A Novel Adaptive Probabilistic Nonlinear Denoising Approach for Enhancing PET Data Sinogram

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 732178, 14 pages http://dx.doi.org/10.1155/2013/732178

Research Article A Novel Adaptive Probabilistic Nonlinear Denoising Approach for Enhancing PET Data Sinogram

Musa Alrefaya1,2 and Hichem Sahli1,3

1 Department of Electronics and Informatics (ETRO-IRIS), Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium 2 Faculty of Information Technology and Computer Engineering, Palestine Polytechnic University, Hebron, Palestine 3 Interuniversity Microelectronics Centre (IMEC), Leuven, Belgium

Correspondence should be addressed to Musa Alrefaya; [email protected]

Received 23 March 2013; Revised 4 May 2013; Accepted 4 May 2013

Academic Editor: Hang Joon Jo

Copyright © 2013 M. Alrefaya and H. Sahli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose filtering the PET sinograms with a constraint curvature motion diffusion. The edge-stopping function is computed in terms of edge probability under the assumption of contamination by Poisson noise. We show that the Chi-square is the appropriate prior for finding the edge probability in the sinogram noise-free gradient. Since the sinogram noise is uncorrelated and follows a Poisson distribution, we then propose an adaptive probabilistic diffusivity function where the edge probability is computed at each pixel. The filter is applied on the 2D sinogram prereconstruction. The PET images are reconstructed using the Ordered Subset Expectation Maximization (OSEM). We demonstrate through simulations with images contaminated by Poisson noise that the performance of the proposed method substantially surpasses that of recently published methods, both visually and in terms of statistical measures.

1. Introduction techniques have been investigated for PET images. Many researchers did explore the application of the well-known Positron Emission Tomography (PET) is an in vivo nuclear Perona and Malik anisotropic diffusion [1] in combination medicine imaging method that provides functional informa- with diverse diffusivity functions on PET images2 [ –6], as tion of the body tissues. The PET image results from recon- well as on sinograms [7–9]. structing very noisy, low resolution raw data, that is, the In [5], authors introduced a postreconstruction adap- sinogram, in which important features are shaped as curved tive nonlinear diffusion (Perona and Malik) filter based on structures. Enhancing the PET image spurred a wide range of varying the diffusion level according to a local estimation of denoising models and algorithms. Some methodologies focus the image noise. Applying the nonlinear anisotropic filtering on enhancing the reconstructed PET image directly, where method on the whole body, PET image and the sinogram others prefer enhancing the sinogram prior to reconstruc- were presented in [4, 8, 10], respectively. Results showed that tion. the anisotropic diffusion filtering algorithm helps improve Although developing an appropriate denoising method the quantitative aspect of PET images. for filtering the PET images has been widely studied in In the study of [9], combining the anisotropic diffusion the last two decades, this problem is still receiving high method with the coherence enhancing diffusion method for attention from researchers trying to improve the diagnosis filtering the sinogram as a preprocessing step was proposed. accuracy of this image. Existing methods may suffer draw- However, the considered cascading approach is time consum- backs such as the careful selection of a high number of ing, and the results are highly dependent on the parameters parameters, smoothing of the important features boundaries, selection criteria. Zhu et al. [11] built the diffusivity function or prohibitive computation. Recently, nonlinear diffusion using fuzzy rules that were expressed in a linguistic form. 2 Journal of Applied Mathematics

The mean curvature and the Gauss curvature diffusion noise distribution of PET sinograms. Further, considering algorithmsforfilteringthePETimagesduringreconstruction the singoram characteristics, we propose a probabilistic edge- were investigated in [12]. An anatomically driven anisotropic stopping function based on Chi-square prior for the ideal diffusion filter (ADADF) as a penalized maximum like- sinogram gradient with a spatially adaptive algorithm for lihood expectation maximization optimization framework calculating the prior odd at each pixel. We show that this was proposed in [2]. This filter enhances a single-photon method is improving and denoising sinogram data which emission computed tomography images using anatomical leads to enhanced reconstructed PET image. information from magnetic resonance imaging as a priori The remainder of the paper is organized as follows, knowledge about the activity distribution. Authors in [3] Section 2 briefly reviews the notions of curvature motion, proposed a reconstruction algorithm for PET with thin edge affected diffusion filtering, and self-snakes. The pro- plate prior combined with a forward-and-backward diffusion posedadaptivefilteringschemeisintroducedinSection 3. algorithm. The main drawback of the above filter, with respect Finally, Section 4 discusses the experimental results, while to sinogram images is that the diffusion produces important theconclusionsandfutureworkaregiveninSection 5. oscillations in the gradient. This leads to a poorly smoothed image [11, 12].Moreover,theadopteddiffusivityfunctionsdo 2. Geometry Driven Scale-Space Filtering not consider the special properties of the sinogram, which are high level of Poisson noise and curved-shape features. This section reviews the formulations for Mean Curvature Happonen and Koskinen [13]proposedfilteringthesin- Motion (MCM), Edge Affected Variable Conductance Dif- ogram in a new domain, that is, stackgram domain, where the fusion (EA-VCD) and self-snakes. Also we recall the Gauge signal along the sinusoidal trajectories can be filtered sep- Coordinates notions. Extensive discussion can be find in [16]. arately. They applied this method using the Gaussian and Let 𝑓 be a scalar image defined on the spatial image domain nonlinear filters. Radon transform is applied for inverse map- Ω, then the family of diffused versions of 𝑓 is given by ping of the filtered data from the stackgram domain to the sinogram domain. The above described method is impracti- 𝑈(𝑓):𝑓(⋅) 󳨀→ 𝑢 (⋅, 𝑡) with 𝑢 (⋅, 0) =𝑓(⋅) , (1) cal for noise reduction of the medical images such as PET, since the corresponding 3D stackgram requires an enormous where 𝑈 is referred to as the scale-space filter, 𝑢 is denoted by + space of computer memory. Furthermore, it is a complex thescale-spaceimage,andthescale𝑡∈R [6]. The denoised method not suitable for clinical applications where timely or enhanced version of 𝑓 is a given 𝑢(⋅, 𝑡) that is the closest reconstructien of the PET image is a very important issue. to the unknown noise-free version of 𝑓.Inthefollowingwe Most of the above studies considered a global image noise denote 𝑢(:; 𝑡) by 𝑢𝑡. level. Local or varying noise level has been used in [14], where a nonlinear diffusion method for filtering MR images with 2.1. Curvature Motion. Mean Curvature Motion (MCM) is varying noise levels was presented. The authors assumed considered as the standard curvature evolution. MCM allows that the MR image can be modeled as a piecewise constant diffusion solely along the level lines. In Gauge coordinates (slowly varying) function and corrupted by additive zero- (see Section 2.4) the corresponding PDE formulation is (𝑘 is mean Gaussian noise. Pizurica et al. [15] proposed a wavelet Euclidian curvature) as follows: domain denoising method for subband-adaptive and spatially ∇𝑢 adaptive image denoising approach. The latter approach is 𝑢 (⋅, 𝑡) =𝑢 =𝑘|∇𝑢| = [ ] |∇𝑢| . based on the estimation of the probability that a given co- 𝑡 VV div |∇𝑢| (2) efficient contains a significant noise-free component called “signal of interest.”The authors of [15] found that the spatially Hence diffusion solely occurs along the V-axis. adaptive version of their proposed method yielded better results than the existing spatially adaptive ones. 2.2. Edge Affected Variable Conductance Diffusion. Variable In this work, based on the following PET sinogram Conductance Filtering (VCD) is based on the diffusion characteristics: with a variable conduction coefficient that controls the rate of diffusion [16].InthecaseofEdgeAffected-VCD(EA- (i) the important features in the sinogram are curved VCD), the conductance coefficient is inversely proportional structures with high contrast values and these repre- to the edgeness. Consequently it is commonly referred to sent the regions of interest in the reconstructed PET as the edge-stopping function (𝑔), in which the edgeness is image, for example, tumor, typically measured by the gradient magnitude. The EA-VCD (ii) the weak edges in the sinogram are the edges that is governed by contain low contrast values, 𝑢 = [𝑔 (|∇𝑢|) ∇𝑢] . (iii) the noise in the sinogram is a priori identified as a 𝑡 div (3) Poisson noise, The above PDE system together with the initial condition we propose filtering the sinogram by means of a curvature given in (1) is completed with homogenous von Neumann constrained filter. The amount of diffusion is modulated boundary condition on the boundary of the image domain. according to a probabilistic diffusivity function that suits Note that the Perona and Malik’s antitropic diffusion1 [ ]isan images contaminated with Poisson noise, being the known EA-VCD. Journal of Applied Mathematics 3

