INDIRECT 4D NONLOCAL MEANS

P. Gonzalez1,3, S. Serriere1, D. Guilloteau1, I. Buvat2 and C. Tauber1

1 UMRS INSERM U930, Universite Francois Rabelais, Tours, France2 CEA/I2BM/SHFJ/LIME, 91402 Orsay, France

3 DCI Universidad Catolica del Maule, Talca, Chile

ABSTRACTIn this work, we present a new approach to enhance the

signal-to-noise ratio of 3D vector-valued images. We extendthe original nonlocal means proposed by Buades to 3Dvector-valued images. In the proposed approach, the simi-larity between vector-valued voxels is calculated indirectlyfrom a smoothed image, avoiding the use of patches andmaking use of the entire spectral information. Moreover, weintroduce weights in the calculus of the similarity to favorlow-noise channels. The weights are estimated automaticallyfrom a wavelet analysis of each channel. Results on real PETDynamic acquisitions and GATE Monte Carlo simulationsillustrate the potential of the proposed method, which led todistinct improvements of figures of merit over several otherapproaches from the literature.

Keywords-Dynamic PET, Image restoration, filtering, nonlo-cal means.

I. INTRODUCTION

Vector-valued images occur in various contexts such ascolor images, hyper-spectral images, medical images ac-quired at subsequent time intervals (e.g. functional MRI,dynamic PET images) or textured images. These imagescan be affected by noise, low spatial resolution and varyingreliability of the different channels that compose the image,which may make their analysis difficult.

Gaussian Filtering (GF) can remove high frequency noisein an effective way, however due to the isotropic nature ofsuch filter, noise and salient features of the image are indis-criminately smoothed, lowering the resolution and increasingthe blurring. In order to avoid such problems, edge preserv-ing filters have been presented in the literature. Perona andMalik [1] proposed an anisotropic diffusion to smooth grey-valued images where edges are preserved by averaging onthe orthogonal direction of the local gradient. Total variationminimization was introduced by Rudin et al. [2] to recoverthe image as the solution of a minimization problem. Bilat-eral Filtering (BF) is a nonlinear filter proposed by Tomasiand Manduchi [3] which is defined as a weighted averageof the nearby voxels by taking into consideration both thegeometric closeness and photometric similarity to preserveedges while smoothing. Nonlocal Means (NLM) formulated

by Buades [4], is based on the redundancy of information inimages to remove noise. The nonlocal selection scheme im-proves the denoising performances, at the cost of increasedcomputational time. Wu et al. [5] present a modification ofNLM using an anisotropic structure tensor to better preservefine details of the image. An robust estimation approachbased on m-estimator reducing the effect of uncorrelatedpatches is presented in [6]. In medical imaging community,NLM have been used for Magnetic Resonance (MR) imagedenoising. A fast version of NLM for 3D MR images ispresented in [7]. An adaptation to rician noise is presentedin [8].

The above methods have been proposed to denoise singlecomponent images and they are not tailored to take intoconsideration the multispectral intrinsic nature of vector-valued images. In parallel, efforts have been made to exploitthe spectral consistency of such data. An edge preservingdenoising approach called vector-based robust anisotropicdiffusion (VRAD) that exploits the spatio-spectral consis-tency was presented by Tauber et al. [11]. Mendrik etal. [12] presents an extention of BF (4DBF) incorporingthe similarity between vector-valued voxels. Extentions ofNLM have also been proposed to denoise vector-valuedimages. Gal et al. [14] exploit the redundancy of informationin temporal sequences of MR images. Chan et al. [15]incorporate anatomical information in PET images. Duttaet al. [16] precompute the similarity between voxels alongtime using the L2 norm differences for all voxels pairsfor dynamic PET images. More recently Dutta et al.[17]extended nonlocal means to process 2D+t dynamic PETimages using spatial and temporal patches.

In this work we propose a new nonlocal means ap-proach for 3D vector-valued image restoration. The proposedmethod called 4D Indirect NonLocal Means (4DINLM)exploits similarities between vector-valued voxels, favoringlow-noise channels considered more reliables. An indirectcalculus is used to avoid the use of patches, decreasing thecomputational complexity of the algorithm.

