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Fast High Resolution Blood Flow Estimation andClutter Rejection via an Alternating Optimization

ProblemDuong-Hung Pham, Adrian Basarab, Jean-Pierre Remenieras, Paul

Rodríguez, Denis Kouamé

To cite this version:Duong-Hung Pham, Adrian Basarab, Jean-Pierre Remenieras, Paul Rodríguez, Denis Kouamé. FastHigh Resolution Blood Flow Estimation and Clutter Rejection via an Alternating Optimization Prob-lem. International Symposium on Biomedical Imaging (IEEE ISBI 2021), Apr 2021, Nice, France.�hal-03124597�

FAST HIGH RESOLUTION BLOOD FLOW ESTIMATION AND CLUTTER REJECTION VIAAN ALTERNATING OPTIMIZATION PROBLEM

Duong-Hung Pham1, Adrian Basarab1, Jean-Pierre Remenieras2, Paul Rodrıguez3, Denis Kouame1

1IRIT, CNRS UMR 5505, Paul Sabatier University, Toulouse 31062, France2UMR 1253, iBrain, Tours University, Tours 37032, France

3Pontifical Catholic University of Peru, Av. Universitaria 1801, San Miguel, Lima 32, Peru

ABSTRACT

This paper introduces a computationally efficient techniquefor estimating high-resolution Doppler blood flow from an ul-trafast ultrasound image sequence. More precisely, it consistsin a new fast alternating minimization algorithm that imple-ments a blind deconvolution method based on robust princi-pal component analysis. Numerical investigation carried outon in vivo data shows the efficiency of the proposed approachin comparison with state-of-the-art methods.

Index Terms— ultrafast ultrasound, Doppler, blood flow,clutter suppression, robust PCA, blind deconvolution.

1. INTRODUCTION

Estimating high-sensitivity and high-resolution blood flowfrom an ultrafast ultrasound (US) sequence has received agreat attention from the medical imaging community overthe last few years. The main objective is to provide a clearand accurate 3D visualization of vascular networks insidethe human body and consequently lead to a better prognosisand treatment of many related diseases such as brain gliomas(tumors) or peri-tumoral infiltration. To this end, the spatio-temporal singular value decomposition (SVD) of the datasetCasorati matrix, able to suppress temporally clutter signalsoriginating from quasi-static tissues from blood flow, is themost widespread method [1]. Unfortunately, the manualchoice of the correlation thresholds for separating the bloodspace from the tissue and noise subspaces partially mitigatesthe applicability of this method. To overcome this drawback,many efforts have been made in the literature, among whichthe most popular relies on the robust principal componentanalysis (RPCA) [2], or its variants, e.g, [3]. Moreover, toaccount for the loss of spatial resolution of the Doppler datacaused by the system point spread function (PSF), recently adeconvolution step was embedded in RPCA-based methods,allowing high-resolution and high-sensitivity blood flow esti-mation [4]. This method requires the knowledge of the PSF,that can be experimentally measured or estimated jointly withthe blood flow, as suggested by the recent blind deconvolution(BD) approach, called BD-RPCA in [5]. Despite its accuracy,BD-RPCA suffers from a high computational cost.

In this paper, we propose a new computationally efficientalgorithm in which the nuclear norm term related to the BD-RPCA model is replaced by a fixed rank constraint, followedby a fast partial SVD-based alternating algorithm. The re-mainder of the paper is organized as follows. The backgroundabout the model and related works is given in Section 2. Theproposed fast blind algorithm of blood flow estimation is de-tailed in Section 3. Finally, numerical results on in vivo ultra-fast US data are regrouped in Section 4, showing the improve-ment achieved by the proposed approach over some existingtechniques.