2.3. Self-Snakes. Self-snakes are a variant of the MCM where The two expressions6 of( ) can be combined linearly in a an edge-stopping function is introduced [17]. The main goal is PDE setting. In this way, an image 𝑢0 is evolved according to preventing further shrinking of the level-lines once they have a weighted combination of these two invariants: reached the important image edges. For scalar images, self- 𝑢 =𝑔 (|∇𝑢|) 𝑢 +𝑔 (|∇𝑢|) 𝑢 snakes are governed by 𝑡 1 VV 2 𝑤𝑤 (9)

∇𝑢 with 𝑢(𝑡 = 0)0 =𝑢 . 𝑢𝑡 = |∇𝑢| div [𝑔 (|∇𝑢|) ]. (4) |∇𝑢| Equation (9) comprises a diffusion modulated by 𝑔1 along the image edges VV (a smoothing term) and a diffusion ad- Thisequationadoptsthesameboundaryconditionas(3). justable by 𝑔2 across the image edges 𝑤𝑤 (a sharpening term). Furthermore, it can be decomposed into two parts [16, 17]: Careful modeling of these terms allows efficiently denoising ∇𝑢 the PET sinograms, whilst keeping their interesting features. 𝑢𝑡 =𝑔|∇𝑢| div [ ]+(∇𝑔)⋅∇𝑢 |∇𝑢| (5) =𝑔𝑘|∇𝑢| + (∇𝑔) ⋅ ∇𝑢. 3. The Probabilistic Curvature Motion Filter

The first part describes a degenerate forward diffusion along The proposed probabilistic curvature motion filters are based the level lines that is orthogonal to the local gradient; it allows on the idea of the probabilistic diffusivity function [19], where preserving the edges. Additionally, the diffusion is limited the diffusivity function is expressed as the probability that the in areas with high gradient magnitude and encouraged observed gradient presents no edge of interest under a suit- in smooth areas. Actually the first term is the constraint able marginal prior distribution for the noise-free gradient. curvature motion. The second term can be viewed as a shock In [19], the probabilistic diffusivity function has been defined filter since it pushes the level-lines towards valleys of high as gradient, acting as Osher’s shock filter. 𝑔 (𝑥) =𝐴(1−𝑃(𝐻 |𝑥)) , pr 1 (10)

2.4. Gauge Coordinates. An image can be thought of as a where the normalizing constant 𝐴 is set to 𝐴 = 1/(1 −1 𝑃(𝐻 | 0)) 𝑔 (0) = 1 𝐻 collection of curves with equal value, the isophotes. At ex- to ensure that pr ,thehypothesis 1 describes the trema an isophote reduces to a point, at saddle points the iso- notion whether an edge element of interest is present given phote is self-intersecting. At the noncritical points; a Gauge the considered noise and 𝐻0 an edge element of interest is coordinate system [18] is defined by two orthogonal axes V absent. For a noisy gradient model 𝑥=𝑦+𝑛,weset and 𝑤, which correspond to the directions of the tangent and 𝐻 :𝑦≤𝜎,𝐻:𝑦>𝜎 normal at the isophote. The second order Gauge derivatives 0 𝑛 1 𝑛 (11) VV 𝑤𝑤 of the image in the and directions have the following 𝑦 𝜎 second-order structures: with being the ideal, noise-free, gradient magnitude, and 𝑛 being the standard deviation of the noise 𝑛.In[19]ithasbeen 2 2 𝑢𝑥𝑥𝑢𝑦 −2𝑢𝑥𝑢𝑦𝑢𝑥𝑦 +𝑢𝑦𝑦𝑢𝑥 demonstrated that 𝑢 = , VV (𝑢2 +𝑢2 ) 1 𝑥 𝑦 𝑔 (𝑥) =(1+𝜇𝜂(0)) pr (12) 2 2 (6) 1+𝜇𝜂(𝑥) 𝑢𝑥𝑥𝑢 +2𝑢𝑥𝑢𝑦𝑢𝑥𝑦 +𝑢𝑦𝑦𝑢 𝑢 = 𝑦 𝑥 . 𝑤𝑤 2 2 with the prior odds defined as (𝑢𝑥 +𝑢𝑦) 𝑃(𝐻 ) These gauge derivatives can be expressed as a product of 𝜇= 1 𝑃(𝐻 ) (13) gradients and a 2×2matrix with second-order derivatives 0 [18]: and the likelihood ratio 𝑢 𝑢 𝑢 2 𝑥𝑥 𝑥𝑦 𝑥 𝑃(𝑥|𝐻 ) 𝑢𝑤𝑤𝑢𝑤 =(𝑢𝑥,𝑢𝑦)( )( ), 1 𝑢𝑥𝑦 𝑢𝑦𝑦 𝑢𝑦 𝜂 (𝑥) = . (14) (7) 𝑃(𝑥|𝐻0) 2 𝑢𝑦𝑦 −𝑢𝑥𝑦 𝑢𝑥 𝑢VV𝑢𝑤 =(𝑢𝑥,𝑢𝑦)( )( ). −𝜆|𝑦| −𝑢𝑥𝑦 𝑢𝑥𝑥 𝑢𝑦 Considering a Laplacian prior 𝑝(𝑦) = (𝜆/2)𝑒 ,wehave [19]: For 𝑢𝑤𝑤 the matrix equals the Hessian 𝐻.For𝑢VV it is det −1 𝜆𝜎 −1 𝐻⋅𝐻 .Notethattheexpressionsareinvariantwithrespect 𝜇=(𝑒 𝑛 −1) , (15) to the spatial coordinates. The above expression can also be obtained as follows [16]: and the parameter 𝜆 can be estimated as 𝑢 =𝑢 𝜃2 +𝑢 𝜃2 −2𝑢 𝜃 𝜃, −1/2 VV 𝑥𝑥 sin 𝑦𝑦 cos 𝑥𝑦 sin cos 𝜆 = [0.52 (𝜎 −𝜎2)] (8) 𝑛 (16) 𝑢 =𝑢 𝜃2 +𝑢 𝜃2 +2𝑢 𝜃 𝜃, 𝑤𝑤 𝑥𝑥 cos 𝑦𝑦 sin 𝑥𝑦 sin cos 2 with 𝜎 denoting the variance of the noisy image and 𝜎𝑛 as ̇ where 𝜃 is given as 𝜃=arctan(𝑢𝑦/𝑢𝑥) and 𝑢V =−𝑢𝑥sin𝜃+ defined above. Due to limited space, the reader is referred to 𝑢𝑦coṡ 𝜃=0. [19] for the detailed expression of 𝜂(𝑥) in (14). 4 Journal of Applied Mathematics

