B-lines, defined as discrete laser-like vertical hyperechoic reverberation artefacts in lung ultrasounds, have been shown to correlate with extravascular lung water in symptomatic and asymptomatic adults and children on dialysis. Recent studies have shown this technique to be useful in children on dialysis. B-line detection in lung ultrasound in children is novel and challenging. This paper presents a novel automatic B-line detection via an inverse problem involving the Radon transform, which is solved using the alternating direction method of multipliers (ADMM). The proposed method restores the lines in the speckle images. Then, the B-lines are automatically detected using a simple local maxima technique in the Radon transform domain, associated with some definitions of the line artefacts. The results show that our proposed method outperforms existing methods by up to 69 %.</p

Achim, Alin

Allinovi, Marco

Anantrasirichai, Pui

Hayes, Wesley

Nantheera Anantrasirichai

Marco Allinovi

Wesley Hayes

Alin Achim

Crossref

Automatic B-line detection in paediatric lung ultrasound

Explore Bristol Research

                          Anantrasirichai, P., Allinovi, M., Hayes, W., & Achim, A. (2016). AutomaticB-line detection in paediatric lung ultrasound. In 2016 IEEE InternationalUltrasonics Symposium, IUS 2016. (Vol. 2016 November). [7728620] IEEEComputer Society. DOI: 10.1109/ULTSYM.2016.7728620Peer reviewed versionLink to published version (if available):10.1109/ULTSYM.2016.7728620Link to publication record in Explore Bristol ResearchPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia IEEE at http://ieeexplore.ieee.org/document/7728620/. Please refer to any applicable terms of use of thepublisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlAUTOMATIC B-LINE DETECTION IN PAEDIATRIC LUNG ULTRASOUNDNantheera Anantrasirichai1, Marco Allinovi2, Wesley Hayes3, Alin Achim11Visual Information Lab, University of Bristol, Bristol, UK2Paediatric Nephrology Unit, Meyer Childrens Hospital, Florence, Italy3Great Ormond Street Hospital, London, UKABSTRACTB-lines, defined as discrete laser-like vertical hyperechoic re-verberation artefacts in lung ultrasounds, have been shown tocorrelate with extravascular lung water in symptomatic andasymptomatic adults and children on dialysis. Recent studieshave shown this technique to be useful in children on dial-ysis. B-line detection in lung ultrasound in children is noveland challenging. This paper presents a novel automatic B-linedetection via an inverse problem involving the Radon trans-form, which is solved using the alternating direction methodof multipliers (ADMM). The proposed method restores thelines in the speckle images. Then, the B-lines are automati-cally detected using a simple local maxima technique in theRadon transform domain, associated with some definitions ofthe line artefacts. The results show that our proposed methodoutperforms existing methods by up to 69 %.Index Terms— ultrasound, lung ultrasound, inverse prob-lem, ADMM, line detection1. INTRODUCTIONFluid balance is an integral component of hemodialysis treat-ments to prevent under- or overhydration, both of which havebeen demonstrated to have significant effects on intradialyticmorbidity and long-term cardiovascular complications. In re-cent years, the use of lung ultrasonography to detect extravas-cular lung water has received growing attention in clinical re-search in adult patients with heart failure, intensive care andchronic kidney disease undergoing hemodialysis (HD) andperitoneal dialysis (PD). Recent studies have shown the bene-fit of lung ultrasound in fluid assessment for children on dial-ysis [1, 2]. A key challenge in the detection of B-lines is op-erator dependency. Identifying and counting B-lines by eyeare variable and open to error between different ultrasoundoperators.During acquisition of lung ultrasonography, the differencein acoustic impedance between the lung and the surroundingtissues will be increased when lung density increases due toextravascular fluid. This results in some vertical narrow basedlines arising from the pleural line to the edge of the ultrasoundscreen, known as B-lines. The presence of a few scattered B-line comets can be a normal variant, as found in healthy sub-jects, whilst multiple B-lines are considered the sonographicsign of lung interstitial syndrome [3].Currently, the observation of B-lines in the lung ultra-sonography is solely done by experts. However, to deal withthe large data and to further analyse or use for detecting theearly stage of some disease conditions, an automatic B-linedetection is required. To