2. BACKGROUND

2.1. Model formulation

Let us consider a temporal in-phase and quadrature (IQ)Doppler ultrafast US sequence, composed of Nt frames ofsizeNx×Nz , with depthNz , probe widthNx and acquisitiontimeNt. S ∈ CNzNx×Nt denotes the related Casorati matrix.The tissue-blood flow model is written as follows [1]:

S = T +B +N , (1)where T ,B,N ∈ CNzNx×Nt are the tissue, blood and ad-ditive noise matrices, respectively. To avoid further nota-tions, the vectorized counterparts of these matrices are, whenneeded, considered hereafter and defined using the same no-tations. In most practical applications, the blood flowB is as-sumed to be sparse, conventionally promoted by the l1-norm,while the tissue T possesses a slight change over time, mod-elled by the nuclear norm ||.||∗. Also, N is assumed to be anadditive Gaussian noise. The main objective of this work is torecover B and T from S based on building an optimizationproblem imposing appropriate constraints on these matrices.

2.2. Blind deconvolved RPCA

An appealing method, called blind deconvolved robust prin-cipal component analysis (BD-RPCA), has been recently pro-posed in [5]. It aims at reconstructing a high resolution bloodflow X simultaneously with the tissue component T and thesystem PSF He from the acquired US Doppler data. The as-sociated optimization problem is formulated as follows:

[X, He, T ] = arg minX,He,T

{||S −He ~X − T ||2F

+ λ||X||1+ρ||T ||∗} , s.t. |F (He)|= H, (2)where ~ denotes the 2D convolution, . the estimated vari-ables, ||.||2F the Frobenius norm and λ, ρ > 0 two hyper-parameters balancing the trade-off between the sparsity of theblood and the low-rankness of the tissue [2, 3]. H is the mag-nitude of the 2D Fourier transform (F ) of the PSF that can bestraightforwardly computed from S− T using homomorphicfiltering [6]. For computational efficiency, the above 2D con-volution, supposed circulant, is rewritten as an element-wisemultiplication in the Fourier domain. To solve (2), a two-stepalternating algorithm was proposed in [5], resumed hereafter.

i) For fixed X and T and using spatio-temporally invariantassumption on PSF He, (2) can be reformulated as:

[He(k+1)

] = arg minHe

{||∑Nt

(S − T (k+1)

)− He ~

∑Nt

(X(k+1))||2F

}, s.t.|F (He)|= H, (3)

where∑Nt

(Z) stands for the temporal average of a 2D ma-

trix Z obtained by taking the mean along the time dimen-sion of the 3D version of this matrix. Then, (3) can besolved by the blind deconvolution (BD) technique [7].

ii) For a fixed He, (2) reads as:

[X(k+1)

, T(k+1)

] = argminX,T

{||S −H(k+1)

e ~X − T ||2F

+ λ||X||1+ρ||T ||∗} . (4)

To solve (4), the alternating direction method of multipliers(ADMM) was used [4, 8]. It consists in performing, at eachiteration (k + 1), the five following main steps:

T(k+1)

= argminT{ρ||T ||∗

+µ

2||T − (S −H(k+1)

e ~X(k) +1

µν(k))||2F

},

z(k+1) = argminz

{λ||z||1+

µ

2||z − (X(k) +

1

µw(k))||2F

},

(5)

(6)

X(k+1)

= argminX

{µ2||S −H(k+1)

e ~X − T (k+1)

+1

µν(k))||2F+

µ

2||X − z(k+1) +

1

µw(k)||2F

},

ν(k+1) = νk + µ(S −H(k+1)e ~X(k+1) − T (k+1)),

w(k+1) = ν(k) + µ(X(k+1) − z(k+1)),

(7)

(8)

(9)

where z is an auxiliary variable equal toX , ν the Lagrangemultiplier and µ the Lagrangian penalty parameter.

Although BD-RPCA has been shown to be an efficientmethod for estimating high-resolution blood flow together

with the tissue and the PSF, it is not computationally attrac-

tive. Indeed, the estimation of T(k+1)

in (5), is a convexproblem that has an analytical solution corresponding to asingular value thresholding (SVT) [9]. This step is computa-tionally expensive since it requires a full SVD decompositionof a large filtered matrix, of the size of the acquired spatio-temporal data.

