Ring artifact and Poisson noise attenuation via volumetric multiscale nonlocal collaborative filtering of spatially correlated noise

Mäkinen, Y.; Marchesini, S.; Foi, A.

doi:10.1107/S1600577522002739

research papers

JOURNAL OF
SYNCHROTRON
RADIATION

ISSN: 1600-5775

Volume 29| Part 3| May 2022| Pages 829-842

https://doi.org/10.1107/S1600577522002739

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Ring artifact and Poisson noise attenuation via volumetric multiscale nonlocal collaborative filtering of spatially correlated noise

Ymir Mäkinen,^a ^* Stefano Marchesini ^b and Alessandro Foi ^a

^aTampere University, Finland, and ^bSLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
^*Correspondence e-mail: [email protected]

Edited by M. Yamamoto, RIKEN SPring-8 Center, Japan (Received 31 August 2021; accepted 10 March 2022; online 21 April 2022)

X-ray micro-tomography systems often suffer from high levels of noise. In particular, severe ring artifacts are common in reconstructed images, caused by defects in the detector, calibration errors, and fluctuations producing streak noise in the raw sinogram data. Furthermore, the projections commonly contain high levels of Poissonian noise arising from the photon-counting detector. This work presents a 3-D multiscale framework for streak attenuation through a purposely designed collaborative filtering of correlated noise in volumetric data. A distinct multiscale denoising step for attenuation of the Poissonian noise is further proposed. By utilizing the volumetric structure of the projection data, the proposed fully automatic procedure offers improved feature preservation compared with 2-D denoising and avoids artifacts which arise from individual filtering of sinograms.

Keywords: BM4D; noise; image processing; tomography; artifact.

1. Introduction

Computed tomography is commonly affected by streak noise in measured raw sinogram data (Jha et al., 2013 ; Artul, 2013 ; Boas & Fleischmann, 2012 ), which can be caused by mis-calibration of detector linear response, beam fluctuations, beam hardening, or dusty or damaged scintillator screens (Haibel, 2008 ; Vidal et al., 2005 ; Anas et al., 2010 ). Streak noise in projections causes ring artifacts in reconstructed volumes, which present as centered circles or half-circles (Croton et al., 2019 ). As the sinogram data are obtained through a photon-counting detector, the statistics of the measured raw data can be further modeled through a Poisson distribution, which may result in high levels of Poissonian noise, commonly attenuated within the reconstruction process through iterative approaches (Mohan et al., 2014 ; Venkatakrishnan et al., 2013 ).

Although ring artifacts can be reduced by scanning protocols (Pelt & Parkinson, 2018 ), high-quality scintillator screens and detectors, it is difficult to completely avoid them and therefore achieve highest-quality reconstruction solely by experimental measures, requiring algorithmic processing of the acquisitions.

Popular methods to reduce ring artifacts include wavelet-FFT filters (Münch et al., 2009 ), combinations of polynomial smoothing filters and careful calibration of the detector response function (Vo et al., 2018 ; Croton et al., 2019), smoothing filters with segmentation in the tomogram domain (Massimi et al., 2018 ), ring removal in the tomogram domain upon polar coordinates transformation (Sijbers & Postnov, 2004 ; Li et al., 2021 ), and iterative algorithms (Paleo & Mirone, 2015 ) that combine regularized reconstruction with denoising. Recently, in Mäkinen et al. (2021), we proposed effective ring artifact attenuation through sinogram-domain collaborative filtering, presenting a multiscale architecture with a Block-matching and 3-D filtering (BM3D) image denoiser for correlated noise (Dabov et al., 2008 ; Mäkinen et al., 2020 ) at the core of the process. To the best of our knowledge, Mäkinen et al. (2021) offers state-of-the-art results in ring attenuation. In particular, it does not cause new artifacts around strong signal features, common to other popular ring removal algorithms. However, being based on a filter for 2-D data¹, applied to individual sinograms, it may cause discontinuities across the third dimension.

In this work, we address both streak reduction and Poissonian noise removal from volumetric stacks of projections. The contribution of this work is threefold: (1) We propose a multiscale streak denoising framework for denoising of volumetric data. In particular, this framework can be seen as an extension of Mäkinen et al. (2021) for filtering of 3-D volumes. (2) After streak noise removal, and before reconstruction, we embed a distinct multiscale denoising step to attenuate the Poissonian noise component of the projections. This allows to apply the reconstruction process using milder regularization and improve the tradeoff between noise reduction and artifact suppression. (3) As a general-purpose algorithmic contribution, the filter used at the core of the multiscale denoising process is an improved version of the BM4D (Maggioni et al., 2012 ) volumetric denoising algorithm. The included enhancements, discussed in Appendix A, allow the long-range noise correlation which characterizes the streaks to be dealt with.

The proposed filtering procedure for both streaks and Poissonian noise is fully automatic and includes self-calibration of the filtering strength. We demonstrate the denoising performance of the proposed approach on real data from the table-top Prisma XRM microCT at Sigray, and from the synchrotron-based microCT at the Advanced Photon Source (APS) in Argonne, available through Tomobank (De Carlo et al., 2018 ).

2. Bright-field normalization

The following normalization of the raw projections and the streak model upon a logarithmic transformation follow that of Mäkinen et al. (2021).

The optical attenuation through the sample is determined experimentally via bright-field corrections through two separate acquisitions, the bright-field and the dark-field (Seibert et al., 1998 ). The bright-field is obtained by the imaging procedure with no sample, and the dark-field is obtained with no beam; both are 2-D arrays the size of effective pixels of the detector. The Beer–Lambert law further relates the X-ray transform through the sample to the optical attenuation by a logarithmic transformation (Swinehart, 1962 ).

Hence, the raw projections P_raw are first normalized as

$[P_{{\rm norm}} = {{P_{{\rm raw}}-I_{{\rm D}}} \over {\,I_{{\rm B}}-I_{ {\rm D}}\,}}, \eqno(1)]$

where I_D is the dark-field and I_B is the bright-field², and then log-transformed as

$[Z = \ln\,\left(P_{{\rm norm}}\right). \eqno(2)]$

2.1. Noise model for normalized projections

Apart from possible completely defective detectors³ we treat the variation in detector response as normally distributed; as such, the streak noise will follow a normal distribution. Furthermore, we model the streak noise as locally stationary, meaning that the variance is presumed constant within the support of the denoising filter. Note that this does not mean that the noise is i.i.d. or white, as it is instead characterized by very long range correlation presenting as streaks.

As the data are obtained through a photon-counting detector, the statistics of the measured raw data can be further modeled through a Poisson distribution with nonstationary parameters after the bright-fielding.

Given the premises of normally distributed streak noise and Poissonian noise, noise in projections normalized by (1) can be formally written as

$[P_{{\rm norm}} = A\left(1+\eta_{{}_{{\rm{P}}}}\right)+\pi = A+A\eta_{{}_{{\rm{P}}}}+\pi, \eqno(3)]$

where A are the underlying noise-free projections, $[\eta_{{}_{{\rm{P}}}}]$ is the normally distributed streak noise component, and π is Poissonian noise which we model as white and zero-mean; all components of (3) are considered as 3-D arrays and multiplications are elementwise.

The natural logarithm of (2) acts as a variance-stabilizing transformation (VST) for the multiplicative noise component $[(1+\eta_{{}_{\rm{P}}})]$ . Hence, we have

$[Z = \ln\left(P_{{\rm norm}}\right) \approx \ln\left(A+{{\pi}\over{1+\eta_{{}_{{\rm{P}}}}}}\right)+\eta_{_{\rm{P}}}, \eqno(4)]$

where the approximation comes from $[\ln(1+\eta_{{}_{{\rm{P}}}})]$ $[\approx]$ $[\eta_{{}_{{\rm{P}}}}]$ .

3. Correlated noise

The denoising is conducted in two steps. First, we aim to estimate the `streak-free' projections

$[{Y = \ln\left(A+{{\pi}\over{1+\eta_{{}_{{\rm{P}}}}}}\right)}, \eqno(5)]$

which are corrupted by white Poissonian noise. Then, as a separate denoising step, we consider the attenuation of the remaining noise originating from π.

