suitable for screening of sensitive anatomical regions. Emission CT is an invasive procedure, which involves the introduction of radionuclides into the body and the toxicity of the resultant ionising radiation, is a cause for concern. Low sensitivity to soft tissue types. There is almost negligible attenuation of X-rays between different soft tissues, which have very subtle linear attenuation coefficient variations, and this poses a problem in tissue discrimination. Selective imaging of specific planes is not possible. Velocity of propagation in the X-ray or gamma ray region is essentially independent of the surrounding environment. The refractive index of all internal structures within the body is unity and the only mechanism of interaction is absorption and scattering. Noisy. Images obtained via Emission CT tend to suffer from noise as scattering and attenuation effects reduce the number of photons reaching the external detectors..3 Nuclear Magnetic Resonance (NMR) NMR involves detection of emissions of electromagnetic energy from the nuclei in isotopes of elements within a body placed in a strong static magnetic field simulated by a relatively weak radio frequency which changes the orientation of the magnetic nuclei relative to the direction of the strong magnetic field. The different magnetic moments of nuclei of different isotopes can be distinguished and various properties of this dynamic behaviour such as relaxation time can be measured..3.1 Advantages High sensitivity to soft tissue types. NMR images have superior soft-tissue contrasts. Non-invasive. Even though NMR subjects the body to relatively intense magnetic fields, it appears to have no toxic effects on the body. Ability to generate multivariate images. By changing operational parameters, NMR 3

4 can generate different images, which emphasise one or more tissue parameters while maintaining reasonable registration among these images..3. Disadvantages of MRI Long scanning time. This results in motion artifacts, which may introduce a considerable amount of noise in the image. Not suitable for all patients. Patients with pacemakers and other metallic devices or implants are not suitable for MRI. Expensive. Among the three techniques, MRI is the most expensive, narrowing the availability of this method to patients from poorer families or countries. 3. Ultrasound Imaging in Detection of Breast Cancers Breast imaging has remained a challenging application for ultrasound. Differences in attenuation of various soft tissue structures in the female breast are small and using ultrasound, it is possible to obtain high contrast details in mammography images. However, several practical issues have to be considered in the usage of ultrasonic imaging for this purpose. Firstly, the results are not highly reproducible. Secondly, the field of view is very small if high resolution is desired. Thirdly, a global image, which enables the comparison of suspicious regions with normal tissues, is not available. Lastly, it may be difficult to accomplish quantitative tissue characterisation with the present system. Generally speaking, the interaction of ultrasonic waves with tissues is a very complex process. Major factors that affect the propagation of ultrasonic waves in tissues include density and elasticity of tissues, specific acoustical impedance, absorption and scattering effects and the parameter of nonlinearity. All these factors are dependent on temperature as well as the frequency of the ultrasonic wave. They are also a function of tissue type and pathological state. Today, ultrasound is the primary adjunctive tool to mammography in the United States, but its role is mainly limited to distinguishing solid from cystic masses. Ultrasound breast scanners have shown great promise in the characterisation of palpable cancers but are unreliable in the detection of smaller masses and microcalcifications. Several recent studies indicate that it is possible for ultrasound to identify nonpalpable masses and masses not visualised on mammography. These studies also suggest that a diagnostic-imaging device that is sensitive to subtle differences in the speed of sound may be useful for enhancing the detection and characterisation of lesions and tumours. All these suggest that ultrasonic imaging can play a bigger role in the management of breast masses and reduce unnecessary biopsies. The immediate task, however, is to improve differentiation of breast cancers from numerous benign conditions of the breast by quantifying the acoustic sound speed. In ultrasonic imaging for medical applications, the target of the ultrasonic waves is a part of the body and it is desired to uncover some information about its internal structure. A known ultrasonic wave is contrived to penetrate into a tissue and the resulting waves from this interaction are measured outside the object. From the known input and the measured output, it is desired to construct an approximate map of one or more of the parameters that describe the interaction. Knowledge of the interaction is based on theoretical notions. The measurements are converted into an image by transforming the acquired data into a meaningful representation of the object. 