It has to be noted that the diffusivity function12 ( ) fits in Starting from the prior odd, (13), we replace the ratio of the cluster of backward-forward diffusivities, and it has no “global” probabilities by a locally adaptive prior, 𝑃(𝐻1 | free parameters to be set. Moreover, for the considered PET 𝑧𝑖)/𝑃(𝐻0 |𝑧𝑖);thatis,𝑃(𝐻1) and 𝑃(𝐻0) are conditioned on sinograms, the noise standard deviation, 𝜎𝑛,in(16)isesti- the local spatial indicator 𝜎2 = (𝑓 ) 𝑓 mated as 𝑛 Var Ln ,wheretheimagenoise, Ln,is 𝑓 𝑃(𝐻1 |𝑧𝑖) 𝑃(𝑧𝑖 |𝐻1) 𝑃(𝐻1) reconstructed via wavelet decomposition of ,fromthetwo = ⋅ =𝜉𝑖 (𝑧𝑖)⋅𝜇, (18) finest resolution levels coefficients, using Daubechies (4) 𝑃(𝐻0 |𝑧𝑖) 𝑃(𝑧𝑖 |𝐻0) 𝑃(𝐻0) wavelet. where (𝜉) is the local likelihood ratio: 3.1. Probabilistic Self-Snakes (PSS). Itcanbedemonstrated 𝑃(𝑧 |𝐻) 𝑃(𝐻 ) 𝜉 (𝑧 )= 𝑖 1 𝜇= 1 . that the diffusion of scalar images via EA-VCD can be 𝑖 𝑖 (19) 𝑃(𝑧𝑖 |𝐻0) 𝑃(𝐻0) decomposed into (5), which can be rewritten as follows [16]: Considering Bayes’ rule, the probability that an “edge of 𝑢 =𝑔(|∇𝑢|) 𝑢 +[𝑔(|∇𝑢|) +𝑔󸀠 (|∇𝑢|) |∇𝑢|]𝑢 𝑡 VV 𝑤𝑤 (17) interest” is present at position 𝑖, 𝑃(𝐻1 |𝑥𝑖),isgivenas consolidating the properties of both the self-snakes and the 𝜇𝜂𝑖 (𝑥𝑖)𝜉𝑖 (𝑧𝑖) 𝑃(𝐻1 |𝑥𝑖)= . (20) EA-VCD into a single diffusion schema. 1+𝜇𝜂𝑖 (𝑥𝑖)𝜉𝑖 (𝑧𝑖) Considering (9) and the probabilistic diffusivity func- 𝑔 (𝑥) =𝑔̇ (𝑥) tion, if we set 1 pr ,andthesharpeningterm, The spatially adaptive probabilistic diffusivity function can 𝑔 (𝑥) =𝑔̇ (𝑥) + 𝑥𝑔󸀠 (𝑥) then be formulated as 2 pr pr ,weobtaintheprobabilisticself- 𝑔 (𝑥 ,𝑧)=𝐴(1−𝑃(𝐻 |𝑥,𝑧)) , snakes (PSS) [7]. By its nature, the PSS enhances the weak apr 𝑖 𝑖 1 𝑖 𝑖 edges and/or features in the sinogram. The second term in 1 (21) (32)allowsthesharpening.Inthisway,weakbutimportant 𝑔 (𝑥 ,𝑧)=(1+𝜇𝜂(0))( ) apr 𝑖 𝑖 edges are enhanced whilst the noise is removed efficiently. 1+𝜇𝜂𝑖 (𝑥𝑖)𝜉𝑖 (𝑧𝑖) PSS proved to be very effective and flexible for the sinogram image where the high contrast regions, which represent with a tumor in the reconstructed PET, should be smoothed wisely 𝑃(𝑥 |𝐻) without blurring the poor edges [7]. Like EA-VCD, the main 𝜂 (𝑥 )= 𝑖 1 . 𝑖 𝑖 𝑃(𝑥 |𝐻) (22) advantage of this filter is that the average gray value of the 𝑖 0 image is not altered during the diffusion process which is a We ensure that 𝑔(0) =, 1 because the minimum of 𝑃(𝐻1 |𝑥) significantissueinthesinogram. is at 𝑥=0and thus (1 − 𝑃(𝐻1 |𝑥))peaks at 𝑥=0. Intuitively, the proposed method considers an “observed 3.2. Adaptive Probabilistic Diffusivity Function Based on a gradient” at a given location as how probable this location Chi-Square Prior for a Sinogram. The probabilistic diffusivity presents useful information compared to its neighborhood, function (12) does not take into account the spatially varying based on noiselevelssuchasthecaseforthesinograms.Ithasaglobal 𝜂 (𝑥 ) threshold parameter, which is related to the image noise (1) the likelihood ratio via 𝑖 𝑖 , 𝑇=𝜎 standard deviation 𝑛. Hence, in such formulation, if (2) a measurement from the local surrounding via 𝜉𝑖(𝑧𝑖), two pixels/voxels have equal gradient magnitude, they will 𝜇 𝑔 (𝑥) (3) the global prior odd via . have the same pr values, no matter the noise level at these pixels. In this work, the probabilistic diffusivity function is The local spatial activity indicator 𝑧𝑖 is defined as improved by considering the local statistical noise at each 1 element. We adopt the ideas of [15] and adapt the estimator 𝑧 = ∑ 𝑥 , 𝑖 𝑁 𝑙 (23) to the local spatial context in the image. 𝑙∈𝑤(𝑖)