Throughout this work, we will represent the volume to be filtered according to the correlated noise model presented in the following subsection. This model will assume different meaning at different parts of the algorithm. First, applying locally to the streaks as a type of long-range correlated noise; second, to the noise arising from the Poissonian component π.

3.1. Correlated noise model

We consider the noisy input $[{z\!:\!X\!\rightarrow\!\bb{R}}]$ to be a combination of underlying data y and additive stationary spatially correlated noise η to be filtered,

$[z(x) = y(x)+\eta(x),\qquad x\in X, \eqno(6)]$

where $[{x\!\in\!X\!\subset\!{\bb{Z}}^{3}}]$ is the coordinate in the finite three-dimensional volumetric domain X and

$[\eta = \nu\,{\bigcirc\kern-9pt\ast}\,\,g,\qquad\nu\,\left(\cdot\right)\sim{\cal N}\left(0,1 \right), \eqno(7)]$

ν being zero-mean i.i.d. Gaussian noise with unit variance, and $[{\bigcirc\kern-9pt\ast}\,]$ denoting 3-D convolution with the kernel g. The kernel g defines the spatial correlation of the noise as well as the noise strength, with $[\|g\|_{2}]$ = $[{\rm{std}}\{\eta\}]$ . An equivalent way of representing correlated noise is by its power spectral density (PSD) Ψ,

$[\Psi = {\rm{E}}\left\{\left|{\cal{F}}\left[\eta\right]\right|^{2}\right\} = { \rm var}\left\{{\cal{F}}\left[\eta\right]\right\} = \left|X\right|\left|{ {\cal{F}}\left[g\right]}\right|^{2}, \eqno(8)]$

with $[{\cal{F}}]$ being the 3-D Fourier transform, and |X| denoting the cardinality (i.e. number of elements) of X. Equivalently, a kernel g satisfying (7)–(8) can be defined from Ψ as

$[g = |X|^{-1/2}{\cal{F}}^{\,-1}\left[{{\rm{std}}\left\{{\cal{F}}\left [\eta\right]\right\}}\right] = |X|^{-1/2}{\cal{F}}^{\,-1} \left[ \sqrt{\Psi} \,\right]\,. \eqno(9)]$

3.2. Estimation of noise standard deviation

When applying the above model to noisy data, it is essential to have knowledge of either the kernel g or, equivalently, the PSD Ψ, as they fully characterize the noise. Assuming g in (7) is known modulo a scaling factor ς from a known kernel g_s, the noise estimation can be simplified to estimating ς. In particular, in order to model the streak and Poissonian noise components arising from the particular composition of noise given in (3), the kernels g_s should induce either very long range correlation or near white noise across each dimension d. The estimation procedure can be built as a direct extension of the one adopted by Mäkinen et al. (2021) to 3-D. To reduce the signal-to-noise ratio (SNR) to acquire a better noise estimate, we convolve z with a 3-D anisotropic kernel g_d that provides either low-pass or high-pass filtering along different dimensions; g_d is designed based on the noise statistics so that it preserves the noise component of interest while attenuating signal contrast. Specific instances are given in Section 4.1.1 and Section 5.3. One can then compute an estimate of the standard deviation of $[{\eta\,{\bigcirc\kern-9pt\ast}\,\,g_{{d}}}]$ via its median absolute deviation (Hampel, 1974 ),

$[\widehat{{\rm{std}}\left\{\eta\,{\bigcirc\kern-9pt\ast}\,\,g_{{d}}\right\}} = 1.4826\,\mathop{\rm{smed}}_{{X}} \left[\big|z{\bigcirc\kern-9pt\ast}\,\,g_{{d}} - \mathop{\rm{smed}}_{{X}}\left(z{\bigcirc\kern-9pt\ast}\,\,g_{{d}}\right)\big|\right], \eqno(10)]$

where smed denotes the sample median and the factor 1.4826 calibrates the estimate with respect to a normal distribution of the noise. As $[{{\rm{std}}\{\eta\,{\bigcirc\kern-9pt\ast}\,\,g_{{d}}\}}]$ = $[\|\varsigma g_{{s}}{\bigcirc\kern-9pt\ast}\,\,{g_{{d}}}\|_{2}]$ , an estimate $[\hat{\varsigma}]$ of ς can be obtained through

$[\hat{\varsigma} = \|g_{{s}}{\bigcirc\kern-9pt\ast}\,\,{g_{{d}}}\|_{2}^{-1} \, \widehat{{\rm{std}}\left\{\eta\,{\bigcirc\kern-9pt\ast}\,\,g_{{d}}\right\}}. \eqno(11)]$

4. Multiscale streak filtering

In the following, we treat the first dimension of the stack of projections as the angular dimension, and the second and third as the horizontal and vertical displacement dimensions.

Because the streaks are inherently low-frequency with respect to the angle, they are filtered entirely at a coarse angular scale; for this task, we extend the multiscale procedure of Mäkinen et al. (2021). The main changes in the proposed procedure arise from replacing the one-dimensional binning operations along the displacement dimension with corresponding 2-D binning operators $[{\cal B}_{2{\rm D}}]$ and $[{\cal B}_{2{\rm D}}^{\,-1}]$ executed across both displacement dimensions. Furthermore, instead of using a direct 3-D extension of the 2-D streak PSD, we adjust the streak model to account for possible long correlation also along the displacement dimensions.

In detail, the multiscale streak attenuation procedure proceeds as follows. We begin by an angular binning $[{\cal B}_{{\alpha}}]$ . The result of the angular binning Z₀ = $[{\cal B}_{{\alpha}}(Z)]$ is binned K times through $[{\cal B}_{2{\rm D}}]$ to obtain Z_K = $[{\cal B}_{2{\rm D}}^{\,K}(Z_{0})]$ . The size of each binned volume is a quarter of the input size. Then, we process each scale in a coarse-to-fine fashion, where progressively for each k = $[\!K,{K-1},\ldots,2,1,0]$ , we obtain an estimate $[\hat{Y}_{k}]$ of $[{\cal B}_{2{\rm D}}^{\,k}\left(Y_{0}\right)]$ = $[{\cal B}_{2{\rm D}}^{\,k}[{\cal B}_{{\alpha}}\left(Y\right)]]$ . We start by taking as noisy input Z^*_K of BM4D the smallest binned volume Z_K; in this way, we obtain from Z^*_K = Z_K the coarsest estimate $[\hat{Y}_{K}]$ , which is taken as initialization for the following recursive steps executed for each scale k = $[{K\!-\!1},\ldots,0]$ :

(1) Replace the coarser-scale components of Z_k by those of the estimate $[\hat{Y}_{k+1}]$ :

$[\eqalign{ Z^{\,*}_{k} & = Z_{k}-{\cal B}^{\,-1}_{2{\rm D}}\big[{\cal B}_{2 {\rm D}}(Z_{k})\big] + {\cal B}^{\,-1}_{2{\rm D}}(\hat{Y}_{k+1}) \cr& = Z_{k}-{\cal B}^{\,-1}_{{\rm 2D}}\left(Z_{k+1}-\hat{Y}_{k+1 }\right)\,.}]$

(2) Denoise Z^*_k with BM4D to produce the estimate $[\hat{Y}_{k}]$ .

Finally, we replace the coarse angular components of the full-size stack Z with those from the finest scale estimate $[\hat{Y}_{0}]$ ,

$[\displaystyle\hat{Y} = Z-{\cal B}_{{\alpha}}^{\,-1}\big[{\cal B}_{{\alpha}}(Z)\big]+{\cal B}_{{\alpha}}^{\,-1}(\hat{Y}_{0}). \eqno(12)]$

4.1. Multiscale noise model

For BM4D denoising, we regard Z^*_k of each scale k as z of the model (6), as

$[\displaystyle Z^{\,*}_{k} = {\cal B}_{2{\rm D}}^{\,k}(Y_{0})+\eta_{k}^{\,*}, \eqno(13)]$

where

$[\eta_{k}^{\,*} = \left\{\matrix{\eta_{K},\hfill & k = K,\hfill \cr \eta_{k}-{\cal B}_{2{\rm D}}^{\,-1}\big[{\cal B}_{2{\rm D}} \left(\eta_{k}\right)\big],\hfill & k\,\lt\,K,\hfill }\right. \eqno(14)]$

and $[\eta_{k}]$ = $[{\cal B}_{2{\rm D}}^{\,k}(\eta_{0})]$ = $[{\cal B}_{2{\rm D}}^{\,k}[{\cal B}_{{\alpha}}(\eta_{P})]]$ .

