This chapter provides an overviews the statistical decision theory and the associated objective image quality assessment. It focuses on the behavior of the noise propagation in computed tomography (CT) imaging systems, with and without in-line phase-contrast. The chapter analyzes the effect on noise of a widely used phase retrieval approach, based on the Transport of Intensity equation (TIE). It presents an approach, based on the noise power spectrum formalism, for quantifying noise in phase retrieved X-ray radiographs, as well as in phase-contrast computed tomography. Using this approach, in-line phase-contrast imaging in combination with a popular TIE-Hom phase retrieval algorithm has been analyzed and compared with conventional imaging in terms of the noise in the reconstructed projections and CT images. A gain factor has been introduced in order to evaluate the improvement of image quality, in terms of the variance of noise, due to phase retrieval.

We perform a theoretical analysis of the mathematical stability and locality of several modes of amplitude and phase contrast computed tomography (CT) suitable for reconstruction of the 3D distribution of complex refractive index in samples displaying weak absorption contrast. We present a general formalism for CT reconstruction in linear shift-invariant optical systems. Examples of such systems include propagation-based and analyser-based CT. We obtain general formulae for CT reconstruction from analyser-based projection data. We also propose a new tomographic algorithm for the reconstruction of the 3D distribution of complex refractive index in a sample from a single propagation-based projection image per view angle, where the images display both absorption and phase contrast. The method assumes that the real and imaginary parts of the refractive index are proportional to each other. Using singular-value decompositions of the relevant operators we show that, in contrast to conventional amplitude-contrast CT, phase-contrast (diffraction) tomography is mathematically well-posed. The presented results are pertinent to biomedical imaging and non-destructive testing of samples exhibiting weak absorption contrast.

Results are presented of a feasibility study of three-dimensional X-ray tomographic mammography utilising in-line phase contrast. Experiments were performed at SYRMEP beamline of Elettra synchrotron. A specially designed plastic phantom and a mastectomy sample containing a malignant lesion were used to study the reconstructed image quality as a function of different image processing operations. Detailed evaluation and optimization of image reconstruction workflows have been carried out using combinations of several advanced computed tomography algorithms with different pre-processing and post-processing steps. Special attention was paid to the effect of phase retrieval on the diagnostic value of the reconstructed images. A number of objective image quality indices have been applied for quantitative evaluation of the results, and these were compared with subjective assessments of the same images by three experienced radiologists and one pathologist. The outcomes of this study provide practical guidelines for the optimization of image processing workflows in synchrotron-based phase-contrast mammo-tomography.

X-ray computed tomography (CT) is an important diagnostic tool in medicine as well as being an essential research technique for animal imaging and bioscience. The key aim of this study is to assess the effectiveness, in both simulation and experiment, of differentiating soft tissue from bone as well as bone densitometry, using energydiscriminating X-ray detection. Polychromatic sources, such as standard X-ray tubes, can produce similar CT numbers for materials with different compositions, making differentiation and quantification of tissue and bone extremely challenging. In addition, 'beam-hardening' which occurs due to the relative increase in the attenuation of low energy photons compared to high energy photons, can create significant CT artifacts. To improve material contrast and eliminate beam hardening, a number of different approaches have been developed. These include dual-energy CT using two different X-ray tube voltages, photon beam filtration, and post-processing of the data. Here we present an alternative approach using the photon counting PiXirad detector. Simulations are used to establish optimal parameters for data acquisition. This is followed by tomographic experiments performed on a phantom and a mouse embryo. The energy discriminating properties of the detector are exploited to avoid beam-hardening artefacts, to differentiate soft-tissue and bone within the mouse embryo, and to quantify bone density. Compared with polychromatic CT using an integrating detector this approach yields a number of significant advantages for materials specific imaging and quantification.