the best of our knowledge, onlytwo automatic approaches have been proposed in the literature[4, 5] and unfortunately their performances are still far frombeing reliable to use with the large data collected from dif-ferent settings, particularly paediatric ultrasound data whichdiffers from adult data given the anatomical differences be-tween children and adults.In this paper, we propose a novel solution to an inverseproblem for line detection in the speckle ultrasound images.This aims to achieve B-line detection in children’s lung ul-trasound, which is more technically challenging due to chestsize and rib positions. We employ a Radon transform, wherea grayscale image is converted to a representation of radiusand orientation as shown in Fig. 1. This inverse problem issolved using the alternating direction method of multipliers(ADMM) [6], offering fast convergence rate. We employ `1regularisation since it gives good results for sparse data, suchas lines in our work. We also propose an automatic robustmethod to detect B-lines, where the strong lines presented cansimply be detected from local peaks in the Radon transformeddomain.The remainder of this paper is organised as follows. Thebackground and related work in this field is stated in Section2. The proposed line detection method is described in Section3. The performance of the method is evaluated in Section 4.Finally, Section 5 presents the conclusions of this work.2. BACKGROUNDWhen lung density increases due to the presence in the lungof transudate, the acoustic mismatch between the lung and thesurrounding tissues is lowered, and the ultrasound beam canbe partly reflected at deeper zones and repeatedly. This phe-nomenon creates some vertical reverberation artefacts knownas B-lines [7]. Similarly appearing artefacts that should notbe confused with B-lines are Z-lines. They are short, broad,ill defined, vertical comet tail artefacts arising from the pleu-ral line but not reaching the distal edge of the screen. Theyare less echogenic than the pleural line and do not erase A-lines. The A-lines are repetitive horizontal echoic lines withequidistant intervals, which are also equal to the distance be-tween skin and pleural line.Two automatic B-line detection techniques have beenproposed in the literature. The first method was proposedby Brattain et al. [4] using angular features and thresholding(AFT). Five features are employed and the B-line is de-tected in a particular image column if each feature exceedsa predefined threshold. This method is not robust to thedifferent machine settings, as noise and intensity of the im-ages can be significantly different thereby requiring differentvalues of the threshold. The second method was proposedby Moshavegh et al. [5] using alternate sequential filtering(ASF). Firstly, the method detects the pleural line using arandom walk technique. The binary mask for the area un-derneath the pleural line is then generated from the absolutevalues of the Hilbert transform of the axial gradient com-ponents. A repeated sequential morphological opening andclosing approach is applied to the mask until possible B-linesare separated. The drawback of this method is that the re-sults always overestimate the number of B-lines because theZ-lines are not discarded. A semi-automatic approach wasproposed by Weitzel et al. [8]. The users define the area be-low the pleural line. The moving average filter is employedand a pre-defined thresholding technique is used to identifythe B-line.3. PROPOSED METHODThis section describes the proposed method of line restorationvia an inverse problem, presented in Section 3.1, and the pro-posed automatic B-line detection method, presented in Sec-tion 3.2.3.1. Line Detection in speckle imagesLines in the noisy ultrasound image are described using themodely = R−1x+ n, (1)where y is the observed ultrasound image, x is the line repre-sented by the orientation θ and distance ρ from the centre ofy. R and R−1 are a Radon transform and an inverse Radontransform, respectively. n is Gaussian noise.In a formulation without a noise term, a Radon transformis described as in Eq. 2, where δ(•) is the delta function. Tooperate with an image, the R and R−1 are discrete versions,which can be implemented as proposed in [9].x =∫R2y(i, j)δ(ρ− i cos θ − j sin θ)didj. (2)Fig. 1. Example of lung ultrasound image y (left column) andits Radon transform Ry (right column), where the horizontalaxis is θ varying from -45◦ to 135◦, the vertical axis is ρ vary-ing from −ρmax to ρmax, and the brighter intensity indicateshigher magnitude of the Radon transform.Lines x can be estimated using `1 regularisation