3D Imaging with Holographic Tomography

Abstract

There are two main types of tomography that enable the 3D internal structures of objects to be reconstructed from scattered data. The commonly known computerized tomography (CT) give good results in the x-ray wavelength range where the filtered back-projection theorem and Radon transform can be used. These techniques rely on the Fourier projection-slice theorem where rays are considered to propagate straight through the object. Another type of tomography called 'diffraction tomography' applies in applications in optics and acoustics where diffraction and scattering effects must be taken into account. The latter proves to be a more difficult problem, as light no longer travels straight through the sample. Holographic tomography is a popular way of performing diffraction tomography and there has been active experimental research on reconstructing complex refractive index data using this approach recently. However, there are two distinct ways of doing tomography: either by rotation of the object or by rotation of the illumination while fixing the detector. The difference between these two setups is intuitive but needs to be quantified. From Fourier optics and information transformation point of view, we use 3D transfer function analysis to quantitatively describe how spatial frequencies of the object are mapped to the Fourier domain. We first employ a paraxial treatment by calculating the Fourier transform of the defocused OTF. The shape of the calculated 3D CTF for tomography, by scanning the illumination in one direction only, takes on a form that we might call a 'peanut', compared to the case of object rotation, where a diablo is formed, the peanut exhibiting significant differences and non-isotropy. In particular, there is a line singularity along one transverse direction. Under high numerical aperture conditions, the paraxial treatment is not accurate, and so we make use of 3D analytical geometry to calculate the behaviour in the non-paraxial case. This time, we obtain a similar peanut, but without the line singularity. IMAGING IN HOLOGRAPHYPeople often regard holography as giving a three-dimensional (3D) image, but actually it has been known for many years that the 3D information in a holographic image is far from complete [1]. This is understandable because a holographic image contains only 2N 2 pixels, where N is the lateral width of the image and the 2 accounts for the modulus and phase information. It is difficult to see how complete 3D information could be recorded or stored on the 2D surface of a photographic film (for a thin hologram) or CCD detector. This information is sufficient to record the surface height of a bulk 3D object, but not the detail inside the 3D object. In consequence, a hologram is often described as providing a 2 1 / 2 D image. It is not desirable to refer to the precision of surface height measurement as "resolution" as resolution refers to resolving the image of two objects, and holography cannot measure the separation of two surfaces. Precision or sensitivity are acceptable descri...

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