To assess the performance of the proposed method, weperformed real dynamics PET acquisitions of rodents. GATEMonte Carlo simulations of dynamic PET images were gen-erated to establish objective comparisons with other image

restoration methods.

II. PROPOSED METHOD

In the continuous domain, a 3D vector-valued image isrepresented by a mapping I : R3 → RN , where N is thenumber of channels. Let x = (x1, x2, x3) ∈ R3 be one voxeland Ik be the kth channel of the image.

Given a noisy image I, the estimated value I4INLM(x)for a vector-valued voxel I(x) is computed as a weightedaverage of voxels of the image as follows:

I4DINLM(x) =∑y∈Nx

wω(I(x), I(y))I(y) (1)

where I(x) is the vector voxel at the position x, Nx ⊆ Iis a rectangular search window centered on the voxel I(x).The weights wω are calculated using the similarities betweenvector-valued voxels as follows:

wω(I(x), I(y)) =e

sω(I(x),I(y))

h2∑y∈Nx

esω(I(x),I(y))

h2

(2)

where h is a constant and acts as a degree of filteringand sω(I(x), I(y)) is an indirect similarity between vector-valued voxels x and y, described in the following section.The weights wω(I(x), I(y)) satisfy the conditions 0 ≤wω(I(x), I(y)) ≤ 1 and

∑y wω(I(x), I(y)) = 1.

II-A. Indirect similarity between vector-valued voxels

Similarity between vector-valued voxels of the noisy im-age I is estimated indirectly from a smoothed image IF .Let IF (x) and IF (y) be two vector-valued voxels of IF , theweighted similarity of the voxels I(x) and I(y) is calculatedas follows:

sω(I(x), I(y))ω =

n∑k=1

ωk ‖ IF (x)k − IF (y)k ‖2 (3)

where ωk is the weighting factor of the kth channel. Itsatisfies the usual conditions 0 ≤ ωk ≤ 1 and

∑nk=1 ωk = 1.

n is the number of channels of I.The smoothed version of the noisy image is obtained as

follow:

IF = Ksρs ∗ (Kv

ρv I) (4)

where Ksρs is a 3D Gaussian kernel of scale ρs, ∗ denotes

the spatial 3D convolution operator, denotes the 1D convo-lution in spectral domain and Kv

ρv is a 1D Gaussian kernel ofscale ρv . Basing the similarity calculus on smoothed versionsof the vector values increases the robustness to noise andavoids the necessity to use patches. Doing so, it diminishesthe computational cost of the method.

II-B. Weighting factorsWe adopt a weighting strategy that relies on a blind

estimate of the noise variance using the discrete wavelettransform of the signal [18]. For vector-valued images,an estimate of the noise variance can be established byanalyzing the distribution of the coefficients at the fine scalesof the decomposition [19]. We define the weighting factorωk for Ik as the inverse of the noise variance estimate σ2

k:

ωk =1

σ2k

=

(0.6745

MAD(|W |)

)2

(5)

where MAD denotes the median absolute deviation andW are the wavelet coefficients in the subband obtained byapplying separable high-pass filters in the three directionsof space, corresponding to the finest resolution scale of themultilevel 3D discrete wavelet transform. The factor 0.6745is chosen for calibration with the normal distribution, asproposed in [19].

III. EXPERIMENTATIONSWe applied the proposed approach to dynamic PET im-

ages which consists in subsequent tomographic reconstruc-tions of a radiotracer concentration map in brain tissues overtime. Time series of uptake values corresponding to eachvoxel in the field of view exhibit the local kinetics of theradiotracer, called time-activity curve (TAC). While yieldingvery valuable functional information, dynamic PET imagessuffer from low spatial resolution, low SNR and the contrastbetween features of interest varies across time.