This definition for $[\eta_{k}^{\,*}]$ , $[{k\,\lt\,K}]$ , follows from considering the coarser-scale estimate $[\hat{Y}_{k+1}]$ as perfectly denoised. Similar to (8), $[{\eta_{k}^{\,*}}]$ is treated as correlated noise with PSD,

$[\Psi_{k}^{\,*} = {\rm var}\left\{{\cal{F}}\left[\eta^{\,*}_{k}\right]\right \} = |X_{k}|\left|{\cal{F}}\left[g_{k}^{\,*}\right]\right|^{2},\qquad K\geq k \geq 0\,, \eqno(15)]$

where g_k^* is a correlation kernel and |X_k| is the pixel size of Z_k. As per (9), the kernel g_k^* can be defined as

$[g_{k}^{\,*} = |X_{k}|^{-1/2} {\cal{F}}^{\,-1}\left[{{\rm{std}}\left\{\cal{F}\left[\eta^{\,*}_{{\rm k}}\right]\right\}}\right]\,. \eqno(16)]$

4.1.1. Adaptive parametric model of $[\Psi_{k}^{\,*}]$

We note that, in addition to the approximately white streak noise, the sinograms may contain streaks with very long range correlation across the displacement dimensions. As this correlation is aligned along the detector axis, it is not clearly observable in individual sinograms, but may create significant noise structure in the full volume. Hence, we approximate the streak noise η₀ through three angularly constant streak noise components distinct in the displacement,

$[\eta_{0} = \eta_{0,w}+\eta_{0,u}+\eta_{0,v}, \eqno(17)]$

where η_0,w is streak noise white across both displacement dimensions, η_0,u is streak noise constant across horizontal displacement, and η_0,v is streak noise constant across vertical displacement.

Let us denote by $[{\eta_{0,p}\!\in\!\{\eta_{0,w},\eta_{0,u},\eta_{0,v}\}}]$ a noise component of η₀. For each η_0,p, we can define a respective scaled correlation kernel

$[\varsigma_{0,p}g_{0,p} = |X|^{-1/2}{\cal{F}}^{\,-1}\left[{{\rm{std}}\left\{ {\cal{F}}\left[\eta_{0,p}\right]\right\}}\right], \eqno(18)]$

where $[\varsigma_{0,p}]$ = $[{\rm{std}}\{\eta_{0,p}\}]$ , and $[\|g_{0,p}\|_{2}]$ = 1. Example realizations, kernels, and PSDs for each of these components as well as η_k are shown in Fig. 1 (top).

Figure 1
Top: example noise η_0,p, the corresponding kernels g_0,p and the root PSD $[{|{\cal{F}}\,[g_{0,p}]|}]$ for each noise component in (17)

with $[\varsigma_{0,w}]$ = 6, $[\varsigma_{0,u}]$ = 5, and $[\varsigma_{0,v}]$ = 8, as well as example noise, kernel, and root PSD corresponding to the compound noise η_k. Bottom: example noise, the corresponding kernels, and root PSD of the corresponding binning residuals $[\eta_{k,p}^{\,*}]$ . For all visualizations, the angular dimension of the data is the vertical dimension in the figure. The DC corner of the Fourier spectra is marked by a circle. Note that all root PSDs are nonzero only on the angular DC plane, and the kernels and the noise consist of repeated planes across the angle.

We note that each $[\eta_{k,p}]$ = $[{\cal B}_{2{\rm D}}^{\,k}(\eta_{0,p})]$ is characterized by a kernel $[\varsigma_{k,p}g_{k,p}]$ = $[2^{k}\varsigma_{0,p}g_{0,p}]$ . In particular, this property arises from the noise structure of the corresponding components: as $[{\cal B}_{2{\rm D}}]$ operates through summation and elimination of adjacent pixels, the operation preserves both noise whiteness and constant noise. The ratio 2^k follows from the summation along two dimensions, meaning that the variance of the coarser scale is four times that of the finer scale.

Disregarding the specific support size of their actual finite realizations, we can identify the stationary random fields as

$[\displaystyle\eta_{k,p} = 2^{k\,}{\rm{std}}\{\eta_{0,p}\}\,\eta_{{\rm G},p} \eqno(19)]$

where η_G,p is noise characterized by g_k,p, and hence $[{\rm var}\{{\eta_{{\rm G},p}}\}]$ = 1. We can then express the residuals of any of the components η_0,p as

$[\eta_{k,p}^{\,*} = \left\{\matrix{2^{K\,}{\rm{std}}\{\eta_ {0,p}\}\,\eta_{{\rm G},p},\hfill & k\! = \!K,\hfill \cr 2^{k\,}{\rm{std}}\{\eta_{0,p}\}\left\{\eta_{{\rm G},p}-{\cal B}^{\,-1}_{2{\rm D}}\left[{\cal B}_{2{\rm D}}(\eta_{{\rm G},p})\right]\right\},\hfill & k\,\lt\,K.\hfill }\right. \eqno(20)]$

Then,

$[\eta_{k}^{\,*} = \eta_{k,w}^{\,*}+\eta_{k,v}^{\,*}+\eta_{k,u}^{\,*}, \eqno(21)]$

where the noise correlation kernel corresponding to a component $[{\eta_{k,p}^{\,*}\!\in\!\{\eta_{k,w}^{\,*},\eta_{k,v}^{\,*},\eta_{k,u}^{\,*}\}}]$ can be written with $[\varsigma_{k,p}]$ = $[2^{k}\varsigma_{0,p}]$ as

$[g_{k,p}^{\,*} = \left\{\matrix{\varsigma_{k,p}\,g_{k,p},\hfill & k = K,\hfill \cr \varsigma_{k,p}\,g_{{\cal B}_{2{\rm D}}}\!{\bigcirc\kern-9pt\ast}\,\,g_{k,p},\hfill & k \,\lt\, K,\hfill }\right. \eqno(22)]$

where $[g_{{\cal B}_{2{\rm D}}}]$ is a 2-D kernel across the displacement dimensions characterizing the residual from 2-D binning of white noise. Specifically,

$[{g_{{\cal B}_{2{\rm D}}}} = {|X_{{\rm G}}|^{-1/2}{\cal{F}}^{\,-1}\left[{{\rm{std}}\left\{{\cal{F}}\left(\eta_{{\rm G,2D}}-{\cal B}^{\,-1}_{2{\rm D}}[{\cal B}_{2{\rm D}}(\eta_{{\rm G,2D}})]\right) \right\}}\right]},]$

where η_G,2D is a two-dimensional white random field. The field size |X_G| is included only for the normalization of the Fourier transform, and the formula holds for an arbitrary size.

Then, the PSD of $[\eta^{\,*}_{k}]$ , $[{K\!\geq\!k\!\geq\!0}]$ , can be written as

$[\eqalignno{ \Psi_{k}^{\,*} & = {\rm var}\left\{{\cal{F}}\left[\eta_{k,w}^{\,*} \right]\right\}+{\rm var}\left\{{\cal{F}}\left[\eta_{k,u}^{\,*}\right] \right\}+{\rm var}\left\{{\cal{F}}\left[\eta_{k,v}^{\,*}\right]\right\} \cr & = |X_{k}|\left(\left|{\cal{F}}\left[g_{k,w}^{\,*}\right] \right|^{2}\,+\,\left|{\cal{F}}\left[g_{k,u}^{\,*}\right]\right|^{2}\,+\,\left| {\cal{F}}\left[g_{k,v}^{\,*}\right]\right|^{2}\right)\,. & (23)}]$

As any $[\eta_{k}^{\,*}]$ is constant in angle, $[\Psi_{k}^{\,*}]$ is non-zero only across the DC plane with respect to the angular dimension. Example realizations, kernels, and PSDs for the residual components are shown in Fig. 1 (bottom).

Although (23) allows for modeling of very long range correlation, the streak noise is likely to contain minor correlation along the displacement not accounted for by this model. To adapt to such deviations, we allow the scaling parameters $[\varsigma_{k,p}]$ $[\geq]$ 0 for each noise component to vary with each scale k by estimating them individually at each scale, effectively accounting for mild local correlation in the noise.