where 𝑥𝑙 is the gradient magnitude at location 𝑙∈𝑤(𝑖). 3.2.1. Spatially Adaptive Probabilistic Diffusivity Function. Assume that all the elements within the small window The most appropriate way to achieve a spatial adaptation isto are equally distributed and conditionally independent. With estimate the prior probability of signal presence 𝑃(𝐻1) adapt- these simplifications, the conditional probability of 𝑧𝑖 given ivelyforeachpixelinsteadoffixingitglobally.Thiscanbe 𝐻1 in a window 𝑤(𝑖) of size 𝑁, which is denoted as 𝑃𝑁(𝑧𝑖 | achieved by conditioning the hypothesis 𝐻1 on a local spatial 𝐻1),isgivenby𝑁 convolutions of 𝑃(𝑥𝑖 |𝐻1) with itself as activity indicator such as the locally averaged magnitude or follows: the local variance of the observed gradient. To estimate the probability that “signal of interest” is 𝑃𝑁 (𝑧𝑖 |𝐻1)=𝑃(𝑥𝑖 |𝐻1) Conv𝑁𝑃(𝑥𝑖 |𝐻1), (24) present at the position 𝑖, we consider a local spatial activity indicator, 𝑧𝑖,ateachposition.𝑧𝑖 is defined as the locally aver- while the conditional probability 𝑃𝑁(𝑧𝑖 |𝐻0) of 𝑧𝑖 given 𝐻0 aged gradient magnitude within a relatively small window, is given by 𝑁 convolutions of 𝑃(𝑥𝑖 |𝐻0) with itself: 𝑤(𝑖), with a fixed size, 𝑁,aroundtheposition𝑖 in the gradient 𝑃 (𝑧 |𝐻) =𝑃(𝑥 |𝐻) 𝑃 (𝑥 |𝐻) . image. 𝑁 𝑖 0 𝑖 0 Conv𝑁 𝑖 0 (25) Journal of Applied Mathematics 5

1 1 0.9 0.8 0.5 0.7 0.6 0 0.5 0.4 −0.5 0.3 0.2 −1 0.1 0 −1.5 −100 −80 −60 −40 −20 0 20406080100 −100 −80 −60 −40 −20 0 20406080100 Gradient 𝑥 Gradient 𝑥 󳰀 Original diffusivity function 𝑔(𝑥) Original diffusivity function [𝑥𝑔(𝑥)] 󳰀 Adaptive diffusivity function 𝑔(𝑥) Adaptive modified diffusivity function [𝑥𝑔(𝑥)] (a) probabilistic Diffusion along the Edge (b) Diffusion accross the edge 𝑔 () 𝑔 () Figure 1: The Adaptive probabilistic diffusivity function apr versus the original probabilistic diffusivity function pr .(a)Diffusivity 󸀠 functions along the edge (𝑔(𝑥)). (b) Diffusivity functions across the edge ([𝑥𝑔(𝑥)] ).

Duetolimitedspace,thereaderisreferredto[19]forthe interval) approaches a Gaussian density function in the case detailed expression of 𝑃(𝑥𝑖 |𝐻0), 𝑃(𝑥𝑖 |𝐻1),and𝜂(𝑥𝑖). of high number of counts [21, 22].Moreover,Milleretal.[21] Figure 1 illustrates the original probabilistic diffusivity showed that the Gaussian approximation is surprisingly ac- functionandadaptiveprobabilisticdiffusivityfunction.The curate, even for a fairly small number of counts. To illustrate adaptive function has lower values, as shown in Figure 1(a), this, we use the logarithmic function to simplify the proof: which allows better preservation of important edges than 𝑆 −𝑚 the original function. The negative peaks of the proposed 𝑚 𝑒 ln 𝑃 (𝑆) = ln ( ). (26) function occur usually at larger gradient magnitude than 𝑆! theoriginalone,whichindicatesstrongeredgeenhancement capability. This property indicates a quick smoothing of the Using Stirling’s formula (for large 𝑆 as we are assuming here) 𝑆 −𝑆 nearly uniform areas while maintaining and enhancing the 𝑆! ≈ 𝑆 ⋅𝑒 √2𝜋𝑆,wehave strong edges. 2 𝑒−(𝑆−𝑚) /2𝑚 𝑃 (𝑆) ≈ . (27) √2𝜋𝑚 3.2.2. A Chi-Square Prior for Noise-Free Sinogram Gradient. The noise in the sinogram is a priori identified as a Poisson Assuming that the sinogram gradient is approximated noise [12, 20]. The magnitude of Poisson noise varies across by absolute difference of neighboring pixel values on a 2- the image, as it depends on the image intensity. This makes connected grid, we demonstrate that the gradient of Poisson removing such noise very difficult. In the original probabilis- random variables follows a Skellam distribution. Then, we tic diffusivity functionSection ( 3), the Laplacian prior was show that the Skellam distribution can be approximated as imposed for the ideal image gradient that is contaminated aGaussiandistribution. with Gaussian noise. In order to take into account the Let 𝑠1 and 𝑠2 be two statistically independent adjacent sinogram’s noise distribution in the filtering scheme, in pixels in the observed sinogram be with a Gaussian dis- this section we aim at redefining the diffusivity function tribution 𝑠1 ∼ Gauss(𝑚1,𝜎𝑠1) and 𝑠2 ∼ Gauss(𝑚2,𝜎𝑠2), for handling Poisson noise. This can be accomplished by respectively. The distribution of the difference, 𝑎=𝑠1−𝑠2, finding an appropriate prior for the ideal noise-free sinogram of two statistically independent random variables 𝑠1 and 𝑠2, gradient. each having Poisson distributions with different expected In the following, we argue that we can represent Poisson values 𝑚1 and 𝑚2, is denoted as the Skellam distribution [23] distribution by a Gaussian distribution as Gauss(0, √2𝑚). and can be given as Afterwards, we demonstrate that the gradient magnitude of 𝑚 𝑎/2 the noise-free sinogram follows a Chi-square distribution, −(𝑚1+𝑚2) 1 PD (𝑎;1 𝑚 ,𝑚2) =𝑒 ( ) 𝐼|𝑎| (2√𝑚1𝑚2) , (28) andfinallywereformulatetheprobabilisticdiffusivityfunc- 𝑚2 tion based on a Chi-square prior for the noise-free sinogram gradient. where 𝐼𝑘(𝑍) is the modified Bessel function of the first kind. In the literature, several studies demonstrated that the The difference between two Poisson variables has the 𝑆 −𝑚 2 2 2 2 Poisson distribution (of probability mass 𝑃(𝑆) =𝑚 𝑒 /𝑆!, following properties: (i) 𝜎𝑠1𝑠2 =𝜎𝑠1 +𝜎𝑠2 =2𝜎 and (ii) with 𝑆 being the number of occurrences of an event and 𝑚 𝑚=𝑚𝑠1𝑠2 =𝑚𝑠1 −𝑚𝑠2 =0. Considering these prop- being the expected number of occurrences during a given erties, the cross-correlation, and the delta function, the 6 Journal of Applied Mathematics

×104 ×104 3.5 3.5 3 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 0 2 4 6 8 10 12 14 16 18 20 0 5 10 15 20 (a) Noise-free histogram (b) Noisy gradient histogram ×104 3.5 3 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 20

Noise-free histogram 2 Estimated 𝜒 prior Estimated Laplacian prior (c) Chi-square and Laplacian prior for the noise free histogram

Figure 2: Chi-square prior versus Laplacian prior. (a) Histogram of the noise-free gradient. (b) Histogram of the noisy gradient magnitude. (c) Laplacian and Chi-square priors for the noisy-free gradient, estimated from the noisy data.