We investigate a variant of the reconstruction technique for the in-line X-ray phase-contrast tomography data. This technique uses a newly introduced quantity, which represents a particular combination of the real and imaginary parts of the complex refractive index n. This quantity coincides with the real part of (1-n) in the case of objects having negligible absorption. The advantage of the proposed approach is in the significantly simplified form of the reconstruction algorithm for the introduced quantity. As demonstrated by our numerical experiments, the newly introduced quantity can be predictably associated with a particular refractive index.

Histopathological analysis is the current gold standard in breast cancer diagnosis and management, however, as imaging technology improves, the amount of potential diagnostic information that may be demonstrable radiologically should also increase. We aimed to evaluate the potential clinical usefulness of 3-D phase-contrast micro-computed tomography (micro-CT) imaging at high spatial resolutions as an adjunct to conventional histological microscopy. Ten breast tissue specimens, 2 mm in diameter, were scanned at the SYRMEP beamline of the Elettra Synchrotron using the propagation-based phase-contrast micro-tomography method. We obtained 1.2 μm pixel size images, which were analyzed and compared with corresponding histological sections examined under light microscopy. To evaluate the effect of spatial resolution on breast cancer diagnosis, scans with four different pixel sizes were also performed. Our comparative analysis revealed that high-resolution images can enable, at a near-histological level, detailed architectural assessment of tissue that may permit increased breast cancer diagnostic sensitivity and specificity when compared with current imaging practices. The potential clinical applications of this method are also discussed.

An uncertainty inequality is presented that establishes a lower limit for the product of the variance of the time-averaged intensity of a mode of a quantized electromagnetic field and the degree of its spatial localization. The lower limit is determined by the vacuum fluctuations within the volume corresponding to the width of the mode. This result also leads to a generalized form of the Heisenberg uncertainty principle for boson fields in which the lower limit for the product of uncertainties in the spatial and momentum localization of a mode is equal to the product of Planck's constant and a dimensionless functional which reflects the joint signal-to-noise ratio of the position and momentum of vacuum fluctuations in the region of the phase space occupied by the mode. Experimental X-ray synchrotron measurements provide an initial verification of the proposed theory in the case of Poisson statistics.

It is shown that in a broad class of linear systems, including general linear shift-invariant systems, the spatial resolution and the noise satisfy a duality relationship, resembling the uncertainty principle in quantum mechanics. The product of the spatial resolution and the standard deviation of output noise in such systems represents a type of phase-space volume that is invariant with respect to linear scaling of the point-spread function, and it cannot be made smaller than a certain positive absolute lower limit. A corresponding intrinsic 'quality' characteristic is introduced and then evaluated for the cases of some popular imaging systems, including computed tomography, generic image convolution and phasecontrast imaging. It is shown that in the latter case the spatial resolution and the noise can sometimes be decoupled, potentially leading to a substantial increase in the imaging quality.

In the present thesis several direct and inverse problems of X-ray in-line phase-contrast imaging and computed tomography are studied. A general method for finding the fundamental solution of the Helmholtz equation subject to Sommerfeld radiation conditions is developed. Unlike the established techniques, this method provides all solutions to the Helmholtz equation before selecting the one that satisfies the chosen valid boundary conditions. Sufficient conditions for the validity of Teague's method for solving the Transport of Intensity Equation are derived and an example of a solution, which cannot be obtained using this method, is provided. Teague's method is also applied to tomography for the reconstruction of the three-dimensional refractive index distribution in a generic sample from in-line X-ray projections. The proposed solution simplifies and stabilises the reconstruction process. A formula is derived for the single-step reconstruction of a newly introduced auxiliary function. This function contains information about both the absorption index and the refractive index decrement. The reconstruction is obtained directly from the intensity measurements, without the intermediate step of phase retrieval for each illumination angle. A precise relationship is established between the newly introduced function and the complex refractive index distribution. The physical meaning of this function is examined for phase objects and for generic objects with slowly varying distributions of absorption index. Some examples of possible applications of our results are discussed in Conclusions.