Estimation of ς_k,w, ς_k,u, and ς_k,v. Based on (22) and (23), the PSD is completely determined by the values assumed by the three parameters ς_k,w, ς_k,u, and ς_k,v and the known kernels g_k,p and $[{g_{{\cal B}_{2{\rm D}}}\!{\bigcirc\kern-9pt\ast}\,\,g_{k,p}}]$ . To adaptively obtain the parameters, we begin by obtaining three noise variance estimates $[\hat{\sigma}_{k,w}^{2}]$ , $[\hat{\sigma}_{k,u}^{2}]$ , and $[\hat{\sigma}_{k,v}^{2}]$ . For each estimate, we define a corresponding filtering kernel g_d^(p) such that $[\hat{\sigma}_{k,w}^{\,2}]$ estimates the variance of high-frequency streaks, $[\hat{\sigma}_{k,u}^{\,2}]$ estimates the variance of horizontally low-frequency streaks, and $[\hat{\sigma}_{k,v}^{\,2}]$ of vertically low-frequency streaks. For this purpose, we define ϕ_d as a 1-D Gaussian function along dimension d, and ψ_d as a 1-D high-pass kernel with Daubechies wavelet `db3' of length 6 along d. Hence, convolution with ϕ_d realizes low-pass filtering, and ψ_d realizes a high-pass filter. Then, g_d^(p) is realized as a tensor product of three one-dimensional kernels across the dimensions d chosen based on the noise statistics through that dimension,

$[\eqalign{ g_{d}^{(w)} & = \phi_{0}\otimes\psi_{1}\otimes\psi_{2} , \cr g_{d}^{(u)} & = \phi_{0}\otimes\phi_{1}\otimes\psi_{2}, \cr g_{d}^{(v)} & = \phi_{0}\otimes\psi_{1}\otimes\phi_{2}. }]$

Specifically, with m₀, m₁, m₂ as the pixel sizes of the three dimensions of Z_k^*, ϕ₀ is a 1-D Gaussian function along the angular dimension with standard deviation m₀/8, and ϕ₁ and ϕ₂ are 1-D Gaussian functions along the two displacement dimensions with standard deviations of m₁/12 and m₂/12, respectively. Through these kernels, we obtain estimates of the three coefficients $[\hat{\sigma}_{k,p}]$ as described in (10) and (11) with g_s as either g_k,p ( k = K) or $[g_{{\cal B}_{2{\rm D}}}\!{\bigcirc\kern-9pt\ast}\,\,g_{k,p}]$ ( $[k \,\lt\, K]$ ).

We note that these three components do not directly correspond to ς_k,w, ς_k,u, and ς_k,v, as the frequencies of the white streak component $[\eta_{k,w}^{\,*}]$ partly overlap with those of $[\eta_{k,u}^{\,*}]$ and $[\eta_{k,v}^{\,*}]$ , i.e. $[\eta_{k,w}^{\,*}]$ includes also some low-frequency streak components. In particular, we have $[{\hat{\sigma}}_{k,u}^{\,2}]$ $[\approx]$ $[\varsigma_{k,u}^{\,2}+\varsigma_{k,w}^{\,2}/m_{1}]$ and $[{\hat{\sigma}}_{k,v}^{\,2}]$ $[\approx]$ $[\varsigma_{k,v}^{\,2}+\varsigma_{k,w}^{\,2}/m_{\,2}]$ . To this end, we can formulate a simple non-negative least-squares optimization as

$[\eqalign{ & \mathop{{\rm argmin}}_{\hat{\varsigma}_{k,w}^{\,2},\hat{\varsigma}_{k,u}^{\,2},\hat{\varsigma}_ {k,v}^{\,2}} \left(\hat{\varsigma}_{k,w}^{\,2}-\hat{\sigma}_{k,w}^{\,2}\right)^{2} \,\,\,+\,\, \left(\hat{\varsigma}_{k,u}^{\,2}+{{\hat{\varsigma}_{k,w}^{\,2}} \over {m_{1}}}-\hat{\sigma}_{k,u}^{\,2}\right)^{2} \cr& \quad+ \left(\hat{\varsigma}_{k,v}^{\,2}+{{ \hat{\varsigma}_{k,w}^{\,2}} \over {m_{2}}}-\hat{\sigma}_{k,v}^{\,2}\right)^{2}. }]$

Finally, we construct the PSD through (22) and (23) with $[\varsigma_{k,p}]$ = $[\hat{\varsigma}_{k,p}]$ .

4.1.2. Nonstationarity of $[\eta_{{}_{{\rm{P}}}}]$

Variance of the streak noise may differ across the sinogram due to changes in photon flux or noise in the bright-field. Thus, it may not be possible to denoise Z^*_k assuming an equal $[\Psi_{k}^{\,*}]$ for all spatial positions without either oversmoothing or leaving noise artifacts in some areas. To adapt to nonstationarity, we further relax the streak model allowing the PSD to vary within each scale k. In particular, before noise estimation and denoising, we split Z^*_k into overlapping, volumetric segments. We apply BM4D separately on each segment of Z^*_k, using a PSD scaled by parameters estimated from the same segment, i.e. we consider each segment as a separate noisy volume z with a corresponding Ψ. After denoising, the segment estimates produced by BM4D are recombined with a windowing function to form the full estimate $[\hat{Y}_{k}]$ .

4.2. Attenuation of extreme streaks

We note that the projections often include several streaks caused by defects in the scintillator. These streaks can be far stronger than that reasonably produced by the distribution of $[\eta_{{}_{{\rm{P}}}}]$ and therefore require a specific pre-processing. To this end, after the bright-fielding and log-transform and before the multiscale denoising procedure, we apply the simple extreme streak attenuation procedure as described by Mäkinen et al. (2021), which applies median filtering on extreme streak values detected through local polynomial fit of angular medians.

5. Poisson denoising

A filter for additive noise is not immediately applicable to the approximately white noise of $[\hat{Y}]$ originating from the Poissonian component π. Firstly, the bright-fielding (1) introduces substantial spatial variability in the Poisson model. As a result, for a given optical attenuation, noise in bright-fielded projections can be stronger or weaker in different parts of the detector, for example around edges in cone-beam acquisition. Secondly, while the logarithm (2) effectively makes the streak noise additive, it also changes the typical affine-variance model of the Poissonian noise to a nonlinear one where the variance is not constant, but asymptotically inversely proportional to the mean. In order to model the noise in $[\hat{Y}]$ through (6), we take care of these two issues as follows.

5.1. Reducing nonstationarity induced by bright-fielding

The Poissonian noise component π originates from a counting process which takes place before bright-fielding (1), and specifically before the division by $[I_{\rm B}\!-\!I_{\rm D}]$ , which introduces a spatially variant scaling of the variances. To undo this scaling, we consider

$[S = \hat{Y}+\ln\left(I_{{\rm L}}\right),\eqno(24)]$

where $[I_{\rm L}]$ = $[I_{\rm B}-I_{\rm D}]$ . Then, S can be treated as the log-scale version of a homogeneous Poissonian process; S is thus subject to signal-dependent noise where the variance of the noise can be expressed as a smooth nonnegative function of the underlying signal,

$[{\rm var}\{S\} = F\left({\rm{E}}\{S\}\right)\,, \eqno(25)]$

where the same F applies to each pixel. In particular, it can be shown that asymptotically for large flux

$[F\left({\rm{E}}\{S\}\right) \, \mathop{\propto}_{{\rm{E}}\{S\}\,\rightarrow\,\infty} \,{{1}\over{{\rm{E}}\{S\}}}.]$

5.2. Stabilization of variance

To turn a model like (25) into (6) we again resort to the use of a VST. As large-flux asymptotics are irrelevant for denoising problems characterized by low signal-to-noise ratio, and to pragmatically accommodate for model uncertainties, we model F as a polynomial with arbitrary data-driven coefficients. The method (Foi, 2009 ; Azzari & Foi, 2014 ) simultaneously identifies the coefficients for an arbitrary signal-dependent noise model where the variance is a positive power of an unknown polynomial, and returns the associated variance stabilizing transformation f as well as the corresponding exact-unbiased inverse VST $[f^{\,-1}_{\rm{EUI}}]$ . An example of an estimated standard deviation function $[({\hat{F}})^{1/2}]$ and the corresponding VST f are illustrated in Fig. 2, where the effectiveness of the stabilization can be deduced by the estimates of $[\widehat{{\rm{std}}\{\,f\,(S)\}}]$ being scattered around 1.