approximated distribution of the sinogram gradient can be degrees of freedom (2 degrees because we are dealing with given as Gauss(0, √2𝑚). 2D images) is given as Based on the assumption that the sinogram gradient 1 follows the Gauss(0, √2𝑚) distribution, we can show that the 𝑃(𝑦)= ⋅𝑒−(1/2)|𝑦|. 2 (31) distribution of this gradient leads a Chi-square distribution as follows: Based on the above, the prior odd, (15), can be reformu- lated considering Chi-square prior instead of Laplacian prior ∇𝑆 ∼ (0, √2𝑚) , 2 (𝑖,𝑗) Gauss and the noise 𝑛∼Gauss(0, 2𝜎𝑛).NotethattheChi-square 󵄨 󵄨 󵄨∇𝑆 󵄨 with 2 degrees of freedom is almost a special case of Laplacian 󵄨 (𝑖,𝑗)󵄨 𝜆=1/2 ∼ Gauss (0, 1) , prior with a rate parameter (scale parameter) .This √2𝑚 (29) parameter determines the “scale” or statistical dispersion of 󵄨 󵄨2 the probability distribution. It indicates that the distribution 󵄨∇𝑆 󵄨 󵄨 (𝑖,𝑗)󵄨 ∼𝜒2. of the ideal gradient is independent of a rate parameter since 2𝑚 the Poisson noise is a pixel dependent (i.e., depends on the number of counts). Therefore, it is more natural to use Chi- The Chi-square distribution is defined by the following square prior for estimating the ideal gradient of Poisson data probability density function: rather than Laplacian prior with a parameter, 𝜆,whichis based on the variances of the image and noise gradients as 𝑦𝜁/2−1 ⋅𝑒−𝑦/2 indicated by (16). 𝑃 (𝑦) = , (30) In Figure 2, we illustrate the estimation of the Laplacian 2𝜁/2𝛾 (𝜁/2) andChi-squarepriorsforthenoise-freegradient.Figures2(a) and 2(b) show the histograms of the gradient magnitudes where 𝛾(𝜁/2) denotes the Gamma function and 𝜁 is a positive for the noise-free and noisy images, respectively. The noise- integer that specifies the number of degrees of freedom. For free gradient histogram is typically sharply peaked at zero the noise gradient model 𝑥=𝑦+𝑛,theChi-squarewith2 since the noise-free images typically contain large portions Journal of Applied Mathematics 7

of relatively uniform regions that produce negligible gradient (iv) Filter the sinogram, 𝑢0 =𝑓,basedon(8) values. Sharp edges and textured regions produce some recursion, for 𝑡=0,...,(number of iterations 1): relatively large gradients, building in this way long tails of the gradient histogram. From the noisy histogram of Figure 2(b), 𝑢(𝑥,𝑦,𝑡+Δ𝑡) we estimate the parameter of the prior for the noise-free sinogram gradient. The results are illustrated in Figure 2(c), =𝑢(𝑥,𝑦,𝑡) which shows the estimated Laplacian (dotted curve) and Chi- 1 square (continues curve) priors in comparison to the ideal, 2 + [𝑔1 ⋅ sin 𝜃⋅𝑢(𝑥−1,𝑦,𝑡) noise-free histogram. Δ𝑡 +𝑔 ⋅ 2𝜃⋅𝑢(𝑥−1,𝑦,𝑡) 3.2.3. Algorithm: The Adaptive Probabilistic Curvature Motion 2 cos Filter Based on Chi-Square Prior. In summary the general 2 +𝑔1 ⋅ sin 𝜃⋅𝑢(𝑥+1,𝑦,𝑡) equation of the proposed Adaptive Modified Probabilistic CurvatureMotion(AMPCM)filterisgivenby 2 +𝑔2 ⋅ cos 𝜃⋅𝑢(𝑥+1,𝑦,𝑡) 𝑢 =𝑔 (|∇𝑢|) 𝑢 +[𝑔 (|∇𝑢|) +𝑔󸀠 (|∇𝑢|) |∇𝑢|]𝑢 . 2 𝑡 apr VV apr apr 𝑤𝑤 +𝑔1 ⋅ cos 𝜃⋅𝑢(𝑥,𝑦+1,𝑡) (32) 2 +𝑔2 ⋅ sin 𝜃⋅𝑢(𝑥,𝑦+1,𝑡) 𝑔 (𝑥 ,𝑧) =𝑔̇ (𝑥 ,𝑧) Let 1 𝑖 𝑖 apr 𝑖 𝑖 ,asgivenby(21), and the sharpen- 󸀠 +𝑔 ⋅ 2𝜃⋅𝑢(𝑥,𝑦−1,𝑡) ing term 𝑔2(𝑥) =𝑔̇ 1(𝑥𝑖,𝑧𝑖)+𝑥𝑔1(𝑥𝑖,𝑧𝑖);theoverallalgorithm 1 cos is given as shown in Algorithm 1. 2 +𝑔2 ⋅ sin 𝜃⋅𝑢(𝑥,𝑦−1,𝑡) Algorithm 1. Adaptive Modified Probabilistic Curvature Mo- −2⋅𝑔 ⋅𝑢(𝑥,𝑦,𝑡)−2⋅𝑔 ⋅𝑢(𝑥,𝑦,𝑡) tion Filter Algorithm 1 2 𝑔 (i) Build the Probabilistic Diffusivity Function: + 2 ⋅ 𝜃 𝜃⋅𝑢(𝑥+1,𝑦+1,𝑡) 2 sin cos (a)computetheprioroddbasedontheChi-Square 𝑔 − 1 ⋅ 𝜃 𝜃⋅𝑢(𝑥+1,𝑦+1,𝑡) prior: 2 sin cos

1 𝑔2 𝑃(𝑦)= ⋅𝑒−(1/2)|𝑦|, + ⋅ sin 𝜃 cos 𝜃⋅𝑢(𝑥−1,𝑦−1,𝑡) 2 2 (33) (1/2)𝜎 −1 𝑔 𝜇=(𝑒 𝑛 −1) . − 1 ⋅ 𝜃 𝜃⋅𝑢(𝑥−1,𝑦−1,𝑡) 2 sin cos 𝑔 (ii) Build the spatially adaptive diffusivity function − 2 ⋅ 𝜃 𝜃⋅𝑢(𝑥+1,𝑦−1,𝑡) 𝑔 (𝑥 ,𝑧) 𝑖=1,..., 2 sin cos apr 𝑖 𝑖 ,foreachpixel size𝑓: 𝑔 + 1 ⋅ 𝜃 𝜃⋅𝑢(𝑥+1,𝑦−1,𝑡) (a) compute the local spatial activity indicator 2 sin cos

1 𝑔2 𝑧 = ∑ 𝑀 , − ⋅ sin 𝜃 cos 𝜃⋅𝑢(𝑥−1,𝑦+1,𝑡) 𝑖 𝑁 𝑙 (34) 2 𝑙∈𝑤(𝑖) 𝑔1 + ⋅ sin 𝜃 cos 𝜃⋅𝑢(𝑥−1,𝑦+1,𝑡)]. (b) compute the likelihood ratio for each window: 2 (38) 𝑃𝑁 (𝑧𝑖 |𝐻1)=𝑃(𝑀𝑖 |𝐻1) Conv𝑁𝑃(𝑀𝑖 |𝐻1), (35) 𝑃 (𝑧 |𝐻)=𝑃(𝑀 |𝐻) 𝑃(𝑀 |𝐻), 𝑁 𝑖 0 𝑖 0 Conv𝑁 𝑖 0 4. Experiments and Discussion