3. PROPOSED FAST BD-RPCA METHODThe main aim of the proposed algorithm is to reduce the com-putational time of BD-RPCA by introducing a fast variant of(4) and subsequently combining it with BD as explained insubsection 2.2. More precisely, instead of imposing the nu-clear norm relaxation in (4), the rank is itself constrained asfollows:

[X(k+1)

, T(k+1)

] = argminX,T

{||S −H(k+1)

e ~X − T ||2F

+ λ||X||1} s.t. rank(T ) = rf , (10)

where H(k+1)e is assumed to be known from the previous it-

eration and rf is a rank hyperparameter that needs to be tuned.Interestingly, the reference value of rf can be efficiently esti-mated using a two-step algorithm called Rank Estimation de-veloped for low-rank matrix completion problem in [10]. Theidea behind this technique is to find the index i correspondingto the rank value that minimizes the following cost function:

R(i) =σi+1 + σ1

√iζ

σi, (11)

where ζ =|CS |√

NzNx×Ntwith |CS | and σi respectively the car-

dinality, and the singular values of the acquired Casorati ma-trix.

To solve (10), we propose herein a two-step alternatingminimization algorithm as follows:

T(k+1)

= argminT

{||(S −H(k+1)

e ~X(k))− T ||2F

}s.t. rank(T ) = rf ,

X(k+1)

= argminX

{||(S − T (k+1)

)−H(k+1)

e ~X||2F

+ λ||X||1} .

(12)

(13)

Subproblem (12) can be rapidly solved based on computinga partial SVD decomposition with only the first rf compo-nents of S−H(k+1)

e ~X(k) [11]. In particular, for small rf ,this decomposition can be performed consistently faster whenusing the Lanczos bidiagonalization algorithm with partial re-orthogonalization through lansvd routine from PROPACK li-brary [12, 13]. Subproblem (13) consisting in a well-knownl1 regularized least absolute shrinkage and selection opera-tor (LASSO) regression can be simply solved efficiently bymeans of an ADMM-based algorithm [8].

The pseudo algorithm associated with the proposedmethod, called fast BD-RPCA, is reported in Algorithm1. The initial values of the blood and tissue are guessed usingSVD for a more efficient convergence of the algorithm [5].

Algorithm 1: fast BD-RPCAInput: Casorati matrix SInitialize: ε = 10−6, [X(0),T (0)] = SVD(S)while ||X(k+1) −X(k)||F> ε do

1. compute temporal average: M (k+1)ST =

∑Nt

(S − T (k)

)2. estimate PSF: H(k+1)

e = BD(M

(k+1)ST

), u. sing Eq. (3)

3. update [X(k+1),T (k+1)] using new fast procedure:

(a) compute a partial SVD and T (k+1):[U,∆, V ] = SVD

(S −H(k+1)

e ~X(k), rf

),

T (k+1) = U∆V T

(b) estimateX(k+1) = LASSO(S − T (k+1),H(k+1)

e

)using Eq. (13)

endOutput: X(k+1), T (k+1) and H(k+1)

e

4. NUMERICAL RESULTS

This section regroups numerical results on in vivo US data todemonstrate the improvement achieved by the proposed ap-proach over other existing methods including SVD [1], ran-domized low-rank & sparse matrix decomposition (GoDec)[14] and BD-RPCA [5]. All the experiments were con-ducted using MATLAB R2019b on a computer with Intel(R)Core(TM) i5-8500 CPU @3.00 GHz and 16GB RAM. 1

4.1. Data acquisitionThe ultrafast Doppler data was acquired in Regional Univer-sity Hospital Bretonneaux of Tours – Department of Neuro-surgery from a patient undergoing brain surgery with openskull whose dura mater had been removed. The AixplorerTM(Supersonic Imagine) ultrasound scanner with the SL10-2probe (192 elements) was used for US acquisition. Theresearch package (SonicLab V12) was used to obtain a par-ticular US sequence of 1000 frames, compounded angles[−5o, 0o,+5o] with pulse repetition frequency PRF=3KHz,frame rate 1KHz, imaging depth [1mm-40mm]. The resultantdataset size was 260× 192× 1000 pixels.