Figure 2
Illustration of the variance stabilization of Fly. Left: scatterplot of the sample standard deviation of S (red) as well as the square root of a polynomial approximation $[\hat{F}]$ of F (black), computed on S scaled to the range [0, 1] as a function of its expectation using a fifth-degree polynomial variance model. Middle: the variance stabilizing transformation f for this $[\hat{F}]$ . Right: sample standard deviation of f(S) as a function of its expectation. The standard deviations and f are both computed using Foi (2009

) and Azzari & Foi (2014

). Every point of the red scatterplot corresponds to the sample mean and sample standard deviation of a narrow segment of the image; the dispersion of the scatterplot is due to the finite sample size of each segment.

5.3. Multiscale denoising of the stabilized Poisson noise

To avoid introducing structured artifacts that are present in the bright-field and dark-field images, we further consider a smoothed version $[I_{{\rm L}}^{\,{\rm{smooth}}}]$ of the field component,

$[I_{{\rm L}}^{\,{\rm{smooth}}} = g_{I}\,{\bigcirc\kern-9pt\ast}\,\,\,{\rm medfilt}(I_{{\rm L }}), \eqno(26)]$

where g_I is a 2-D Gaussian kernel, and medfilt denotes a 2-D median filter. The median filter is adopted in order to remove extreme outliers (e.g. from broken pixels), and the convolution with the Gaussian ensures a smooth result. Then, $[I_{{\rm L}}^{\,{\rm{smooth}}}]$ can be used for approximate correction for the bright-field induced nonstationarity with $[\hat{Y}+\ln(I_{{\rm L}}^{\,{\rm{smooth}}})]$ .

The stabilized noisy stack can then be written as

$[{\tilde{Z}}{} = f\,[\hat{Y}+\ln(I_{{\rm L}}^{\,{\rm{smooth}}})] \approx \tilde{Y}+\tilde{\pi}, \eqno(27)]$

where $[\tilde{\pi}]$ corresponds to the stabilized noise and $[\tilde{Y}]$ to the signal upon stabilization.

We consider π white, and assume the streak denoising procedure to remove all streak noise frequencies, including those of π. Hence, we treat $[\tilde{\pi}]$ as missing the streak frequencies, i.e. with a PSD,

$[\Psi_{{\tilde{Z}}} = \left\{\matrix{0,\hfill & {\rm{on\ angular\ DC\ plane}}\,,\hfill \cr c,\hfill & {\rm{elsewhere}}\,,\hfill }\right. \eqno(28)]$

where c is a constant such that $[{\rm var}\{\tilde{\pi}\}]$ = $[c|X|^{-1}(m_{0}-1)\,m_{0}^{-1}]$ .

For multiscale denoising of the Poisson component, we define three-dimensional binning and debinning functions as $[{\cal B}_{3{\rm D}}]$ = $[{\cal B}_{2{\rm D}}\circ{\cal B}_{{\alpha}}]$ and $[{\cal B}_{3{\rm D}}^{\,-1}]$ = $[{\cal B}_{{\alpha}}^{\,-1}\circ{\cal{B}}_{2{\rm D}}^{\,-1}]$ , and obtain K_Poi scales of binned noisy volumes as $[{\tilde{Z}}_{k}]$ = $[{\cal B}_{3{\rm D}}({\tilde{Z}})]$ , $[{k\!\in\!\{0,\ldots,K_{{\rm Poi}}\}}]$ . Then, unlike the progressive denoising of the streaks, we begin by BM4D denoising of $[{\tilde{Z}}_{k}]$ of each scale k; at each scale, we model the noise through a PSD of the form (28). This way, we obtain an initial estimate $[\hat{\tilde{Y}}_{k}]$ of the corresponding noise-free volume $[\tilde{Y}_{k}]$ at each scale. Then, starting from Mathematical equation , we combine only the denoised volumes of each scale by recursively replacing the low-scale components of $[\hat{\tilde{Y}}_{k}]$ , $[{k = \{K-1,\ldots,0\}}]$ , by those of the lower scale,

$[\hat{\tilde{Y}}_{k}^{\,*} = \hat{\tilde{Y}}_{k}-g_{{\cal G}}\,{\cal B}_{3 {\rm D}}^{\,-1}\big[{\cal B}_{3{\rm D}}(\hat{\tilde{Y}}_{k})-\hat{\tilde{Y} }_{k+1}\big], \eqno(29)]$

where $[g_{{\cal G}}]$ denotes a 3-D Gaussian kernel. Although the low frequencies are obviously denoised more effectively in the coarser scale, the higher frequencies of the coarser scale are commonly estimated worse than the respective estimate of the finer scale (Facciolo et al., 2017 ). As such, $[g_{{\cal G}}]$ realizes a low-pass filtering which selects only low frequencies of the coarser estimate to be used in the full estimate.

To account for possible remaining nonstationarity and slight correlation of the noise, we perform the denoising in segments similar to as described in Section 4.1.2 for streak noise, and estimate a separate scaling parameter $[\varsigma_{{\tilde{Z}}}^{\,2}]$ in construction of the PSD at each scale. In particular, we estimate $[\varsigma_{{\tilde{Z}}}^{\,2}]$ as described in Section 3.2 with g_d = $[\psi_{0}\otimes\psi_{1}\otimes\psi_{2}]$ and g_s = $[{\cal{F}}^{\,-1}[({\Psi_{{\tilde{Z}}}/\|\Psi_{{\tilde{Z}}}\|})^{1/2}]]$ defining the unscaled noise correlation kernel, and finally construct the PSD through (28) with c = $[m_0(m_0-1)^{-1}\varsigma_\tilde{Z}^{\,2}]$ .

The final estimate of the underlying stack of projections can be obtained by applying $[f^{\,-1}_{\,\rm{EUI}}]$ to the finest scale estimate $[\hat{\tilde{Y}}_{0}^{\,*}]$ and then removing the field $[I_{{\rm L}}^{\,{\rm{smooth}}}]$ ,

$[\widehat{\ln(A)} \,=\, f^{\,-1}_{\,\rm{EUI}} \left(\hat{\tilde{Y}}_{0}^{\,*}\right) - \ln\left(I_{{\rm{L}}}^{\,{\rm{smooth}}}\right). \eqno(30)]$

As (30) negates the field correction, we note that had we used the non-smooth field I_L in (27) [and respectively in (30)], any noise or spurious structures present in I_L could be introduced into $[\widehat{\ln(A)}]$ , as they might have been denoised by BM4D and hence not preserved in $[\hat{\tilde{Y}}_{0}^{\,*}]$ .

Upon variance stabilization, Poissonian data become asymptotically normal (Curtiss, 1943 ). Due to the additional Gaussianization induced by the binning and by the linear transformations operated by the filter, the assumption of normality in (7) can be adopted for denoising of the Poissonian component in this work even for low-count data.

The full denoising process is shown in Fig. 3.

Figure 3
The full denoising process, requiring as inputs the noisy projections P_raw and the bright- and dark-fields I_B, I_D, (1)

and producing as the output an estimate $[\widehat{\ln(A)}]$ (30)

of the underlying stack of projections ln(A) (3)

. As an intermediate output, an estimate $[\hat{Y}]$ (12)

of the streak-free yet noisy stack of projections Y (5)

is also produced.

6. Experiments

We test our pipeline on synthetic data as well as two real acquisitions displaying ring artifacts and Poisson noise.

As a comparison, we show results for Mäkinen et al. (2021) available on PyPI as bm3d-streak-removal, the proposed algorithm embedding the conventional BM4D denoiser (Maggioni et al., 2012), as well as two leading streak-removal procedures from the tomopy Python library (Gürsoy et al., 2014 ): Münch et al. (2009) and Vo et al. (2018). In particular, for the latter we combine `Algorithm 3', `Algorithm 5', and `Algorithm 6', which is demonstrated by Vo et al. (2018) to attenuate a variety of different streaks. These streak denoising algorithms are run with the default parameters provided by the software library. To evaluate the benefit of the proposed Poisson denoising procedure with reconstruction which includes further regularization of the data, we include experments with the iterative Total Variation (TV) reconstruction (Goldstein & Osher, 2009 ) of Marchesini et al. (2020).