(c) compute the normalizing constant 𝐴: For the analysis of the proposed denoising approaches, we use a simulated thorax PET phantom, containing three hot 𝐴=1+𝜇𝜂(0) (36) regions of interest (tumors) of activities 1.18, 1.8, and 1.57. 100 6 realizations (noisy sinograms) with added noise of 1×10 (d) compute 𝑃(𝐻1 |𝑥𝑖) as defined by (20) coincident events have been generated. Each sinogram has a size of 256×256 pixels and its spacing is 2×2 (mm/pixel),with (iii) compute the diffusivity function 𝑔(𝑥𝑖,𝑧𝑖): 128 detectors and 128 projection angles. Figure 3(a) illustrates the noise-free sinogram, and Figure 3(b) illustrates a noisy 𝑔 (𝑥 ,𝑧)=𝐴(1−𝑃(𝐻 |𝑥,𝑧)) , 1 𝑖 𝑖 1 𝑖 𝑖 realization. (37) For reconstructing the PET images, we adopt the Ordered 𝑔 (𝑥 ,𝑧)=𝑔 (𝑥 ,𝑧)+𝑥𝑔󸀠 (𝑥 ,𝑧). 2 𝑖 𝑖 1 𝑖 𝑖 1 𝑖 𝑖 Subset Expectation Maximization (OSEM) reconstruction 8 Journal of Applied Mathematics

(a) (b)

Figure 3: Experimental dataset. (a) Noise-free simulated sinogram. (b) Noisy sinogram. ((c)-(d)) Corresponding reconstructions. algorithm [24]. One way to represent the imaging system is For the experimental assessment of the proposed diffu- with the following linear relationship: sion filtering, we use two sets of evaluation measures. The firstsetstemsfrommeasuringthequalityofthefiltering 𝑓=𝐻𝑓 +𝑛, techniques on the sinogram, whilst the second set originates PET (39) from validating the quality of the PET reconstruction, after filtering the sinogram observations. As ground-truth infor- where 𝑓 is the set of observations (sinogram), 𝐻 is the known mation, the former uses the noise-free sinogram, while the system model (projection matrix), 𝑓 is the unknown PET latter needs prior identification of the important areas by a image, and 𝑛 isthenoise.Thegoalofreconstructionisto medical professional. use the data values 𝑓 (projections through the unknown A fundamental issue with scale-spaces induced by dif- object) to find the image 𝑓 . Under the Poisson assumption, PET fusion processes, as the ones proposed in this paper, is counts of all the detectors around the object are considered the automatic selection of the most salient scale. For PET as independent Poisson variables. OSEM groups projection sinogram denoising application, we use an earlier proposed data into an ordered sequence of subsets and progressively optimal scale selection approach [25], where the maximum processes every subset of projections in each iteration process correlation method has been adopted: [24]. The OSEM method results are highly affected by the number of iterations and subsets used. The iterations should be ended before the noise becomes too dominant and not 𝑡 = [Ĉ (𝑡)] too early to avoid losing important information. In our opt argmax mp experiments, the parameters of the OSEM algorithm are set 16 4 (40) as follows: we use subsets and run them for iterations. 𝜎[𝑛𝑜 (𝑡0)] Such PET reconstructions are illustrated in Figures 3(c) and = argmax [𝜎 𝑡[𝑢 ]+ 𝜎[𝑛𝑜 (𝑡)]] 𝜎[𝑢𝑡 ] 3(d) for the noise-free and noisy sinograms, respectively. 0 Journal of Applied Mathematics 9

(a) AMPCM (b) PSS

𝑢𝑡 Figure 4: Optimal enhanced scale ( opt ) of the noisy sinogram of Figure 3(b).

with 𝑛𝑜 being the so-called outlier noise estimated using For the qualitative assessment of the denoised sinogram, wavelet-based noise estimation. 𝑡0 is the zeroth scale; thus 𝑢𝑡 ,withrespecttothenoise-freeimage𝐼,weadoptthe 𝑢 =𝑓 𝑛(𝑡 ) opt 𝑡0 and 0 represents the initial amount of noise. following measures [25]:

DQ1 The Peak Signal to Noise Ratio (PSNR) is a statistical 4.1. Sinogram Denoising Evaluation. In this work, we are in- measureoferror,usedtodeterminethequalityofthe terested in comparing the proposed AMPCM filter with filter- filtered images. It represents the ratio of a signal power ing approaches from the literature: the Probabilistic self- to the noise power corrupting. Obviously, one sees snakes (PSS) [7], Perona and Malik filter (P-M) [1], and the that the higher the PSNR, the better the quality: Noise-Adaptive Nonlinear Diffusion Filtering (NAF)14 [ ]. 2 2 The Lorentzian diffusivity function 𝑔(𝑋) = 1/(1 +𝑋 /𝑘 ) Card (Ω) is used for applying the P-M filter. This function was proposed PSNR (𝑡 )=10log 󵄨 󵄨. opt 10 󵄨 󵄨 (41) ∑𝑝∈Ω 󵄨𝐼(𝑝)−𝑢𝑡 (𝑝)󵄨 by Perona and Malik [1] and widely used in the literature. The 󵄨 opt 󵄨 contrast parameter 𝑘 iscalculatedbasedonthevaluegiven by a percentage of the cumulative histogram of the gradient magnitude [16]. The same diffusivity constant is used for all DQ2 The correlation (C𝑚𝜌) between the noise-free and the filters (𝑑𝑡 = 0.2). filteredimage.Thehigherthiscorrelationis,thebetter Figure 4 illustrates the optimal enhanced scale of the the quality is: sinograms for all the considered filtering methods. The filtered sinogram by AMPCM and PSS shows better enhanced C𝑚𝜌 (𝑡 )=𝜌[𝐼,𝑢𝑡 ]. (42) results especially the curvy shape features. opt opt 10 Journal of Applied Mathematics

(a) AMPCM (b) AD-AMPCM

(e) P-M (f) AD-PM

Figure 5: Mean of the filtered sinograms (left column). Absolute difference between the mean of the filtered sinograms and the noise-free image (right column). Journal of Applied Mathematics 11

Table 1: Denoising quality measures. Table 2: Optimal number of Iterations.