4. Tomographic Imaging with Ultrasound Tomography involves cross-sectional imaging of an object from either transmission or reflection of data collected by illuminating the object from many different directions. The resultant projection data are results of interactions between the entity used for imaging and the internal structure of the object. An image of the object s interior is reconstructed from these measurements. When ultrasound is used for Tomographic imaging, diffraction effects have to be taken into consideration, especially when the size of inhomogenities in the object becomes comparable to or smaller than a wavelength. 4

5 Measured forward scattered field Fourier Transform y v x u Object Incident wave plane Space Domain Frequency Domain Figure 1: Fourier diffraction projection theorem 4.1 Fourier Diffraction Theorem When an object is illuminated with a plane wave as shown in Figure 1, the Fourier transform of the forward scattered fields measured on a line perpendicular to the direction of propagation of the wave gives the -D Fourier transform of the object along a circular arc as shown in the figure. This is known as the Fourier diffraction projection theorem and it is only valid when inhomogenenities in the object are weakly scattered. According to the Fourier diffraction theorem and the projection theorem, by illuminating the object in many different directions and measuring the diffracted projection data, one can in principle fill up the Fourier space with samples of the Fourier transform of the object over an ensemble of circular arcs and then reconstruct the object by Fourier inversion using a variety of techniques. When a 3-D object is assumed to vary only slowly along one of the dimensions, -D theory can be readily applied to the object. There are two practical reasons for adopting this approach. Firstly, the fundamental idea behind the theory is easier to understand and visualise in -D. Secondly, using the current technology, it is still not realistic to implement large 3-D transforms that are required for direct 3-D reconstruction of the object. In addition, direct display of 3-D objects is not easy. The current trend, however, is to rely on digital data processing technology to obtain a 3-D view from the acquired -D image of the object. The basic principle of diffraction tomography for application with ultrasound is now presented. Since acoustic energy does not travel along straight lines and projections are not line integrals, the flow of energy will be described by a wave equation. Since there are no direct methods for solving the problem of wave propagation in an inhomogeneous medium, approximate formalisms such as the Born and Rytov approximations are used so that the theory of homogenous medium wave propagation can be applied to generate solutions in the presence of weak 5

6 inhomogeneities. This results in a simplified expression that relates the measured scattered field to the object. This relationship will then be inverted for several measurement geometries to obtain an estimate of the object as a function of the scattered field. In the next few sections, various reconstruction techniques together with their merits and limitations will be presented. The restrictions imposed on the reconstruction techniques by the Born and Rytov approximations as well as other various factors will also be reviewed. 