For the synthetic experiments, we replicate the noise generation setup of Mäkinen et al. (2021) on a stack of projections ( 238×181×238 pixels) obtained from a 3-D BrainWeb phantom (Cocosco et al., 1997 ) obtained through a padding and Radon transform upon a sign change and an exponential transformation. Specifically, we regard this stack as the underlying projections A and generate noise according to (3) with g as a constant of size m₀×1×1 (equal to g_0,w of Fig. 1). To obtain streak noise of different strengths, the streak noise component $[{(1\!+\!\eta_{{}_{{\rm{P}}}})}]$ is generated with $[{\rm{std}}\{\eta_{{}_{{\rm{P}}}}\}]$ = 0.005, 0.01, 0.02, 0.05. Next, to generate noisy measurements with different SNR levels for the Poisson component, we separately scale A to the ranges [2560, 5120] (higher SNR), [1280, 2560], and [640, 1280] (lower SNR) and generate a Poisson variate with mean and variance $[{A(1\!+\!\eta_{{}_{{\rm{P}}}})}]$ , thus defining the Poissonian noise π as the difference between this Poisson variate and $[{A(1\!+\!\eta_{{}_{{\rm{P}}}})}]$ . Furthermore, we include experiments with π = 0 (infinite SNR), thus resulting in a total of 16 combinations of Poisson and streak noise strengths. We do not simulate extreme streaks or the bright-fielding ( $[I_{{\rm B}}]$ = 1 and $[I_{{\rm D}}]$ = 0). For the streak removal, we consider $[\ln[A+\pi/(1\!+\!\eta_{{}_{{\rm{P}}}})]]$ as the streak-free yet noisy stack Y.

The results of the phantom experiments⁴ for streak attenuation are collected in Table 1, and, for full denoising, evaluating the reconstructed volumes, in Table 2 using iterative regularized TV reconstruction with optimized regularization parameter strength r. The experiments for both streak and Poisson denoising are illustrated in Figs. 4 and 5. All reconstructions are performed upon a sign change.

Table 1
Average signal-to-noise ratio for attenuation of streaks in the BrainWeb phantom subject to mixed streak and Poissonian noise as in (3), with different combinations of $[{{\rm{std}}\{\eta_{{}_{{\rm{P}}}}\}}]$ and peak values of A, with `peak' = ∞ being the limiting case for which π = 0

Left-to-right: noisy stack of projections Z (4), and estimates $[\hat{Y}]$ of the stacks of projections Y (5) denoised by the proposed procedure (12) (`new BM4D'), proposed procedure embedding BM4D of Maggioni et al. (2012) (`old BM4D'), Mäkinen et al. (2021), Münch et al. (2009), and Vo et al. (2018). As the table compares only streak removal, the SNR values are calculated with respect to the streak-free yet noisy projections Y = $[\ln[A+\pi/(1\!+\!\eta_{{}_{{\rm{P}}}})]]$ as $[{\rm{SNR}}(\hat{Y})]$ = $[10\log_{10}\{ \mathop{\rm{svar}}_{X} \{Y\}/\mathop{\rm{smean}}_{X} [(\hat{Y}\!-\!Y)^{2}]\}]$ , where svar and smean denote sample variance and sample mean, respectively. Each value of the table is the average SNR over 10 different noise realizations.

Peak	std{η_P}	Z	Proposed (new BM4D)	Proposed (old BM4D)	Mäkinen et al.(2021)	Munch et al. (2008)	Vo et al. (2018)
∞ (π = 0)	0.005	27.80	35.10	32.53	33.70	11.28	26.30
	0.01	21.85	32.60	28.97	29.66	11.26	25.33
	0.02	16.06	29.40	23.40	24.96	11.17	23.10
	0.05	8.82	24.44	14.64	18.22	10.61	18.53
5120	0.005	27.87	32.88	28.80	32.19	11.11	26.65
	0.01	21.91	31.00	26.91	29.00	11.09	25.49
	0.02	16.09	28.38	22.87	24.74	11.00	23.16
	0.05	8.83	24.04	14.64	18.25	10.47	18.55
2560	0.005	27.93	31.63	27.23	31.06	10.96	26.58
	0.01	21.97	30.03	25.55	28.38	10.94	25.36
	0.02	16.13	27.71	22.40	24.49	10.86	23.07
	0.05	8.85	23.69	14.63	18.23	10.34	18.54
1280	0.005	28.07	30.04	25.66	29.52	10.71	26.21
	0.01	22.09	28.72	23.79	27.44	10.69	25.03
	0.02	16.22	26.77	21.57	24.09	10.62	22.85
	0.05	8.90	23.16	14.61	18.21	10.15	18.49

Table 2
Average SNR for the reconstructed volumes of the BrainWeb phantom for the set of experiments shown in Table 1

Each reconstruction is performed with TV, either on the estimate $[\hat{Y}]$ of the streak-free projections Y (5), or the estimate $[\widehat{\ln(A)}]$ of the underlying stack of projections ln(A) (4). Left-to-right: reconstructed volumes of estimates produced by the proposed full procedure with improved BM4D, proposed streak removal with improved BM4D, proposed full procedure with old BM4D, Mäkinen et al. (2021), Münch et al. (2009), and Vo et al. (2018). Each regularization parameter r of TV is optimized individually for best SNR for each realization and method. The SNR values are each computed against the ground-truth noise-free phantom.

		Proposed (new BM4D)	Proposed (new BM4D)	Proposed (old BM4D)	Mäkinen et al. (2021)	Munch et al. (2008)	Vo et al. (2018)
Peak	std{η_P}	TV $[\widehat{\ln(A)}]$	TV $[(\hat{Y})]$	TV $[\widehat{\ln(A)}]$	TV $[(\hat{Y})]$	TV $[(\hat{Y})]$	TV $[(\hat{Y})]$
∞ (π = 0)	0.005	23.15	23.15	19.59	20.26	8.32	15.89
	0.01	21.05	21.05	17.09	18.01	8.25	15.39
	0.02	18.38	18.56	13.73	15.23	8.27	14.39
	0.05	14.23	14.75	11.29	11.05	8.04	12.11
5120	0.005	18.02	16.37	16.75	15.99	8.11	14.80
	0.01	17.45	16.00	15.70	15.41	8.10	14.49
	0.02	16.47	15.28	13.95	14.10	8.08	13.78
	0.05	14.03	13.41	11.02	10.82	7.92	11.87
2560	0.005	17.15	15.30	16.14	14.97	7.99	14.12
	0.01	16.75	15.03	15.21	14.53	7.99	13.82
	0.02	15.92	14.46	13.82	13.45	7.97	13.23
	0.05	13.76	12.85	10.94	10.67	7.87	11.67
1280	0.005	16.24	14.32	15.23	14.01	7.90	13.62
	0.01	15.95	14.13	14.71	13.71	7.90	13.40
	0.02	15.28	13.70	13.59	12.90	7.88	12.90
	0.05	13.41	12.43	10.87	10.50	7.77	11.43

Figure 4
Denoising of the 3-D BrainWeb phantom with noise as in (3)

with $[{\rm{std}}\{\eta_{{}_{{\rm{P}}}}\}]$ = 0.02 and signal peak 2560, displaying a single sinogram extracted from the stack. Left: a sinogram of ln(A) (3)

and the corresponding streak-free but noisy sinogram of Y (5)

. Right: on top, noisy sinogram Z (4)

and the comparison of estimates: $[\hat{Y}]$ for Münch et al. (2009)

, Vo et al. (2018)

, Mäkinen et al. (2021)

, the proposed framework using old BM4D and using improved BM4D, and $[\widehat{\ln(A)}]$ with the proposed framework and new BM4D. Below, corresponding estimation errors. Note that the errors $[|\hat{Y}-Y|]$ show only the effectiveness of streak removal, as the compared algorithms are designed for streak attenuation, whereas $[|\widehat{\ln(A)}-\ln(A)|]$ shows the complete denoising error, scaled separately. Notably, the proposed algorithm offers superior performance, but only when embedded with the improved BM4D.