Table 1. Optimal hyperparameter settingSVD Tc = 100, Tb = 150

GoDec rG = 41, τG = 5× 103

BD-RPCA λB = 0.0051, µB = 0.0403, ρB = 1

fast BD-RPCA λf = 8× 10−5, rf = 41

4.2. Performance comparisonTo ensure fair comparison, the hyperparameters correspond-ing to each tested method were tuned by cross-validation to

1The Matlab scripts implementing the method and generating all figuresare available at github.com/phamduonghung/fast_BDRPCA

the best possible values. Table 1 regroups the optimal hyper-parameters associated with SVD, GoDec, BD-RPCA, and fastBD-RPCA. Moreover, for visualization purpose, the PowerDoppler image (in dB) was used and defined from the esti-mated blood flowX , for a given position (x, z), as follows:

IPD(x, z) = 10 log 10

(1

Nt

Nt∑k=1

X(x, z, k)2

).

In Fig. 1, we depict the Power Doppler image resultsobtained by the four methods carried out on peritumoral data.Visually examining these plots, one may remark that thePower Doppler images estimated by both SVD and GoDecare very noisy and blurred while those given by BD-RPCAand fast BD-RPCA, overall, provide clear and similar picturesof vessel structures with high resolution. To quantitativelycompare these methods, contrast ratio (CR) was used [15]:

CR[dB] = 20 log 10

(µR2

µR1

),

where µR1(resp. µR2

) is the mean value of intensities in thewhite (resp. green) rectangular patch R1 (resp. R2) character-izing the background (resp. the blood signal). Furthermore,for each Power Doppler image, R1 is kept fixed as a referencepatch while R2 is moved around in a such way that it occupiesall non-overlapping positions of the image (more details thereader refers to [5]). The resulting CR values are representedby a boxplot as in Fig. 2. From this boxplot, one can noticethat the fast proposed algorithm exhibits an equivalent resultto BD-RPCA, and outperforms SVD and GoDec.

Moreover, Table 2 shows the running time correspondingto each studied method on the peritumoral dataset. It is no-ticeable that as expected, the computational times obtained byboth SVD and GoDec are the lowest while the one given bythe fast proposed algorithm is significantly lower than BD-RPCA. A much lower computational time of SVD or GoDecstems from the fact the the blind deconvolution approach isat the expense of much more running time because of the al-gorithmic complexity, but ensures considerably higher reso-lution results.

To conclude, the above numerical in vivo results plead infavour of using fast BD-RPCA, not requiring PSF measure-ments and well balancing the trade-off between the low com-putational cost and the high quality blood flow reconstruction,compared to the other studied methods.

Table 2. Running times in s for each methodSVD GoDec BD-RPCA fast BD-RPCA10.38 3.7 219.9 89.2

5. CONCLUSIONIn this paper, we introduced a novel fast algorithm for estimat-ing the blood flow and the tissue from an ultrafast sequence ofUS images, based on improving the model of BD-RPCA. Theproposed technique not only enabled to handle the limitationrelated to the high computational cost of the latter but alsoproviding similar recovery performances. Numerical simu-lations carried out on in vivo peritumoral data showed the

R1

R2

-10 0 10

X [mm]

10

20

30

40

Z [

mm

]

(a)

-10 0 10

X [mm]

10

20

30

40

Z [

mm

]

(b)

-10 0 10

X [mm]

10

20

30

40

Z [

mm

]

(c)

-10 0 10

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Z [

mm

]

(d)

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5

10

15

20

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30

35

Z [

mm

]

-35

-30

-25

-20

-15

-10

-5

0

Fig. 1. Power Doppler images obtained by (a): SVD; (b): GoDec; (c) BD-RPCA; (d): fast BD-RPCA.

SVD GoDec BD-RPCA fast BD-RPCA

-10

0

10

20

30

40

50

60

CR

[dB

]

Fig. 2. CR measurements for the different tested methods.

the benefits of using the proposed algorithm. Future workcould be devoted, for instance, to evaluate the clinical contri-bution of the proposed method in the prognosis and treatmentof blood-related diseases, or to extend the proposed methodto a spatio-temporally invariant PSF model.