approach asx = arg minx{12||y−R−1x||22 +α||x||+β||∇R−1x||}, (3)where α and β are regularisation constants. The first twoterms are similar to the method proposed in [10]. The thirdterm is included in the estimation because the output of theRadon transform is quantised by a discrete pre-defined rangeof orientation Θ. ∇g is the gradient of g, defined as ∇g =(gxxg2y − 2gxgygxy + gyyg2x)/(g2x + g2y), where gx and gy are∂g∂x and∂g∂y in horizontal and vertical directions, respectively.This regularisation term is similar to that in a super-resolutionapproach for compressed video [11], which is used to sup-press quantisation noise.3.1.1. Alternating direction method of multipliersThe alternating direction method of multipliers (ADMM) [6]is employed to solve the problem in Eq. 3. It is a variant ofthe augmented Lagrangian scheme that uses partial updatesfor the dual variables. It is simple to implement by splitting alarge problem into a series of subproblems as follows.minimize f(u) + g(x),subject to u− x = 0. (4)wheref(u) =12||y −R−1u||22, (5a)g(x) = α||x||+ β||∇R−1x||. (5b)Then, the Augmented Lagrangian for (4) isLρ(u, x, z) =12||y −R−1u||22 + β||∇R−1x||+ α||x||+ zT (u− x) + γ2||u− x||22,(6)where z is the dual variable or Lagrange multiplier. γ > 0 isa penalty parameter and zT indicates the transport of z.The ADMM technique allows this problem to be solvedapproximately by first solving for u with x fixed, and thensolving for x with u fixed. The computation process con-sists of three-step iterations, namely i) u-minimisation, ii) x-minimisation, and iii) dual update, as follows.uk+1 := arg minuLγ(u, xk, zk), (7a)xk+1 := arg minxLγ(uk+1, x, zk), (7b)zk+1 := zk + γ(uk+1 − xk+1). (7c)where k is an internal iteration counter.3.1.2. ImplementationThe problem in (7a) is a quadratic function about u, whichcan be solved as in Eq. 8, where I is the identity matrix.uk+1 = arg minu12||y −R−1u||22 + (zk)T (u− xk)+γ2||u− xk||22,= ((R−1)TR−1 + γI)−1((R−1)T y + γxk − zk).(8)To solve Eq. (7b), we first ignore β||∇R−1x|| for sim-plicity, then we havexk+1 = arg minxαγ||x||+ 12||uk+1 − x+ zkγ||22,= Sα/γ(uk+1 +zkγ).(9)where Sλ(•) is a soft thresholding described asSλ(a) = sign(a) max(|a| − λ, 0). (10)Subsequently, we apply a smooth function to suppress thequantisation noise, occurring due to a discrete pre-definedrange of orientation Θ. Denoting y˜k+1 as the line image ofxk+1, y˜k+1 = R−1xk+1. Then, xk+1 is updated as describedin Eq. 11.xk+1 = R (y˜k+1 + β∇y˜k+1) (11)Note that if the step size for θ is very small, e.g. ∆θ < 0.25◦,this smooth term can be removed.The last step in each iteration is for updating z, which iszk+1 = zk + γ(uk+1 − xk+1). (12)3.2. B-line DetectionThe exact number of B-lines must be sought on the pleuralline, i.e. it is equivalent to the number of points from whichlines originate. Therefore, the procedure starts with detectingthe pleural line in the restored x, using the fact that it is alwaysthe brightest line. The search area of the maximum value islimited to Θp ∈ [70◦, 110◦] (0◦ starts from the x axis), andlimiting the range of ρ to ρ > 0 since the pleural line is al-ways located on the top half of the image. Then the A-lines,physiological horizontal lines below the pleural line, are de-tected using ΘA = Θp and the fact that they are equidistant.The lines are detected using local maxima technique proposedin [12]. The B-lines are detected at ΘB ∈ [−20◦, 20◦]. TheA-lines are erased by the B-lines, so any vertical artefacts withthe presence of the A-lines are removed – these artefacts arecalled Z-lines which are not used for diagnosis. Any detectedvertical lines that merge at the pleural line are counted as asingle B-line.4. RESULTS AND DISCUSSIONWe tested our methods with 50 paediatric lung ultrasoundimages. They were acquired using linear array transducersbecause they were best suited to small body size. If con-vex transducers are used, the images have to be transformedto rectangular images, which can be done using affine trans-form. We set α, β and γ equal to 1 throughout the experi-ment. The algorithm stopped with the convergence criterion||xk+1 − xk||/||xk|| < 10−3, resulting in the process endingin <25 iterations.The subjective results in Fig. 2 show the detected pleural,A-, B- and Z-lines (red = pleural line, blue = A-lines, yellow= B-lines, green = Z-lines). Fig. 3 shows the comparisonof our method with two existing approaches for automatic B-line detection, which are i) angular features and thresholding(AFT) [4] and ii) alternate sequential filtering (ASF) [5]. Thex-axis is the number of B-lines identified by the expert andy-axis is the results marked by automated