4. Co-ordinate Transformation for Interpolation In diffraction tomography, only a finite number of projections are possible and each projection is measured over a finite set of sampling points only. Direct Fourier reconstruction basically involves the reconstruction of the object o(x, y) from a given set of sample points of its Fourier transform O(u, v) distributed over a grid that is formed by arcs of radius k 0. Note that each arc on the grid corresponds to the transform of a single projection. This is illustrated in Figure below. Figure : Fourier transform of an object determined over a circular arc grid When an object is illuminated with a plane wave travelling in the positive y-direction, the Fourier transform of the forward-scattered fields gives values along the arc as shown in Figure 1. Therefore, if an object is illuminated from many different directions, we can in principle fill up a disk of radius of K 0 with samples O(u, v) which is the Fourier transform of the object and then reconstruct the object by direct Fourier inversion. Note that K 0 is the wave number. Since it is inversely proportional to the wavelength λ, as the frequency of the wave, f increases, λ decreases and the radius of the circular arc in the object s frequency domain increases. Therefore, we can conclude that diffraction tomography determines the object up to a maximum angular spatial frequency of K 0. However, the loss of resolution as a result of this bandwidth limitation is negligible. As shown in Figure., the measured data are available on circular arc grids whereas for the convenient of display and reconstruction, it is desirable to have samples over a rectangular lattice. Thus, it is necessary to select parameters representing each grid and derive a relationship between them. Co-ordinate transformation equations are given in Appendix A. 5. Image Reconstruction There are several methods for direct Fourier reconstruction from the diffracted projection data. The various computational strategies can be classified into two categories, namely interpolation techniques in the frequency domain (such as nearest neighbour interpolation and bilinear interpolation) and interpolation techniques in the space domain (such as filtered backpropagation method). 5.1 Nearest Neighbour Interpolation Technique (NN) For each rectangular grid point (u, v) at which an interpolated value of the transform is desired, the corresponding (κ 1,φ 1 ) and (κ,φ ) values are first calculated using (19) and (0) or (1) and () as shown in Appendix A. Recall grids generated by portion OB and AO will be denoted by (κ 1, φ 1 ) and (κ, φ ) respectively. For grids generated by the former, κ varies from 0 to K 0 and φ varies from 0 to π while for the latter, κ varies from -K 0 to 0 and φ varies from 0 to π. The two nearest neighbours of (κ 1,φ 1 ) and (κ,φ ) on the two arc grids are determined and the closer of these two grid points is retained and its value is assigned to the point (u,v). Q(κ,φ) is computed at each rectangular grid point and the object O(x, y) can be reconstructed by -D inverse Fast Fourier Transform (FFT). 6

7 5. Bilinear Interpolation Technique Given Nκ N φ uniformly located samples, the bilinearly interpolated values of Q( κ, φ) are computed at each rectangular grid point using the following formulae: i= Nκ j= Nφ Q( κ, φ) Q( κi, φj) h1( κ κi) h( φ φj) and = i= 1 κ h1( κ ) = 1 κ h1( κ ) = 0 φ h ( φ ) = 1 φ h ( φ ) = 0 j= 1 for otherwise for κ otherwise κ φ φ where φ and κ are sampling intervals of φ and κ respectively. After calculating Q(κ,φ) at each point on the rectangular grid, the object O(x, y) can be reconstructed by -D inverse FFT. 5.3 Zero-Padding The results obtained with NN interpolation and bilinear interpolation can be considerably improved if the sampling density in the (κ,φ) plane is first increased. This can be done by zero-extending the -D inverse FFT of the Q(κ,φ) matrix. The -D inverse FFT of the N κ N φ matrix consisting of Q(κ,φ) values is taken. The resultant N ω N φ matrix is zeroextended to mn ω nn φ and FFT of the new array is taken. The result is an mn-fold increase in the samples in the (κ,φ) plane. This method is computationally efficient and the minimal sampling is determined by the bandlimit of fluctuations in the Q(κ,φ) function in the (κ,φ) plane. 5.4 Filtered Backpropagation Technique The idea is to employ inverse diffraction to produce an image of the wave-field everywhere and in particular to compute the wave-field data along an appropriate straight (1) () (3) line from the measured boundary values of the wave-field on the circular arc. Ideally, reconstruction of the object can be obtained by simply summing the reconstruction from each angle until the entire frequency domain is filled. As the name implies, there are two stages to this technique. For the first stage, each projection represents a nearly independent measurement of the object and gives the Fourier transform of the object along a circular arc as shown in Figure 1. The filtering part in this stage can be visualised as simply weighting of each projection in the frequency domain so that each projection takes up an equal proportion on the image plane. In the second stage called the