Method 𝑓 AMPCM PSS P-M NAF AMPCM PSS P-M NAF 𝑡 30.8 PSNR ( opt)15.1 29.73 22.8 29.1 16 16 22 21 𝑡 0.02 NV ( opt) 0.11 0.0223 0.225 0.0261 C 𝑡 0.9955 0.9956 OSEM-contrast recovery curve 𝑚𝜌 ( opt) 0.92 0.9950 0.9911 1.5

1.4 DQ3 The calculated variance of the noise (NV) describes the remaining noise level. Therefore, it should be as 1.3 small as possible: 1.2 󵄨 󵄨 (𝑡 )= (󵄨𝐼−𝑢 󵄨). NV Var 𝑡 (43) gain Contrast 1.1 opt 󵄨 opt 󵄨 1 We are interested in comparing the maximum of each measure for different filtering approaches. 0.9 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Table 1 lists the above measures for each of the considered ×10−3 filtering approaches. The best performance (maximum of the Variance measure)isdisplayedinbold.Asitcanbeseenthebest AMPCM Perona and Malik performing filtering is achieved when using the AMPCM and PSS CCM-Sapiro PSS. Furthermore, we notice that for all measures, AMPCM andPSSoutperformtheNAFand(P-M)filters.Theperfor- Figure 6: The contrast recovery curves for reconstructed PET by mance of the smoothing algorithms proposed in this paper OSEM. is remarkable in discriminating a true edge from image noise (see Figure 4). We also notice the improved performance of the probabilistic curvature motion algorithms as compared to 𝑓 (𝑡). For evaluating the effect of sinogram predenoising the standard diffusion algorithms. Table 2 gives the number PET on the PET reconstruction, we use the contrast recovery of iterations to reach the optimal scale 𝑡 as defined in (40). opt method which aims to measure the quality of 𝑓 (𝑡) by The absolute difference between the mean of the filtered PET measuring the contrast recovery of the interesting objects sinograms and the noise-free one is another indication of (ROIs)intheimage.Thecontrastrecoveryiscomputedbased the behavior and stability of the filtering methods. The mean on the contrast gain. The latter measures how much the results of the filtered realizations by the PSS and AMPCM ROIs (i.e., tumor) in the PET image are discriminated from filters are close to the ideal image, as shown in Figure 5. the background by sharp edges and on the variance of the Comparing the mean results between the AMPCM and PSS contrast gain for different realizations. Further, the contrast with the other methods clearly demonstrates the effect of the gain evaluates the stability of the applied algorithm. The sharpening/enhancing term which yields a better enhanced contrast recovery curve is estimated using the set of regions edges. On the other hand, P-M keeps edges unsmoothed, of interest that were identified by a medical professional. This which is appearing as sharper edges in Figure 5. Figure 5 curve is widely used in the literature for evaluating the quality shows more noise remained, represented as larger values in of the reconstructed PET images [26]. the absolute difference images of P-M and NAF.Furthermore, 3 using AMPCM and PSS, the absolute difference approaches In Figure 3(c),thereare white spots that represent zero (black image), indicating that the noise has been effec- tumors. We calculate the contrast gain CG𝑖 for each real- 𝑓𝑖 ,𝑖 ∈ [1,𝑁] 𝑁 = 100 tively and adaptively reduced from the noisy sinograms. ization PET ( denotes the number 𝜎2 𝑅= The heavy noise is clearly eliminated without destroying of realizations), and its overall variance CG.Let {𝑟 ,𝑟 ,...,𝑟 } =3 the texture (curves) by the probabilistic curvature motion 1 2 no (no in our case) be the set of identified filter. From the above and other various examples, we have ROIs, and let 𝐵 be a representative background tissue area observed that the proposed algorithm has ability to eliminate then the Poisson noise. No stair-casing has been observed, nor 1 1 severe blurring is introduced. Figure 4(c) shows that P-M (𝑡) = ∑ [ ∑𝑓(𝑖) (𝑝, 𝑡) −𝐶 ], CG𝑖 PET bkg filter performs well for relatively low levels of noise, while no 𝑟∈𝑅 Card (𝑟) 𝑝∈𝑟 itresultsinpoorqualityofimagesforhighnoiselevels. However, Perona’s operator does not work well when applied 1 𝑁 𝜎2 (𝑡) = ∑ ( (𝑡) − ), to very noisy images because the noise introduces important CG 𝑁 CG𝑖 CG oscillations of the gradient. 𝑖=1 (44) 4.2. PET Reconstruction Evaluation. In this section, we use 𝑝 𝑡 the smoothed sinograms, 𝑢𝑡, for PET reconstruction via the where denotes a pixel, the scale number, and CG the =(1/𝑁)∑𝑁 (𝑡)2 𝐶 OSEM algorithm. The reconstructed PET image is denoted as contrast mean, CG 𝑗=1 CG𝑗 . bkg represent 12 Journal of Applied Mathematics

(Var-image (noisy PET))

(a) Reconstructed after AMPCM (b) Variance image of (a).

Figure 7: Evaluation of the PET reconstruction using enhanced sinograms.

𝐶 = the mean intensities inside background, defined as bkg In order to perform, under the same conditions, the (𝑖) (1/Card(𝐵)) ∑ 𝑓 (𝑝, 𝑡). comparison of the contrast curves of the different diffusion 𝑝∈𝐵 PET 𝑡 Plottingthecontrastgaininfunctionofthevariancegiven scheme, one should identify the scales, , emanating from the smoothness parameter, which is in this work, the scale different diffusion schemes, having similar information con- parameter 𝑡 yields the contrast recovery curve [26]. tent. For this, we use the scale synchronization proposed by Journal of Applied Mathematics 13