approaches. Ourmethod achieves the best mean square error (MSE) which is0.45, whilst the MSE of the AFT and ASF are 1.44 and 1.02,respectively. Both existing techniques overestimate the num-ber of B-lines when it should be 0 or 1, since A-lines are nottaken into account, resulting in Z-lines being misclassified.They underestimate when the number of B-lines is between 2-4, since the shadow of the ribs in children causes unclear/fadelaser-like artefacts, unlike those in the adult cases.5. CONCLUSIONThis paper presents two novel algorithms for automated de-tection of B-lines in lung ultrasound images. The first methodrestores the lines in the speckle images by solving an inverseproblem based on the Radon transform. The proposed methodoffers a simple and fast implementation via alternating direc-tion method of multipliers. With the accurate line detectionresults from the first method, the second method identifies B-lines automatically from the local maxima in the Radon trans-(a) (b) (c) (d) (e)Fig. 2. Lung ultrasound images with (a) three B-lines and two Z-lines, (b) one B-line and one Z-line, (c) two B-lines and threeZ-lines, (d) one B-line and several A-lines, and (e) Z-line and several A-lines. Note that the pleural line is red, the B-lines areyellow, the Z-lines are green, and the A-lines are blue.Fig. 3. Estimation of the number of B lines. MSE of AFT,ASF and the proposed method are 1.44, 1.02 and 0.45, re-spectivelyform domain. The results show that our proposed method out-performs two existing automatic B-line detection up to 69 %.Future work will combine despeckling [13, 14] and PSF esti-mation as inverse problems for the case when the input is araw radiofrequency image.6. REFERENCES[1] M. Allinovi, M.A. Saleem, O. Burgess, C. Armstrong, and W. Hayes,“Lung ultrasound: a novel technique for detecting fluid overload inchildren on dialysis,” Nephrol. Dial. Transplant., 2016.[2] M. Allinovi, M.A. Saleem, O. Burgess, C. Armstrong, and W. Hayes,“Finding covert fluid: methods for detecting volume overload in chil-dren on dialysis,” Pediatr Nephrol., 2016.[3] G. Soldati, R. Copetti, and S. Sher, “Sonographic interstitial syndrome:The sound of lung water,” J. Ultrasound Med., vol. 28, pp. 163–174,2009.[4] L. J. Brattain, B. A. Telfer, A. S. Liteplo, and V. E. Noble, “AutomatedB-line scoring on thoracic sonography,” J. Ultrasound Med., vol. 32,no. 12, pp. 2185–2190, Dec. 2013.[5] R. Moshavegh, K. L. Hansen, H. Møller Sørensen, M. C. Hemmsen,C. Ewertsen, M. B. Nielsen, and J. A. Jensen, “Novel automatic de-tection of pleura and B-lines (comet-tail artifacts) on in vivo lung ul-trasound scans,” in SPIE Medical Imaging: Ultrasonic Imaging andTomography, 2016, pp. 1–7.[6] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Dis-tributed optimization and statistical learning via the alternating direc-tion method of multipliers,” Foundations and Trends in Machine Learn-ing, vol. 3, no. 1, pp. 1–122, 2011.[7] D. A. Lichtenstein, G. A. Mezie`re, J.-F. Lagoueyte, P. Biderman,I. Goldstein, and A. Gepner, “A-lines and b-lines: Lung ultrasoundas a bedside tool for predicting pulmonary artery occlusion pressure inthe critically ill,” Chest, vol. 136, no. 4, pp. 1014–1020, 2009.[8] W.F.Weitzel, J.Hamilton, X.Wang, J.L.Bull, A.Vollmer, A.Bowman,J.Rubin, G.H.Kruger, J.Gao, M.Heung, and P.Rao, “Quantitative lungultrasound comet measurement: Method and initial clinicalresults,”Blood Purif, vol. 39, pp. 37–44, 2015.[9] B. T. Kelley and V. K. Madisetti, “The fast discrete Radon transform:I. Theory,” IEEE Transactions on Image Processing, vol. 2, no. 3, pp.382–400, Jul 1993.[10] N. Aggarwal and W. C. Karl, “Line detection in images through reg-ularized Hough transform,” IEEE Transactions on Image Processing,vol. 15, no. 3, pp. 582–591, March 2006.[11] N. Anantrasirichai and C. N. Canagarajah, “Spatiotemporal super-resolution for low bitrate H.264 video,” in IEEE International Con-ference on Image Processing, 2010, pp. 2809–2812.[12] D. G. Lowe, “Distinctive image features from scale-invariant key-points,” International Journal of Computer Vision, vol. 60, no. 2, pp.91–110, 2004.[13] A. Achim, A. Bezerianos, and P. Tsakalides, “Novel bayesian multi-scale method for speckle removal in medical ultrasound images,” IEEETransactions on Medical Imaging, vol. 20, no. 8, pp. 772–783, Aug.2001.[14] N. Anantrasirichai, L. Nicholson, J. E. Morgan, I. Erchova, K. Mort-lock, R. V. North, J. Albon, and A. Achim, “Adaptive-weighted bilat-eral filtering and other pre-processing techniques for optical coherencetomography,” Computerized Medical Imaging and Graphics, vol. 38,no. 6, pp. 526–539, Sep. 2014.