backpropagation stage, the -D inverse Fourier transform of each weighted projection is summed. The computational procedure in reconstructing an image may be presented in the following steps: Step 1. Filter each projection with a separate filter for each depth in the image frame. For illustration purpose, only nine depths are chosen in Figure 3 and hence only nine different filters need to be applied to the diffracted data. Step. For each pixel (x, y) on the image frame, a value of the filtered projection that corresponds to the nearest depth line is allocated. Step 3. The preceding two steps are repeated for all projections. As a new projection is taken up, its contribution is added to the current sum at pixel (x, y). Diffracted projection η 1 η η 3 η 4 η y η Figure 3: Filtered Backpropagation Technique x Image frame 7

8 For each η constant line, the diffracted projection must be filtered with a transfer function. The reconstructed equation of the object can be expressed as π 1 o( x, y) = φ( xsinφ y cosφ, π where Π φ is the filtering operation on the projection. 6. Discussions 0 xsinφ + y cosφ) dφ (4) 6.1 Limitations There are many factors that limit the accuracy and precision of diffraction tomography reconstruction. These limitations are caused by both approximations that must be made in the derivation of the reconstruction process and experimental factors Mathematical Limitations The Born and Rytov approximation limit imposes severe mathematical limitations. These approximations are fundamental in the reconstruction process and limit the range of the object that can be examined. One major restriction is that in the forward process of diffraction tomography, it is necessary to assume that the object is weakly scattering in order for the approximation to be valid. Within limits of the Fourier Diffraction Theorem, simulation studies have shown that the reconstruction of a 1λ object (or smaller) with a small refractive index are similar for both approximations and correspond closely to the simulated object. However, the two approximations differ for objects that have a large refractive index or have a large radius. Born approximations are generally good for objects with large refractive index provided the phase shift of the incident field is less than π radians. The Rytov approximation is very sensitive to the refractive index of the object but produces excellent reconstruction even for objects as large as 100λ. Simulation studies also show that Rytov approximation is numerically superior to the Born approximation in the context of noise-free measurements. Geometrically, the Born approximation is expected to be poor when the scattering object is very extended and the Rytov approximation may be expected to be compromised when the scattering object has sharp edges. Depending on the noise level in the measurements, the Born approximation may prove to be simpler to implement. This is because the Born approximation only requires the measurement of the complex amplitude of the scattered field while the Rytov approximation requires unwrapped phases of the scattered field. Higher order of the Born or Rytov series does provide a better model of the scattering process but reconstruction algorithms based on these series have yet to be developed. Nonlinearity in the higher series could be a serious complication in formulating the reconstruction procedures Experimental Limitations In addition to the restrictions imposed on the reconstruction by the Born and Rytov approximation, the following experimental limitations must also be taken into account. A. Limitations caused by ignoring evanescent waves. Evanescent waves have a complex wave-number and they attenuate rapidly over a short distance of only a few wavelengths. This limits the highest received wave-number to K max = π/λ. It also imposes a serious limit on the propagation process and can only be improved by using a higher frequency (hence shorter wavelength). B. Effects of finite sampling intervals. After the wave has been scattered by the object and propagated to the receiver line, it must be measured. It is not possible to sample every single point on the receiver line and so a non-zero sampling interval must be chosen. This introduces measurement errors into the reconstruction process. C. Effects of a finite receiver line. There is a limitation on the amount of data that can be collected because samples of the received waveform will be collected only at finite points on the receiver line. This is 8