Niessen et al. [27], being a Shannon-Wiener entropy used to problem of poor and discontinuous edges which are common compare nonlinear scale-space filters: in PET sinograms. Applied as a preprocessing step before PET reconstruc- 𝜎2 (𝑓 (0))−𝜎2 (𝑓 (𝑡)) tion, we demonstrated via qualitative measures (the contrast R [𝑓 (𝑡)] ≜ PET PET , (45) PET 𝜎2 (𝑓 (0))−𝜎2 (M (𝑓 )) curve, the variance, and the mean images) that the adaptive PET PET probabilistic diffusion function has a great capability and stability to detect and enhance the important features in the M where represents the averaging operator that maps an reconstructed PET image, which make it a reliable and practi- 𝑓 image PET onto a constant image, where each pixel equals cal approach as it has no free parameters to be optimized. All the average feature vector. parameters are image based and are automatically estimated 6 The contrast gain and the variance were computed for andprovedtogivethebestresults. scales, for each filtering methods, with similar information content, selected based on Niessen et al. [27]approach. Figure 6 shows the contrast recovery curves for the con- Acknowledgment sidered dif-fusion schemes AMPCM, PSS, P-M, and CCM- The authors would like to thank Professor M. Defrise, Sapiro [17]. The best quality PET reconstruction is situated Division of Nuclear Medicine at UZ-VUB, for providing them in the upper, that is, high contrast gain, left, that is, high the PET phantom images, the discussions, and feedback on stability, area of the plot. The figure shows that AMPCM has the Poisson noise and Chi-square prior, as well as on the better contrast recovery of the interesting objects (ROIs) in evaluation of the PET reconstruction results. the reconstructed PET images. Figure 7 illustrates the reconstructed PET starting from the enhanced sinogram, by the AMPCM, along with the References variance images of the reconstructed PET, the mean of [1] P. Perona and J. Malik, “Scale-space and edge detection using the reconstructed PET images, and the absolute difference anisotropic diffusion,” IEEE Transactions on Pattern Analysis between the mean of the filtered PET and the noise-free and Machine Intelligence,vol.12,no.7,pp.629–639,1990. image. We immediately notice that the background seems [2] D. Kazantsev, S. R. Arridge, S. Pedemonte et al., “An anatom- much flatter when adopting the OSEM reconstruction. ically driven anisotropic diffusion filtering method for 3D Experiments results show that combining the probabilis- SPECT reconstruction,” Physics in Medicine and Biology,vol.57, tic diffusivity function with the curvature motion diffusion no. 12, pp. 3793–3810, 2012. produces a powerful nonlinear filtering method that is [3] Z. Quan and L. Yi, “Image reconstruction algorithm for appropriate for the unique characteristics of the PET images. positron emission tomography with Thin Plate prior combined The AMPCM filter preserves the boundaries of the curvy with an anisotropic diffusion filter,” Journal of Clinical Rehabil- shape features and wisely smooths the regions of interest as itative Tissue Engineering Research,vol.15,no.52,2011. well as the other regions. Figure 7 demonstrates that the edge [4] O. Demirkaya, “Post-reconstruction filtering of positron emission tomography whole-body emission images and attenuation enhancement around image data is significantly improved. maps using nonlinear diffusion filtering,” Academic Radiology, The effect of edge strengthening is even more pronounced for vol. 11, no. 10, pp. 1105–1114, 2004. weaker edges in PET images or in image areas affected by a [5] D. R. Padfield and R. Manjeshwar, “Adaptive conductance high level of noise. filtering for spatially varying noisein PET images,” Progress in Biomedical Optics and Imaging,vol.7,no.3,2006. [6] J. Weickert, Anisotropic Diffusion in Image Processing,Euro- 5. Conclusions pean Consortium for Mathematics in Industry, B. G. Teubner, Stuttgart, Germany, 1998. Adaptive probabilistic curvature motion filter (AMPCM) [7] M. Alrefaya, H. Sahli, I. Vanhamel, and D. Hao, “A nonlinear for enhancing PET images is developed and discussed in probabilistic curvature motion filter for positron emission this work. The filter is applied on the 2D sinogram pre- tomography images,” in Scale Space and Variational Methodsin reconstruction. For considering the special characteristics of Computer Vision ,vol.5567ofLecture Notes in Computer the sinogram data, a Chi-square is used as a marginal prior Science, pp. 212–2223, 2009. for noise-free sinogram gradient in the diffusivity function. [8] O. Demirkaya, “Anisotropic diffusion filtering of PET attenua- The qualitative evaluation results results show that, tion data to improve emission images,” Physics in Medicine and among other diffusivity functions, the probabilistic adap- Biology,vol.47,no.20,pp.N271–N278,2002. tive function seems more effective in smoothing all the [9] W. Wang, “Anisotropic diffusion filtering for reconstruction homogenous regions that contain high level of noise. Further, of poisson noisy sinograms,” Journal of Communication and AMPCMretaininanaccuratewaythelocationoftheedges Computer,vol.2,no.11,pp.16–23,2005. that defines the shape of the represented structures in the [10] C. Negoita, R. A. Renaut, and A. Santos, “Nonlinear anisotropic diffusion in positron emission tomography kinetic imaging,” in sinogram. It preserves the boundaries of the curvy shape SIAM Conference on Imaging Science, Salt Lake City, Utah, USA, featuresandwiselysmoothstheregionsofinterestaswellas 2004. the other regions. The application of the proposed diffusion [11] H. Zhu, H. Shu, J. Zhou, C. Toumoulin, and L. Luo, “Image scheme results in well-smoothed images and preserves the reconstruction for positron emission tomography using fuzzy edges. It gains the advantages of the curvature motion nonlinear anisotropic diffusion penalty,” Medical and Biological diffusion and the shock filter. Further, it deals better with the Engineering and Computing,vol.44,no.11,pp.983–997,2006. 14 Journal of Applied Mathematics

[12] H. Zhu, H. Shu, J. Zhou, X. Bao, and L. Luo, “Bayesian algorithmsforPETimagereconstructionwithmeancurvature and Gauss curvature diffusion regularizations,” Computers in Biology and Medicine,vol.37,no.6,pp.793–804,2007. [13] A. P. Happonen and M. O. Koskinen, “Experimental inves- tigation of angular stackgram filtering for noise reduction of SPECT projection data: study with linear and nonlinear filters,” International Journal of Biomedical Imaging,vol.2007,Article ID 38516, 2007. [14] A. A. Samsonov and C. R. Johnson, “Noise-adaptive nonlinear diffusion filtering of MR images with spatially varying noise levels,” Magnetic Resonance in Medicine,vol.52,no.4,pp.798– 806, 2004. [15] A. Pizurica, P. Scheunders, and W. Philips, “Multiresolution multispectral image denoisingbased on probability of presence of features of interest,” in Proceedings of Advanced Concepts for Intelligent Vision Systems (Acivs ’04),2004. [16] I. Vanhamel, Vector valued nonlinear diffusion and its application to image segmentation [Ph.D. thesis],VrijeUniversiteit Brussel, Faculty of Engineering Sciences, Electronics and Infor- matics (ETRO), Brussels, Belgium, 2006. [17] G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, Cambridge, UK, 2001. [18] A. Kuijper, “Geometrical PDEs based on second-order derivatives of gauge coordinates in image processing,” Image and Vision Computing,vol.27,no.8,pp.1023–1034,2009. [19] A. Pizurica,ˇ I. Vanhamel, H. Sahli, W. Philips, and A. Katartzis, “A Bayesian formulation of edge-stopping functions in nonlinear diffusion,” IEEE Signal Processing Letters,vol.13,no.8,pp. 501–504, 2006. [20] A. Teymurazyan, T. Riauka, H. S. Jans, and D. Robinson, “Properties of noise in positron emission tomography images reconstructed with filtered-back projection and row-action maximum likelihood algorithm,” Journal of Digital Imaging,vol. 26, no. 3, pp. 447–456, 2013. [21]G.Miller,H.F.Martz,T.T.Little,andR.Guilmette,“Using exact poisson likelihood functions in Bayesian interpretation of counting measurements,” Health Physics,vol.83,no.4,pp.512– 518, 2002. [22]H.A.Gersch,“SimpleformulaforthedistortionsinaGaussian representation of a Poisson distribution,” American Journal of Physics, vol. 44, no. 9, pp. 885–886, 1976. [23] J. G. Skellam, “The frequency distribution of the difference between two Poisson variates belonging to different popula- tions,” JournaloftheRoyalStatisticalSocietyA,vol.109,no.3, p. 296, 1946. [24] X. Lei, H. Chen, D. Yao, and G. Luo, “An improved ordered subsets expectation maximization reconstruction,” in Advances in Natural Computation,vol.4221,pp.968–971,2006. [25]I.Vanhamel,C.Mihai,H.Sahli,A.Katartzis,andI.Pratikakis, “Scale selection for compact scale-space representation of vector-valued images,” International Journal of Computer Vision,vol.84,no.2,pp.194–204,2009. [26] C. Comtat, P. E. Kinahan, J. A. Fessler et al., “Clinically feasible reconstruction of 3D whole-body PET/CT data using blurred anatomical labels,” Physics in Medicine and Biology,vol.47,no. 1, pp. 1–20, 2002. [27] W.J. Niessen, K. L. Vincken, J. A. Weickert, and M. A. Viergever, “Nonlinear multiscale representations for image segmentation,” Computer Vision and Image Understanding,vol.66,no.2,pp. 233–245, 1997. Copyright of Journal of Applied Mathematics is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.