UWB Tomography via Simulated Optical Systems #Ahmed Dia Hassanein1, David J. Edwards2, Christopher J. Stevens3

1,2,3 Engineering Science Department, University of Oxford, OX1 3PJ, UK. ahmed.hassanein@seh.ox.ac.uk

Abstract

Microwave imaging is being actively investigated for medical and process tomography. The formation of tomographic images from UWB signals is often problematic due to limitations arising from diffraction, reflection and refraction as well as attenuation. Many approaches to diffraction tomography rely on complex iterative approaches to solve the inverse problem of determining the object from scattered signals, or on some assumed foreknowledge of the dielectric of the environment / object being probed. Here we demonstrate both theoretically and experimentally how to apply a new method that achieves tomographic imaging without either of these problems. The method applies a forward propagation rather than back propagation and this requires no foreknowledge of the object’s properties or its environment.

Index Terms — Tomography, Frequency Domain Analysis, Image Reconstruction. 1. Introduction This paper addresses the issues of image formation from diffracting radiation which has been propagated through or scattered from, a target object. The recent interest in microwave imaging for early cancer detection have created a need to better establish image formation techniques for these diffracting waves [1]. Medical imaging by UWB radiation is currently receiving much attention as a potentially valuable medical diagnostic tool [2]. Driving these developments are several factors, primary being the potential for high contrast available between tissue types [2]. For typical human tissues the dielectric constants range from 4.4 (breast fat) to 64.5 (vitreous humor) and the conductivity between 0.3 (bone marrow) to 8.1 1.S m− (Cerebro-Spinal fluid) [3]. Of secondary, but significant importance is the relative safety of microwave based techniques as compared to x-ray based topographies. The high cost of magnetic resonance Imaging, Positron emission Tomography and x-ray techniques is another important factor in the development of this technology. Whilst conventional x-ray tomography is enabled by relatively simple back projection algorithms, when the radiation being used suffers diffraction simple ray methods break down. A method of image formation is required and is proposed in this paper which meets the following criteria:

1) No foreknowledge of the objects shape or composition is required to obtain good images. 2) No iterative solutions to the inverse problem for back-propagation should be required. 3) Cross sectional images should be formed using a single view direction rather than requiring

multiple views as in CAT techniques. 2. Method: The Virtual Microscope Transform Fig.1a shows a diagram of the basic concept of our imaging method. To form cross sectional images we first determine the focal length and location required for a lens to form an image on it's own axis,

of each point in the required object field ( ),x z′ ′ . We then position a virtual lens (not a real physical lens but a lens simulated in software) at this location. The wave fronts recorded, arriving from an object, in plane Ul are computationally propagated through this perfect thin lens (L1) and result in a new set of wavefronts in plane Ul'. The computation deals with the phase transformation through the lens which generates the focusing action [4]. The phase change for the waves passing through the lens

assumes the lens has no attenuation or reflection properties. To then obtain the image of point ( ),x z′ ′ the new wavefronts are propagated to the appropriate image plane of the lens and the complex amplitude of that point recorded. The propagation transformation from one plane to another uses the Huygens-Fresnel principal (using the first Rayleigh-Sommerfeld solution) [4]. The lens is moved

vertically to calculate the image of the object along the focal line(x-axis) and focal length adjusted to interrogate the depth planes (z-axis). By using this technique 3D tomographic images would be achieved from single or a small numbers of view directions. We use the phase transformation for a perfect thin lens and the Huygens-Fresnel principal for wave propagation to build an algorithm using matlab (TM). Two variants on the image formation algorithm were tested, both based on the same mathematical method. One is using a single frequency (monochromatic) to form images and the other is combining many frequencies over a wide bandwidth (UWB) into a single image. The UWB method combines images using incoherent summation investigated as described later.