Figure 5
Comparison of reconstructions of the 3-D BrainWeb phantom corrupted with streak and Poisson noise as in (3)

, corresponding to the sinograms shown in Fig. 4

. Top: ground truth volume, and reconstructions of ln(A) (3)

and Z (4)

obtained through filtered back-projection. Bottom: comparison of TV reconstruction of estimates with various regularization strengths r, where the percentage implies a multiplier to the regularization optimized to maximize SNR, i.e. 100% means `SNR-optimal' regularization. Top-to-bottom: proposed full estimate ln(A), proposed streak-free estimate $[\hat{Y}]$ , and streak-free estimate of $[\hat{Y}]$ (Vo et al., 2018)

, each with 100%, 50%, and 150% relative regularization strengths. Proposed estimates are computed embedding the improved BM4D. Notably, the full filtering offers improved reconstruction quality, and is also less sensitive to variations in the regularization parameters.

The Fly dataset consists of 180 projections with 50 s exposure (detector pixel size 27 µm, demagnified to 15.7 µm by cone-beam geometry) collected using a Sigray Prisma X-ray micro-tomography instrument at 34 kV; the detector size is 512×512 pixels. The denoising results for two different sinograms, as well as a corresponding tomogram after streak attenuation, are shown in Fig. 6. A comparison of denoising on a vertical slice of the stack of tomograms is shown in Fig. 7, and a comparison for fully denoised reconstructions is shown in Fig. 8.

Figure 6
Denoising of the stack of projections of Fly, showing two sinograms of the noisy stack of projections Z (4)

and the corresponding estimates $[\widehat{\ln(A)}]$ of the underlying stack of projections ln(A) (3)

obtained with the proposed framework (top), and the tomograms of the second sinogram (bottom), obtained with filtered back-projection using cone-beam geometry (Feldkamp et al., 1984

), for both the noisy data Z and the proposed estimate $[\hat{Y}]$ of the streak-free projections Y (5)

. The tomogram for $[\widehat{\ln(A)}]$ is shown in Fig. 8

. The first sinogram shows significant model nonstationarity in both streaks and the Poissonian component due to the bright-fielding.

Figure 7
Comparison of resulting stack of tomograms after different denoising procedures on Fly, each obtained with filtered back-projection, where individual tomograms are horizontal lines. Displayed are slices of tomograms, reconstructed from, top-to-bottom: noisy projections Z (4)

, estimates $[\hat{Y}]$ of the streak-free projections Y (5)

denoised with Mäkinen et al. (2021)

, $[\hat{Y}]$ from the proposed streak denoising, and the result of the proposed full denoising $[\widehat{\ln(A)}]$ . Note the horizontal `streaks' in the estimate produced by Mäkinen et al. (2021)

, which arise from differences in estimates for consecutive slices; the proposed method is not prone to such artifacts, as it considers the full stack of projections in the denoising.

Figure 8
Comparison of fully denoised tomograms of Fly. Top-to-bottom: tomogram reconstructed from a noisy sinogram of the stack of projections Z (4)

, from the estimate of stack of underlying projections $[\widehat{\ln(A)}]$ with proposed procedure with FBP reconstruction, and tomograms of the estimates for streak-free stacks $[\hat{Y}]$ of Münch et al. (2009)

with TV reconstruction, and $[\hat{Y}]$ of Vo et al. (2018)

with TV reconstruction. TV regularization was tuned visually, balancing residual noise and smoothing of signal. Compared with the reference methods, the proposed procedure manages to remove most noise without creating shadowlike artifacts common to Münch et al. (2009)

and Vo et al. (2018)

We also test the algorithm on a soft tissue sample 00072 displaying severe ring artifacts freely available in TomoBank (De Carlo et al., 2018). The data contain 1500 projections with 1.43 µm pixels, obtained at the Advanced Photon Source, 2-BM beamline; other experimental parameters are X-ray energy of 20 keV, 10 µm LuAG scintillator, and sample-to-detector distance of 15 mm. The detector size is 2160×2560 pixels. Included are ten samples for bright- and dark-fields, which are averaged to obtain a single bright-field and dark-field. A sinogram and a corresponding tomogram from the denoising results for streak removal are shown in Fig. 9, and slices of the stack of tomograms are compared in Fig. 10. Reconstructions of fully denoised projections are further compared in Fig. 11.

Figure 9
Denoising of the stack of projections of 00072. Top-to-bottom: display of a single sinogram of the noisy stack of projections Z (4)

, the corresponding estimate of the underlying projections $[\widehat{\ln(A)}]$ from the proposed procedure, and the corresponding tomograms of Z and the estimates $[\hat{Y}]$ of streak-free stacks Y (5)

, respectively, obtained with filtered back-projection. The tomogram for $[\widehat{\ln(A)}]$ is shown in Fig. 11

. Although the data present challenges through inconsistent noise intensities across the angular dimension, most streak noise and Poissonian noise is attenuated without notable loss of signal.

Figure 10
Comparison of central slices of resulting tomograms after different denoising procedures on 00072, where individual tomograms are horizontal lines each obtained with filtered back-projection. Displayed are slices of tomograms, reconstructed from, top-to-bottom: noisy projections Z (4)

, estimates $[\hat{Y}]$ of the streak-free projections Y (5)

denoised with Mäkinen et al. (2021)

, $[\hat{Y}]$ from the proposed streak denoising, and the result of the proposed full denoising $[\widehat{\ln(A)}]$ . Note the loss of signal near the central part of the Mäkinen et al. (2021)

estimate, not observed in the proposed results.

Figure 11
Comparison of fully denoised tomograms of 00072, corresponding to the sinogram in Fig. 9

. Top-to-bottom: tomogram reconstructed from a noisy sinogram of the stack of projections Z (4)

, of the estimate of stack of underlying projections $[\widehat{\ln(A)}]$ with the proposed procedure with FBP reconstruction, and tomograms of the estimates for streak-free stacks $[\hat{Y}]$ of Münch et al. (2009)

with TV reconstruction, and $[\hat{Y}]$ of Vo et al. (2018)

with TV reconstruction. TV regularization was tuned visually, balancing residual noise and smoothing of signal. Compared with the reference methods, the proposed procedure manages to remove most streaks without significant loss of detail, as well as most Poissonian noise without excess smoothing of the signal.

The proposed method achieves superior SNR values in streak removal in all simulated noise experiments. Although the difference to the 2-D implementation of Mäkinen et al. (2021) is not immediately visually obvious from individual sinograms or tomograms, the displayed vertical slices of the reconstructed objects show clear improvement in both signal preservation and avoiding discontinuity between different tomograms. Compared with Münch et al. (2009) and Vo et al. (2018), the proposed method avoids creation of shadow artifacts around strong signal features. Furthermore, performing the Poisson denoising through the proposed framework allows application of standard filtered back-projection reconstruction to data originally corrupted by Poisson noise, but can also improve quality of iterative TV reconstruction.

6.1. Parameters

For streak attenuation, we calculate K following the formula of horizontal binning from Mäkinen et al. (2021), using as the base the size of the smallest displacement dimension. As a result, we use K = 5 for 00072, K = 3 for Fly, and K = 2 for the phantom. These values were found to offer a reasonable compromise between denoising wide streaks versus preserving low-frequency signal components. Other processing parameters are adjusted for the smaller block size and processing neighborhood of BM4D. For angular binning, we use $[m_{\alpha}]$ = $[\lceil m/\lceil m/32\rceil\rceil]$ $[\approx]$ 32 pixels, where m is the original angular size and m_α the output size; the resulting size is half of that used by Mäkinen et al. (2021). For segmentation of the streak denoising, we use a window of size $[{\lceil m_{\alpha}/2\rceil\!\times\!19\!\times\!19}]$ pixels. For the Poisson denoising, we use K = 1 and m×19×19 segments. For variance stabilization, we use the implementation ClipPoisGaus (Azzari & Foi, 2015 ) of Foi (2009) and Azzari & Foi (2014), and use a quadratic polynomial for the variance model F.