6. COMPLIANCE WITH ETHICAL STANDARDSThe data acquisition was done with a clinical research proto-col (ELASTOGLI) approved by the institutional review board(CCP: ‘Comite de Protection des Personnes’,N o 123748) andlocal ethical committee. It strictly complies with the ethicalprinciples for medical research involving human subjects ofthe World Medical Association Declaration of Helsinki.

7. ACKNOWLEDGMENTSThis work was partly funded by France Life Imaging (grantANR-11-INBS-0006).

8. REFERENCES

[1] C. Demene et al., “Spatiotemporal clutter filtering of ul-trafast ultrasound data highly increases doppler and ful-trasound sensitivity,” IEEE Trans. Med. Imag., vol. 34,no. 11, pp. 2271–2285, Nov. 2015.

[2] J. Wright et al., “Robust principal component analysis:Exact recovery of corrupted low-rank matrices via con-vex optimization,” Neural Inf. Process. Syst, 2009.

[3] M. Bayat and M. Fatemi, “Concurrent clutter and noisesuppression via low rank plus sparse optimization fornon-contrast ultrasound flow doppler processing in mi-crovasculature,” in IEEE Int. Conf. Acoust., Speech Sig-nal Process. (ICASSP), Calgary, Canada, April 2018.

[4] H. Shen et al., “High-resolution and high-sensitivityblood flow estimation using optimization approacheswith application to vascularization imaging,” in EEE Int.Ultrason. Symp. (IUS), 2019, pp. 467–470.

[5] D. H. Pham, A. Basarab, I. Zemmoura, J. P. Remenieras,and D. Kouame, “Joint blind deconvolution and robustprincipal component analysis for blood flow estimationin medical ultrasound imaging,” IEEE Trans. Ultrason.,Ferroelect., Freq. Control, pp. 1–1, 2020.

[6] T. Taxt, “Restoration of medical ultrasound imagesusing two-dimensional homomorphic deconvolution,”IEEE Trans. Ultrason., Ferroelect., Freq. Control,vol. 42, no. 4, pp. 543–554, 1995.

[7] O. Michailovich et al., “Iterative reconstruction of medi-cal ultrasound images using spectrally constrained phaseupdates,” in IEEE 16th Int. Symp. Biomed. Imag. (ISBI),April 2019, pp. 1765–1768.

[8] S. Boyd et al., “Distributed optimization and statisticallearning via the alternating direction method of multi-pliers,” Found. Trends Mach. Learn., vol. 3, no. 1, pp.1–122, 2010.

[9] J.-F. Cai, E. J. Candes, and Z. Shen, “A singular valuethresholding algorithm for matrix completion,” SIAM J.Optim., vol. 20, no. 4, pp. 1956–1982, Jan. 2010.

[10] R. H. Keshavan, A. Montanari, and S. Oh, “Matrixcompletion from noisy entries,” J. Mach. Learn. Res.,vol. 11, p. 2057–2078, August 2010.

[11] P. Rodriguez and B. Wohlberg, “Fast principal compo-nent pursuit via alternating minimization,” in 2013 IEEEInt. Conf. on Image Process., 2013, pp. 69–73.

[12] R. M. Larsen, “Lanczos bidiagonalization with partialreorthogonalization,” DAIMI Report Series, vol. 27, no.537, Dec. 1998.

[13] ——, “Propack,” available at: http://sun.stanford.edu/∼rmunk/PROPACK/, March 2004.

[14] T. Zhou and D. Tao, “Godec: Randomized low-rank& sparse matrix decomposition in noisy case,” ser.ICML’11. Madison, WI, USA: Omnipress, 2011.

[15] A. Rodriguez-Molares et al., “The generalized contrast-to-noise ratio,” in IEEE Int. Ultrason. Symp. (IUS).IEEE, Oct. 2018.