III-A. Simulated PET imagesTo assess the proposed 4DINLM approach, we performed

realistic Monte Carlo simulations [20] of Philips GeminiGXL PET 4D acquisitions, using the Zubal head phantom asa voxelized brain source [21]. This phantom constitutes theground truth which was used for objective validation. TACswere generated according to a three compartment model[22]. Two dynamic PET images with different signal-to-noiseratios (SNR) were reconstructed and respectively denotedas mid SNR and low SNR simulations (simulation 1 andsimulation 2 respectively). We used a fully 3D OP-OSEM(ordinary Poisson ordered subset expectation maximization)iterative method to reconstruct images into 20 frames of2.2 × 2.2 × 2.8 mm3 voxels. For simulation 1, the recon-struction was performed using 5 iterations and 8 subsets. Forsimulation 2, we used 10 iterations and 16 subsets.

III-B. Real PET images4DINLM was also applied to a real [18F]DPA-714 acqui-

sition of a rat brain. [18F]DPA-714 is a radiotracer of thetranslocation protein (TSPO), which is over expressed underpathologic neuroinflammatory conditions [24], [25]. The im-ages were acquired on a microPET-CT GE Vista in list-modeand reconstructed using 2D-OSEM method using 8 iterations

and 16 subsets, with corrections for attenuation, random andscattered coincidences. 27 time frames of 175 × 175 × 61voxels of 0.39× 0.39× 0.78 mm3 were reconstructed overa period of 50 minutes according to the following protocol:4×10s, 4×20s, 6×60s, 10×80s, 3×600s. The inflammationwas induced by performing unilateral quinolinic acid lesionsin the right striatum of the rat.

Finally 4DINLM was applied to a real [18F]LBT−999 ac-quisition of a rat brain. The sequence was acquired on a GEExplore Vista microPET/CT scanner using 27 consecutiveframes of following durations: 4 × 10s, 4 × 20s, 4 × 60s,14×180s and 1×120s. The image was reconstructed using aniterative 2D OSEM method using 2 iterations and 16 subsets,with voxel sizes of 0.2×0.2×0.2 mm3. The dynamic imageswere registred into Paxinos coordinates with Pmod 3.3 usingthe rat brain FDG template. The Schiffer Atlas provided anindication of the expected localization of the striata, althoughnot constituting an absolute ground truth [26].

III-C. Comparison of 4DINLM to other methods

We compared our results with three denoising approachesof the literature that we describe in this section.

Nonlocal Means (NLM): Proposed by Buades in [4],NLM estimate the value of the voxel x as a weighted averageof the voxels of the noisy image. A marginal treatment usingNLM in each channel is defined as follows:

INLM(x)k =∑y∈Nx

w(I(x)k, I(y)k)Ik(y) (6)

The weights w(x,y) depend on the similarity of the vectorsof intensity levels level v(Nx) and v(Ny), where Nn calledpatches, denotes a square neighborhood of voxel size andcentered at a voxel n (smaller than the search windowWx ⊆ I centred in x). The similarity is measured asa decreasing function of the weighted euclidean distance‖ v(Nx)−v(Ny) ‖22,a, where a > 0 is the standard deviationof the Gaussian kernel.

The weights w are defined as follows:

w(I(x), I(y)) =e−

‖Ka(v(Nx)−v(Ny))‖22h2∑

y∈Wxe−

‖Ka(v(Nx)−v(Ny))‖22h2

(7)

h acts as a degree of filtering. It controls the decay of theexponential function.

Bilateral Filter: Proposed by Tomasi [3], Bilateral Filter(BF) is a nonlinear filter that gained interest in imageprocessing and specially in medical images due to its edgepreserving capabilities. An extention of BF to 4D images(4DBF) is presented in [12]. The equation that describe the4DBF is defined as follows:

I4DBF (x) =∑

y∈N(x)

w(I(x), I(y))I(y) (8)

where N(x) denotes the set of neigbouring voxels aroundx. The weights w are based on the product of two separatesimilarities acting in the 3D spatial and the spectral domains:

w(I(x), I(y)) =e

(− (x−y)2

2σ2s

)e

(‖I(x)−I(y)‖2

2σ2v

)

∑y∈N(x) e

(− (x−y)2

2σ2s

)e

(‖I(x)−I(y)‖2

2σ2v

) .