3. Results and discussion

3.1. Monochromatic Imaging In order to test the algorithms, we generated two test objects from which to determine initial wavefronts for the method. The object planes used for our images contain either a single point source (shown in fig.2a) or an array of point sources. In either case, the point sources have uniform amplitude equal to one and they are either phase locked together, or in a later case, have phases determined by assuming they are scatterers reflecting signals from a separate point source illuminator located outside the field of view. Images are formed using a pixel dimension of 6x4mm in the horizontal plane. To obtain input wavefronts for testing the algorithm we forward propagate the object's wavefronts into a common plane following the same assumptions of the Huygens-Fresnel principal. The objects of fig.2a are used to generate test wavefronts in the input planes of lenses as shown. Images are calculated on the far side of the lens from the object. Images calculated for the object using monochromatic signals with a wavelength of 0.02m are shown in figs.2a and 2b. The single point image from the source shown in fig.2a is very clearly imaged in fig.2b with a conical halo surrounding it as a consequence of the defocusing of its image as the lens is scanned through the image space. The angle of this conical halo is the focal cone of the lens itself. Fig.3a shows the image computed under identical conditions for an array of nine phase locked sources arranged in a matrix of 3 by 3. This is a much less clear image than the single source of fig.2b with significant distortion and interference arising from the coherent nature of the sources being imaged. This is a general problem when imaging using coherent waves and is not easily solved unless a source of extremely low coherence can be obtained. With a single frequency this is impossible to avoid. Single frequency imaging is equivalent to photography using laser illumination where coherence of the signals leads to interference across the image. The coherent imaging problem using a single frequency can be addressed using broad band signals that are the microwave equivalent to photography using white light bulb illumination. Laser illumination will generate interference features across the image formed whilst the white light (ie non-coherent) illumination will not. The solution to this is presented in the next section.

3.2. Ultra Wide Band Imaging Using UWB signals where the fractional bandwidth is greater than 50% of the centre frequency, interference effects for each of the frequencies occur on a range of different spatial scales because of the range of available wavelengths. Consequently, by combining image information from each frequency, these interference effects can be reduced. In the subsequent figures, the wavelengths used range from 0.02 to 0.04m which corresponds to a UWB frequency range 7.5 to 15 GHz. The object plane used is an array of nine point sources arranged in a matrix of 3 by 3. In order to get a final image we must combine all the individual single frequency images to form a single multiple frequency image. In practice the phase of each point scatterer in an object will not be identical. In general one would illuminate the object with a single source outside of the imaged field and the phase of signals scattered by each point will vary according to the net optical path and secondary illumination from scattered signal between the illuminating source and each point. White light interference only forms constructive interference peaks when the phase of the signals sourced at each point are identical. If the points are illuminated remotely then the phase of the scattered signals varies with frequency and the coherent summation fails. One way to deal with this is compute the expected initial phase for each frequency at each point in the object plane but this would require foreknowledge of the optical path between the illuminator and each point. Since this is a requirement we would prefer to avoid we apply incoherent summation techniques instead. Incoherent summation can be applied using the equation:

( )2eiI a φ= ∑ Where I is the intensity of the final UWB image, a is the amplitude of the each point

considered and eiφ represents the phase of each point. Here the phase of each individual frequency