9 complicated by the fact that spatial frequencies of the scattered wave vary with position along the receiver lines. The highest measured frequency is inversely proportional to the distance of the receiver line from the point of scatter and directly proportional to the length of the receiver line. A balance must be established so that the unmeasured data can be safely ignored. D. Limited view of the object. It is not possible to generate and receive plane waves in all directions and this creates voids in regions where there is no estimate of the Fourier transform of the object that will ultimately degrade the reconstruction process. 6. Comparison of Reconstruction Techniques In terms of simplicity, the NN and the bilinear interpolation methods are comparatively much simpler than the filtered backpropagation method. Without considering zero-padding, for most N x N reconstruction, the filtered backpropagation method will require approximately N FFT as compared to 4N FFT for the interpolation technique. Thus, the filtered backpropagation method is computationally very demanding and the processing time for the image generated by the filtered backpropagation method is expected to be much longer than that of the interpolation method. The quality of the image produced by the bilinear method is comparable to that obtained by the filtered backpropagation method. The image from the NN method is comparatively less satisfactory with less distinct boundaries, poorer contrast and presence of artifacts. Artifacts are also present in the image from the filtered backpropagation method and the bilinear method, though they are much lesser, particularly in the latter. A sharper, clearer image can be obtained by zero-padding techniques (increasing sampling density) for the interpolation method and the additional computational overhead is relatively small. For the filtered backpropagation method, modifications can be made to the filter function such that localised accuracy can be obtained in the image. At the expense of poorer overall image quality, computational time is drastically reduced. 7. Conclusions The promising nature of ultrasound in medical applications such as breast screening is largely due to the fact that it is safe, non-invasive and is sensitive to soft tissue types. However, ultrasound does not travel along straight paths within an object such as the human body and thus reconstruction methods that take into account the diffraction effects of ultrasound have to be employed in CT. There are also several limitations associated with such reconstruction methods in terms of mathematics and procedural set-up and all of them have to be taken into consideration if high quality and accurate image is desired. References: [1] P.A. Andre, H.S. Janee, P.J. Martin, G.P. Otto, B.A. Spivey, and D.A. Palmer, High Speed Data Acquisition in a Diffraction Tomography System Employing Large-Scale Toroidal Arrays, John Wiley and Sons Inc, [] S.X. Pan S.X. and A.C. Kak, A Computational Study of Reconstruction Algorithms for Diffraction Tomography : Interpolation Versus Filtered Backpropagation, IEEE Trans. Acoustic, Speech and Signal Processing, Vol. ASSP- 31, 1983, pp [3] A.C. Kak, Principles of Computerised Tomographic Imaging, IEEE Press, [4] S. Leeman, A.J. Healey, M. Betts and J.P. Jones, Diffraction Tomography Revisited, Plenum Press, [5] T.C. Leondes, Medical Imaging, System Techniques and Applications- Computational Techniques, Gordon and Breach Science Publishers, [6] A. Macovski, Medical Imaging Systems, Prentice Hall, [7] H.J. Lim and C.H. Wong, Acoustical Chaotic Fractal Images for Medical Imaging, School of Electronic and Electrical Engineering,

10 Appendix A Measured forward scattered field ξ η y U S, φ (κ) η = l x Object Illuminating wave plane s 0 φ Figure A1: An arbitrary object being illuminated by a plane wave. A -D object is being illuminated by a plane wave propagating along a unit propagation vector s 0 as shown in Figure A1 below. The object is represented as a distribution function o(x, y) in space and it has a -D Fourier transform jπr = o( r) e dr O( w) (1) where w =(u, v) and r = (x, y) The illuminating plane wave is assumed to be monochromatic and the object is assumed to be immersed in a water medium. In the absence of the object, the plane wave propagate through the water medium obeying the following equation: ui(r) = U 0 e -jk0s0.r () where U 0 is the complex amplitude of the illuminating wave at the origin of the space domain; K 0 is the wave number associated with the water medium and is equal to π/λ where λ is the wavelength. The total field, u(r) at any position can be modelled as the superposition of the incident field, u i (r) and the scattered field, u s (r). u(r) = u i (r) + u s (r) (3) When the illuminating wave is incident at an angle φ as shown in Figure A1 above, a rotated co-ordinate system (ξ,η) is set up such that the η axis coincides with the unit propagation vector s 0 and measure the scattered field along the line η = l. The incident and scattered field will now be denoted by u i,φ (r) and u s,φ (r) respectively. The inhomogeneous Helmholtz equation must be solved in order to obtain a relationship between the scattered field and the scattering object. ( + k 0 )u(r) = -k 0 o(r)u(r) (4) It is impossible to solve the equation exactly. Under the assumption that the inhomogenities 10