(9)where ‖ I(x) − I(y) ‖ is the euclidean distance betweenvector valued voxels x and y. Two scale factors σ2

s and σ2v

control geometric and spectral spreads respectively.Vector-based Robust Anisotropic Diffusion (VRAD):

Proposed by Tauber et al. [11], VRAD performs nonlineardiffusion in a vector fashion based upon weighted Euclideandistances between neighbouring vectors. The associated dif-fusion problem is∂I∂t − div(c(||∇I||)∇I) = 0, everywhere in Ω, 0 < t ≤ T,

∂I∂n |∂Ω = 0, ∀t ∈ [0, T ] (boundary conditions),

0I(x) = I(x, 0) = (initial conditions),(10)

where the coefficient of diffusion c(||∇I||) is based uponrobust statistics and enables to completely stop the diffusionacross identified edges in order to better preserve functionalboundaries.

III-D. Quantitative criteriaThe following quantitative criteria were measured:• The Total Signal to Noise Ratio (TSNR) was defined

as:

10log10

(||Itruth||

||Itruth − Ires||

)2

, (11)

where Itruth is the ground truth and Ires the imageresulting from the filtering process.

• The Structural Similarity Index (SSIM) is a recent im-age quality metric that can offer a good approximationof perceived image distortion [23]. It returns a numberbetween 0 and 1, with 1 being a perfect match. We usedthe same parameters as the one found in the seminalpaper of Wang et al.

• Pratt’s figure of merit (PFOM) returns a number be-tween 0 and 1 that increases with the quality of the edgepreservation and enhancement. The PFOM is based onedge detection and localization. An automatic Cannydetector from Matlab v2010a was applied on eachimage as a prior step for objective evaluation. ThePFOM was defined as:

PFOM(A,B) =1

max(NA, NB)

NB∑i=1

1

1 + αd2i, (12)

where NA and NB are respectively the number of theactual and detected edge voxels, di denotes the distance

from the ith-detected edge voxel to the nearest actualedge voxel and α is a scaling constant set to 1/9 as inPratt’s work (Pratt 1977).

IV. RESULTS

(a) (b) (c)

(d) (e) (f)

Fig. 1. Representative results for simulation 1 (mid SNR):(a) Ground Truth (GT), (b) Unprocessed simulated image(UI), (c) NLM, (d) VRAD (e) 4DBF and (e) 4DINLM.

(a) (b) (c)

(d) (e) (f)

Fig. 2. Representative results for simulation 2 (low SNR):(a) Ground Truth (GT), (b) Unprocessed simulated image(UI), (c) NLM, (d) VRAD (e) 4DBF and (e) 4DINLM.

Representative results with the four approaches on bothsimulations are presented in figures 1 and 2 respectively. Avisual inspection suggests that VRAD, 4DBF and 4DINLMwere better able to denoise the original image than mono-component NLM. In simulation 1, all filters improved theSNR, however 4DINLM was better at denoising whileperserving resolution. In simulation 2 results obtained forVRAD, 4DBF and 4DINLM were relatively similar in the

central part of the field of view (FOV), with functionalboundaries more sharpened with VRAD. However VRADfailed at correctly denoising images at the extremities ofthe axial FOV (figure 2(d)). Due to inherent properties ofPET imaging, regions of the image at extremities of theaxial direction (upper and lower parts of UI, figure 2(b))are more impaired by noise than the center of the field ofview. 4DINLM was much more robust to this effect andshowed more homogeneous denoising ability throughout theFOV. The result obtained with 4DINLM appeared betterfiltered from noise. Figure 3 shows representative results of

(a) (b) (c)

(d) (e) (f)

Fig. 3. Representative results of automatic Canny edgedetection for simulation 2 (low SNR): (a) Ground Truth(GT), (b) Unprocessed simulated image (UI), (c) NLM, (d)VRAD (e) 4DBF and (e) 4DINLM.

