Tomography refers to cross-sectional imaging of an object, either from trans-mission or reflection data collected by illuminating the object from varying directions. We consider parallel-beam tomography and cone-beam tomography systems in which images of an object are recorded at different angles over a 180-degree space using a 2-D detector. The object is penetrated by X-rays, which are attenuated as they pass through the material. The resulting inten-sity of each beam is recorded by the detector, and can be used to reconstruct a volume of the object. In parallel-beam tomography, the X-rays fired at an angle are each parallel to each other, and such penetrate the object from the same direction. In cone-beam tomography, the rays are divergent, forming a
cone spreading towards the imaging target, allowing faster scanning speed and higher resolution due to the cone geometry (Kumar et al. 2015).
In terms of correlated noise, X-ray microtomography, designed for imaging of very small objects, is especially interesting due to the high levels of streak noise in the projection data. In a microtomography setup, the measurement of the attenuated X-rays is commonly done by first converting the beam to visible light, then capturing the light with a digital camera. The conversion to light is done through a scintillator, from which the visible light is routed into the camera chip through an optics system. The process produces a 3-D stack of projections, which have the displacement of the pixel in two dimensions (i.e. the pixel coordinate of a single capture), and the angle of the capture as the third. A slice of this stack, containing each angle for a single pixel row of the detector is called a sinogram due to its sinusoid-like form arising from the rotation. (Stock 1999)
The optical attenuation of the X-rays through the sample is determined ex-perimentally via field corrections by two additional inputs, the bright-field and the dark-bright-field (Seibert et al. 1998). The bright-bright-field is an acquisition obtained by the imaging procedure with no sample, and the dark-field is ob-tained with no beam; both are 2-D arrays the size of effective pixels of the detector. The bright-field and dark-field corrections are applied as
P = Praw−ID
IB−ID , (2.32)
where Praw are the raw sinogram projections, andID andIB are the dark-field and and bright-field, respectively, both size of the 2-D detector and thus repli-cated across each angle. Furthermore, all processing is commonly done upon logarithmic transformation of the projection data due to the Beer–Lambert law relating the X-ray transform to the measured optical attenuation (Swinehart 1962).
The sinograms are usually not analysed directly; instead, a reconstruction algorithm is applied to construct a 3-D volume of the object of interest, ap-plied on the log-data. The standard algorithm for tomographic reconstruc-tion is filtered back-projection, although iterative reconstruction methods have gained popularity recently, including the use of convolutional neural networks (Schofield et al. 2020; Willemink and No¨el 2019). The filtered back-projection
consists of an inverse Radon transform (Radon 1917) mapping the line integrals of the sinograms into slices of the 3-D volume, and a ramp filter applied be-fore the transform to reduce blurring (Kak and Slaney 1988; Shepp and Logan 1974).
2.7.1 Noise in X-ray microtomography; sinogram streaks and ring artifacts
Errors in the measurement can come from multiple parts of the detector, such as non-linear response of the sensor chip or dust on lens or mirrors of the op-tics system. Still, the main source of defects is often the scintillator: it is very challenging to produce high-quality high-resolution scintillators sufficiently tol-erant to radiation; even then, long exposures may damage the micro-structures of the scintillator or release particles obstructing the surface (Vo et al. 2018).
Although the components may be cleaned or replaced, it can be costly, or impossible to resolve in the middle of an experiment.
Since each sinogram consists of multiple captures across different angles by the same setup across a time span, each of the aforementioned defects affecting the sensitivity of detection in either a single pixel or a wider area result in streak noise in the projections across the angular dimension. Apart from possible defective pixels of constant intensity, most streak sources cause a multiplicative shift in the pixel intensity. These errors tend to change very slowly, resulting in very long-range noise correlation across the angle. In the reconstructed volume, sinogram streaks will present as concentric circles or half-circles obstructing the signal. These circular noise patterns are called ring artifacts. (Artul 2013; Boas and Fleischmann 2012; Jha et al. 2013)
Due to their potentially large intensity, amplified by the reconstruction pro-cess, the attenuation of ring artifacts is often essential for interpretation and processing of the reconstructed volumes (Davis and Elliott 1997). They are particularly harmful for automatic systems such as defect detection, where ar-tifacts may both obstruct the defect but also otherwise mislead the algorithm.
Furthermore, algorithms such as segmentation are often very sensitive to dis-tortions caused by simple filtering of the artifacts (M¨unch et al. 2009; Sijbers and Postnov 2004). Another challenging setup is medical imaging of soft
tis-sue, where signal components can be particularly weak compared to the ring artifacts, and getting accurate pixel data can be essential (Croton et al. 2019).
Streak noise is not the only noise present in the sinograms; they are partic-ularly affected by Poissonian noise arising from the limited photon-counting.
Other artifact sources include intensity bias, and blurring due to movement, hardware constraints, or reconstruction (Schl¨uter et al. 2014). Beam-hardening, which is caused by preferential absorption of the low energy photons of the X-ray beam increasing the mean beam energy, may result in especially hard-to-remove artifacts in the reconstruction, such as cupping, streaks, dark bands, or flare artifacts (Barrett and Keat 2004).
2.7.2 Methods for ring artifact removal
As the bright-fielding procedure effectively performs normalization through constant pixel gain variations, it will often partly suppress streak noise. Still, the bright-fielding will usually not result in perfect streak correction (Davidson et al. 2003); the sensitivity of different components may change during the experiment, meaning that the streaks are not necessarily fully constant across the angular dimension. As the fields nevertheless leverage physically relevant data to obtain the attenuation, they are commonly applied on the sinograms as the first step of processing. To reduce the error in bright-fielding, the fields used for normalization are often computed by averaging multiple fields obtained e.g., both before and after the actual experiment.
Many denoising algorithms have been proposed specifically for ring arti-fact removal, either by attempting to remove the streak noise from the sino-grams, or by removing the ring artifacts directly from the reconstructions.
A very simple way of sinogram-based streak removal separates the streaks through averaging and a high-pass filtering (Kowalski 1978). More sophisti-cated methods include total-variation regularization (Salehjahromi et al. 2019), transform-domain methods utilizing either Fourier transform, or wavelet trans-forms (M¨unch et al. 2009; Raven 1998), and combinations of polynomial smooth-ing filters (Vo et al. 2018).
Algorithms which work in the reconstructed domain commonly apply polar coordinate transformations to convert the rings into streak-like artifacts. They
may then be removed with methods very similar to those executed in the pro-jection domain, e.g., subtracting median of each column with sufficiently small variance (Sijbers and Postnov 2004), combining median and low-pass filters (Prell et al. 2009), total variation regularization (Liang et al. 2017), smooth-ing decomposed images (Chen et al. 2009), and wavelet-decomposition with Fourier-domain filtering (Wei et al. 2013). Recently, neural networks have also been applied for ring removal (Chao and Kim 2019). Other approaches to ring artifact removal include calibration of detector response function (Croton et al.
2019), and iterative algorithms (Paleo and Mirone 2015) combining regularized reconstruction with denoising.
Although a variety of methods are successful in suppressing most ring arti-facts, many are prone to introducing new artifacts. In particular, many meth-ods tend to create new streaks around high-frequency edges (Vo et al. 2018), and large-scale distortions due to shifting the mean in areas with strong signal features.