image pixel is lost on the summation. This is essentially the same process as is occurring in an optical camera using white light where the intensity is summed over all frequencies. Fig.3b shows the image formed using the point source array with 21 frequencies combined over a bandwidth from 7.5 to 15GHz. We can see an intensity of distortions and some black lines between the points of maximum intensity. Although fig.3b is clear enough to show the point sources imaged, the oval haloes between them degrade the clarity. The origins of these halos are the effective depth of field of the virtual lens that we have modeled. Using a virtual lens with a larger field of view would reduce the volume around each point contaminated by the halo by allowing the signal to spread more rapidly from the focal point. Alternatively, a larger field of view can be synthesized by recording a wavefront from another view point and then combining images recorded from the two different directions. As an example, the point source in fig.2a is imaged from an orthogonal angle to the one used in fig.2b. Multiplying the two orthogonal angles is suggested. Fig. 2c shows the result of multiplying two orthogonal views of the point source of fig. 2a. The distortions from both angles will weaken each other more than summation because small numbers (distortions) when multiplied produce much smaller numbers. Points of maximum amplitude (imaged point sources) from orthogonal angles when multiplied will produce much larger amplitudes. 4. Practical considerations to virtual confocal microscope technique Thus far we have shown that a method based on the principals of physical optics can generate cross sectional images of arrays of point sources. All the sources used to test our algorithms were phase locked as though excited by the same signal source with no apparent time delay between them. This is quite an unphysical situation and thus unlikely to be the case in a real imaging system. We tested our technique experimentally by replacing the point source with a scatterer (a single vertical metal rod) that is being illuminated by a remote vivaldi antenna outside of the field of view in an anechoic chamber as shown in fig.1b. This imaging system is composed of two parts: firstly the physical imager (fig.1b) consisting of an illuminating antenna (the Tx antenna), and a wavefront sensor (the RX antenna). Secondly, the virtual imager consists of the virtual optical system to generate the images. The virtual system has no real existence but is entirely simulated within a computer. A PC and Vector Network Analyzer (VNA) are used to send and receive data from the antennas in the chamber. The phase of signals originating at the scatterer is now a function of the distance between each and the illuminating antenna as well as of the wavelength of the illuminating radiation. There is also a phase accumulation between the signal source and the illuminating antenna through the cables connecting them and this adds an extra contribution to the phase of each point scatterer. Thus there can no longer be any kind of phase lock between the scatterers in either space or frequency terms. The frequency ranged used is 5 to 10 GHz divided into 1601 bins. In order to dela with this, we incoherently sum the amplitudes of the 1601 images formed to generate a final UWB image. The result is shown in fig.3c. In this fig. we have included the value of the cable length to be 11m where the phase of the illuminating antenna is varying with frequency as well as the spatial variation from the various object points to transmitter antenna. The image is surprisingly good showing a fairly clear location of the point source as an “X” shape with maximum amplitude at the center. The “X” shape matches the effective focal cone of our simulated optical system exactly. 5. Conclusion We have shown that a method of microwave image formation based on wave front sensing and physical optics techniques can be implemented using a simple algorithm. Our aim to achieve cross sectional imaging of an object field without using back propagation, using single view angles and without requiring foreknowledge of the objects environment has been mostly achieved. We have investigated and found that it is possible to form images utilizing UWB frequency ranges. The UWB approach is successful for incoherent method of image combination without knowledge of the

propagation characteristics of the environment and object plane. In a practical realization of this method incoherent summation of images gives us good result and limits the effect to the uncertain initial phase differences of each point in the object plane. For a single view angle we find that images contain a significant halo effect due to the focal cone of the lens used in the computations. Only simple objects were used in this paper to illustrate that the new method basically works.

Fig. 1 A diagram on the left (a) showing the basic concept of the proposed imaging method. The

illustration on the right (b) demonstrates how our method could be practically applied.

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shown is distorted by interference. (b) 21 images combined using incoherent summation. (c) 1601 images combined using incoherent summation over 5-10GHz. References [1] A. C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, 1988. [2] X. L. Elise C. Fear, Susan C. Hagness, Maria A. Stuchly, "Confocal Microwave Imaging for Breast Cancer Detection: Localization of Tumors in three Dimensions," IEEE Transactions on Biomedical Engineering, vol. 49, August 2002. [3] S. Gabriel, R. W. Lau, C. Gabriel, “The dielectric Properties of Biological Tissues: III. Parametric Models for the Dielectric Spectrum of Tissues”, Phys. Med. Biol., vol. 41, 1996, p. 2271-2293. [4] J. W. Goodman, Introduction to Fourier Optics, 2nd ed: McGraw-Hill International Editions, 1996.