Canny automatic edge detection in the mid-saggital view forsimulation 2. 4DINLM obtained better edge detection withless spurious edges than the other methods. Figure 4 and 5

0 5 10 15 201

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

CHANNEL

TSNR

UINLM4DBFVRAD4DINLM

(a) TSNR

0 5 10 15 20

0.2

0.25

0.3

0.35

0.4

CHANNEL

SSIM

UI

NLM

4DBF

VRAD

4DINLM

(b) SSIM

0 5 10 15 200.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

CHANNEL

PFOM

UI

NLM

4DBF

VRAD

4DINLM

(c) PFOM

Fig. 4. Quantitative criteria for simulation 1: (a) TSNR (b)SSIM and (c) PFOM.

show TSNR, SSIM and PFOM scores as a function of thechannel for both simulations 1 and 2 respectively. Table Iand II shows quantitative results averaged over the channelsfor both simulations. All the filters improved the quality ofthe images with respect to the different figures of merit. TheVRAD approach obtained second best results on average inmost of results but failed at recovering the FOV extremities(figure 2(d)). 4DINLM is more robust to variations of noisealong the FOV, with a better recovery of the peripherical

0 5 10 15 200

1

2

3

4

5

6

CHANNEL

TSNR

UI

NLM

4DBF

VRAD

4DINLM

(a) TSNR

0 5 10 15 200.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

CHANNEL

SSIM

UI

NLM

4DBF

VRAD

4DINLM

(b) SSIM

0 5 10 15 200.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

CHANNEL

PFOM

UI

NLM

4DBF

VRAD

4DINLM

(c) PFOM

Fig. 5. Quantitative criteria for simulation 2: (a) TSNR (b)SSIM and (c) PFOM.

slices. Overall, 4DINLM most improved the average TSNR,SSIM and PFOM scores. A representative result of 4DINLM

Table I. Figures of merit averaged over the channels forsimulation 1.

Method TSNR SSIM PFOMUI 1.49± 0.42 0.19±0.01 0.44±0.01NLM 3.26±0.33 0.22±0.02 0.56±0.014DBF 4.43±0.34 0.31±0.01 0.63±0.01VRAD 5.06±0.39 0.32±0.01 0.65±0.014DINLM 5.42±0.26 0.36±0.00 0.78±0.00

Table II. Figures of merit averaged over the channels forsimulation 2 .

Method TSNR SSIM PFOMUI 1.22± 0.36 0.22±0.02 0.42±0.01NLM 2.67±0.41 0.31±0.02 0.51±0.014DBF 3.86±0.48 0.35±0.02 0.53±0.01VRAD 3.69±0.51 0.38±0.03 0.54±0.014DINLM 4.57±0.59 0.59±0.02 0.70±0.01

on the real PET image with [18F]-DPA−714 is presentedin figure 6, where the lesion is indicated by an arrow.Despite the high number of reconstruction iterations whichinduces high levels of noise, a notable increase in SNR wasobserved in the filtered image, in which the striatal lesionappears more clearly. Figure 7 shows a representative resultobtained with 4DINLM filter on [18F]LBT−999 dynamicPET acquisition of a rat. A salient high uptake region,consistent with the expected morphology of the striata regionwas obtained.

V. CONCLUSIONWe have proposed a new approach for restoring 3D

vector-valued images. Our approach make use of the entirespatial and vectorial information available in each voxelto restore the image. In our experiments, 4DINLM led toimproved figures of merit compared to three other filtersof the literature, confirming the potential of the proposedapproach.

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(a) (b)

(c) (d)

Fig. 6. [18F]-DPA−714 dynamic PET acquisition brain ofa quinolonic acid lesion (indicated by an arrow). Top row:Axial in frame 15: (a) Unprocessed image, (b) 4DINLM.Bottom row: Coronal view: (c) Unprocessed image and (d)4DINLM.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Fig. 7. Brain PET image of a rat with [18F]-LBT-999. Toprow: Axial view: (a) Unprocessed Image, (b)4DINLM, (c)4DINLM with superimposed atlas of the striatum. Middlerow: Sagittal view: (d) Unprocessed Image, (e) 4DINLM, (f)4DINLM with superimposed atlas of the striatum. Bottomrow: Coronal view: (g) Unprocessed Image, (h) 4DINLM,and (i) 4DINLM with superimposed atlas of the striatum.

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