7. Discussion and conclusions

We have presented a framework for three-dimensional attenuation of streak noise extending the 2-D framework of Mäkinen et al. (2021), as well as a BM4D denoiser utilizing the algorithmic improvements of Mäkinen et al. (2020). Furthermore, we have included a denoising step for Poisson noise in the sinogram domain through BM4D and the adaptive variance stabilization of Foi (2009) and Azzari & Foi (2014).

We test the algorithm on both synthetic and real data, demonstrating superior SNR compared with other popular streak removal algorithms, and showing improvements in streak attenuation over Mäkinen et al. (2020). Furthermore, we compare the results with those which use the conventional BM4D for correlated noise, demonstrating that the included improvements in BM4D for correlated noise are essential for successful streak attenuation. The included Poisson denoising allows for full sinogram-domain denoising within the framework. By operating fully in the 3-D stack of projections, the 3-D structure of the data can be leveraged for more accurate noise removal. The proposed procedure is fully automatic and does not require extra input parameters.

To compare different methods under their own ideal conditions, we have specifically selected the TV regularization parameter values that provide the best reconstruction quality. However, in real-world applications, these values cannot be identified precisely, and too small or too large parameter values may lead to residual noise or excess smoothing of the reconstructions. Inclusion of the proposed Poisson denoising step allows for weaker regularization, but notably also reduces the effects of relative shifts in the parameter values, meaning that the reconstruction can be safely deployed even when the regularization cannot be precisely tuned.

To consider the computational cost, we note that both denoising steps of Fly ( 181×512×512 pixels) run single-threaded on an AMD Ryzen 7 1700 processor each take about one hour. The computational cost is mostly due to the BM4D denoising in CPU. Although the adopted implementation is single-threaded, the algorithm is embarrassingly parallel, and thus a highly parallel GPU-based implementation is expected to reduce the total run time to the scale of seconds (Davy & Ehret, 2020 ).

The Poissonian noise attenuation can also be performed without the preceding ring reduction step on data which do not display ring artifacts. In such case, $[\Psi_{{\tilde{Z}}}]$ should be replaced by a flat PSD, as the Poissonian noise is approximately white prior to streak attenuation, whereas (28) considers the streak noise frequencies removed. Running the full denoising procedure in the absence of either streak or Poisson noise will lead to very small estimates for the corresponding noise components, meaning that no significant denoising will be performed for that noise.

We note that although we have focused on the full denoising of the projections, typical reconstruction pipelines, such as the iterative TV, provide further noise attentuation. For best results in combining the proposed denoising procedure with such pipelines, it may be necessary to adjust the filter strength for the denoising of Poissonian noise, e.g. for reduced attenuation of high-frequency noise, as it is further attenuated within the reconstruction. Likewise, integration of the proposed procedure within an iterative alternating reconstruction is left for future study.

APPENDIX A

Collaborative filtering and the BM4D denoising algorithm

A1. Collaborative filtering

The rationale of transform-domain filtering is to work with a representation of the signal where most of the signal is compacted to only a few coefficients, whereas the remaining coefficients mostly comprise noise. Hence, by attenuating the coefficients with a non-linear shrinkage operator, it is possible to attenuate noise while keeping most of the signal intact. Nonlocal collaborative filters utilize this property in the context of collective transform coefficients of groups of similar patches extracted from the input. One of the most popular collaborative filters is the Block-Matching and 3-D filtering (BM3D) (Dabov et al., 2007 ) denoising algorithm, which performs denoising on groups of blocks extracted from a 2-D image. In the BM4D volumetric denoiser (Maggioni et al., 2012), the patches are 3-D volumes extracted from the volumetric data.

All operations of collaborative filters are made with regard to a reference patch moving through the volume. For each position of the reference patch, the following steps are executed:

(1) Collect similar patches into a group through patch-matching.

(2) Obtain a group transform spectrum by collectively transforming the group of patches.

(3) Perform shrinkage.

(4) Transform the shrunk spectra back to patch estimates and aggregate them to the original locations from which they were collected.

For details about the algorithm in arbitrary dimensionality, we refer the reader to Mäkinen et al. (2020). In the following section, we describe special considerations for the implementation of the algorithmic improvements in Mäkinen et al. (2020) for volumetric denoising.

A2. Improved BM4D for correlated noise

Most of the improvements described by Mäkinen et al. (2020) are directly applicable to the 3-D denoiser. In this section, we consider extensions which are not immediate from the inclusion of an extra dimension.

A2.1. Shrinkage parameters

For the selection of shrinkage parameters λ and μ², we embed the parameter selection subroutine of Mäkinen et al. (2020), which is based on the 2-D input PSD; for simplicity, we adopt directly also the pre-computed features and parameters computed for a set of 2-D PSDs. To utilize this system with a 3-D PSD, we include a simple procedure which obtains a 2-D projection of the 3-D PSD by preserving the two largest principal components, aiming to preserve the characterizing features of the PSD shape. This projection is then used to compute features as described for a 2-D PSD by Mäkinen et al. (2020) for the estimation of suitable λ and μ².

A2.2. Fast implementation

We consider the fast implementation as suggested by Mäkinen et al. (2020). In particular, we perform all operations on a downscaled PSD of size N_f×N_f×N_f and compute exactly only the K_f first volumes of the 4-D spectrum and approximate the rest using the conventional variances. Furthermore, Fourier symmetries and sparsity of the transformed arrays can be exploited to reduce computational cost similar to the 2-D case.

A2.3. Refiltering

As noted by Mäkinen et al. (2020), even with exact modeling of the collaborative transform-domain noise spectrum, the accuracy of collaborative filtering is limited by the systemic factors arising from the used transforms, both in size and possible symmetries of the transform spectrum which may limit the modeling of the global PSD. As a result, the denoising may attenuate excess signal, leading to oversmoothing in some frequencies; Mäkinen et al. (2020) proposes the mitigation of these systemic issues through an extra filtering step performed on the denoising residual. The three-dimensional spectra are not exempt from these limitations, and as such we adopt the global Fourier thresholding and refiltering procedure through a 3-D FFT.

Footnotes

¹Despite its name, BM3D is a filter for 2-D images, where the third dimension originates from the nonlocal prior leveraged by the algorithm.

²As P_raw is a 3-D array, the pixels of I_B and I_D are replicated through the angle dimension for the operations in (1).

³Extreme streak noise arising from defective detectors is addressed separately in Section 4.2.

⁴For the purpose of visualization and objective SNR comparison, we have corrected the intensity response of every estimate through a cubic polynomial such that it matches the ground truth. This is done particularly to improve the results of those methods such as Münch et al. (2009) whose intensity response significantly deviates from the ground truth.

Acknowledgements

We are very thankful to Dr Sheraz Gul of Sigray Inc. for providing us with the experimental Fly data.

Funding information

This work was supported by the Academy of Finland (project No. 310779) and by the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, which is supported by the US Department of Energy, Office of Science, under Contract No.DE-AC02-76SF00515. Partial support by Hong Kong RGC 14302920.

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JOURNAL OF
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Volume 29| Part 3| May 2022| Pages 829-842

https://doi.org/10.1107/S1600577522002739

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Ring artifact and Poisson noise attenuation via volumetric multiscale nonlocal collaborative filtering of spatially correlated noise

1. Introduction

2. Bright-field normalization

2.1. Noise model for normalized projections

3. Correlated noise

3.1. Correlated noise model

3.2. Estimation of noise standard deviation

4. Multiscale streak filtering

4.1. Multiscale noise model

4.1.1. Adaptive parametric model of

4.1.2. Nonstationarity of

4.2. Attenuation of extreme streaks

5. Poisson denoising

5.1. Reducing nonstationarity induced by bright-fielding

5.2. Stabilization of variance

5.3. Multiscale denoising of the stabilized Poisson noise

6. Experiments

6.1. Parameters

7. Discussion and conclusions

APPENDIX A

Collaborative filtering and the BM4D denoising algorithm

A1. Collaborative filtering

A2. Improved BM4D for correlated noise

A2.1. Shrinkage parameters

A2.2. Fast implementation

A2.3. Refiltering

Footnotes

Acknowledgements

Funding information

References

research papers

4.1.1. Adaptive parametric model of $[\Psi_{k}^{\,*}]$

4.1.2. Nonstationarity of $[\eta_{{}_{{\rm{P}}}}]$