11 in object is weakly scattering i.e. u s (r) << u i (r), the Born approximation may be used which consist of replacing u by u i on the right hand side of (4). The Fourier transform of the scattered fields measured along line η = l is U U s, φ s, φ -jκξ ( κ) = us, φ( ξ)e dξ k0 U ( κ) = jγ 0 e jγl Q φ( κ) (5) which is related to the Fourier transform of the (6) object by where γ = (k 0 - κ ) ½ (7) and Q ( κ) = φ o( r)e j[ κξ+ ( γ k 0) η] dr (8) is the Fourier transform of the object evaluated along semicircle of radius k 0 centred at k 0 s 0 as shown in Figure A below. v k 0 s κ k 0 k 0 s u -k 0 s 0 s 0 φ Figure A: The thick lined arc is the k 0 semicircle. The centre of this semicircle is at -k 0 s 0 and the ensemble of all semicircles for different s 0 creates the k 0 disk. An alternative way is to solve (4) using the Rytov approximation under the assumption that the inhomogenities in the object are weakly scattering. For Rytov approximation, the total field is expressed as u(r) = U 0 ejk0s0.r + ϕs(r) (9) The scattered field is taken into account with an additional term in the complex phase. Fourier transform of the scattered field ψ s (r) ϕ ( κ) = s, φ ϕ s, φ ( ξ)e - jκξ dξ measured on line η = l is which is related to the Fourier transform of the object by ϕ s, φ k 0 U ( κ) = j γ 0 e j( γ k 0 ) l Q φ( κ) (10) (11) 11

12 For the Born approximation, jγ jγl Q φ( κ) = Q( κ, φ) = e k U0 For the Rytov approximation, jγ j( γ k Q φ( κ) = Q( κ, φ) = e k0 U0 Us, φ( κ ) 0 0) l ϕs, φ( κ ) In most cases the transmitted data will be uniformly sampled in space and a discrete Fourier transform of these sampled data will generate uniformly spaced samples of U S,φ (κ) in the κ domain (Note that U S,φ (κ) represents the Fourier transform of the transmitted data when the object is illuminated by a plane wave travelling in the positive y direction and φ indicates the illumination angle). Since Q φ (κ) is the Fourier transform of the object along the circular arc and since κ is the projection of a point on the circular arc on the tangent line CD, the uniform samples of Q φ in κ will translate into non-linear samples along the circular arc AOB as shown in Figure A3 below. v Sampling along the arc (1) (13) transformation between the (κ, φ) parameters and the (u, v) co-ordinates. Grids generated by portion OB and AO will be denoted by (κ 1, φ 1 ) and (κ, φ ) respectively. For grids generated by the former, κ varies from 0 to K 0 and φ varies from 0 to π. For grids generated by the latter, κ varies from -K 0 to 0 and φ varies from 0 to π. For portion OB, (u, v) is first expressed in polar co-ordinates (w, θ) as shown in Figure A4. w = (u + v ) ½ (14) θ = arc tan (v/u) (15) As shown in Figure B, angle β 1 is the angular position on the arc of the point having parameter κ 1. The relationship between β 1 and κ 1 is κ 1 = k 0 sin β 1 (16) The relationships between the polar coordinates (w, θ) and the parameters β 1 and κ 1 are β 1 = arc sin (w/k 0 ) (17) φ 1 ½β 1 = θ + ½π (18) k u κ1 = k0 sin ( arc sin u + v k 0 ) (1 Figure A3: Uniformly sampling the projection in the space domain leads to uneven spacing of samples along the circular arc. Each point on arc AOB will be designated by its (κ, φ) parameters. The rectangular coordinates in the frequency domain will be denoted by (u,v). As φ is varied from 0 to π, arc AOB generates a double coverage of the frequency domain that is undesirable when establishing a 1 to 1 v u + v π φ1 = arctan + arcsin + (0) u k0 From (14) (18), the following transformation equations between (κ 1, φ 1 ) and (u, v) are obtained. Similarly using the same approach, the following transformation equations between (κ, φ ) and (u, v) are also obtained. κ = - k0 sin ( arc sin v φ = arctan arcsin u u + v k0 u + v k0 ) (1) 3π + () 1

13 v k 0 (u, v) s 0 φ 1 κ 1 θ w u k 0 β 1 Figure A4 Relationship between the (κ 1, φ 1 ) parameters and the (w, θ) polar co-ordinate system. 13