Anantrasirichai, P., Allinovi, M., Hayes, W., & Achim, A. (2016).Automatic B-line detection in paediatric lung ultrasound. In 2016 IEEEInternational Ultrasonics Symposium, IUS 2016 (Vol. 2016November). Article 7728620 IEEE Computer Society.https://doi.org/10.1109/ULTSYM.2016.7728620Peer reviewed versionLink to published version (if available):10.1109/ULTSYM.2016.7728620Link to publication record on the Bristol Research PortalPDF-documentThis is the author accepted manuscript (AAM). The final published version (version of record) is available onlinevia IEEE at http://ieeexplore.ieee.org/document/7728620/. Please refer to any applicable terms of use of thepublisher.University of Bristol – Bristol Research PortalGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/brp-terms/AUTOMATIC B-LINE DETECTION IN PAEDIATRIC LUNG ULTRASOUNDNantheera Anantrasirichai1, Marco Allinovi2, Wesley Hayes3, Alin Achim11Visual Information Lab, University of Bristol, Bristol, UK2Paediatric Nephrology Unit, Meyer Childrens Hospital, Florence, Italy3Great Ormond Street Hospital, London, UKABSTRACTB-lines, defined as discrete laser-like vertical hyperechoic re-verberation artefacts in lung ultrasounds, have been shown tocorrelate with extravascular lung water in symptomatic andasymptomatic adults and children on dialysis. Recent studieshave shown this technique to be useful in children on dial-ysis. B-line detection in lung ultrasound in children is noveland challenging. This paper presents a novel automatic B-linedetection via an inverse problem involving the Radon trans-form, which is solved using the alternating direction methodof multipliers (ADMM). The proposed method restores thelines in the speckle images. Then, the B-lines are automati-cally detected using a simple local maxima technique in theRadon transform domain, associated with some definitions ofthe line artefacts. The results show that our proposed methodoutperforms existing methods by up to 69 %.Index Terms— ultrasound, lung ultrasound, inverse prob-lem, ADMM, line detection1. INTRODUCTIONFluid balance is an integral component of hemodialysis treat-ments to prevent under- or overhydration, both of which havebeen demonstrated to have significant effects on intradialyticmorbidity and long-term cardiovascular complications. In re-cent years, the use of lung ultrasonography to detect extravas-cular lung water has received growing attention in clinical re-search in adult patients with heart failure, intensive care andchronic kidney disease undergoing hemodialysis (HD) andperitoneal dialysis (PD). Recent studies have shown the bene-fit of lung ultrasound in fluid assessment for children on dial-ysis [1, 2]. A key challenge in the detection of B-lines is op-erator dependency. Identifying and counting B-lines by eyeare variable and open to error between different ultrasoundoperators.During acquisition of lung ultrasonography, the differencein acoustic impedance between the lung and the surroundingtissues will be increased when lung density increases due toextravascular fluid. This results in some vertical narrow basedlines arising from the pleural line to the edge of the ultrasoundscreen, known as B-lines. The presence of a few scattered B-line comets can be a normal variant, as found in healthy sub-jects, whilst multiple B-lines are considered the sonographicsign of lung interstitial syndrome [3].Currently, the observation of B-lines in the lung ultra-sonography is solely done by experts. However, to deal withthe large data and to further analyse or use for detecting theearly stage of some disease conditions, an automatic B-linedetection is required. To the best of our knowledge, onlytwo automatic approaches have been proposed in the literature[4, 5] and unfortunately their performances are still far frombeing reliable to use with the large data collected from dif-ferent settings, particularly paediatric ultrasound data whichdiffers from adult data given the anatomical differences be-tween children and adults.In this paper, we propose a novel solution to an inverseproblem for line detection in the speckle ultrasound images.This aims to achieve B-line detection in children’s lung ul-trasound, which is more technically challenging due to chestsize and rib positions. We employ a Radon transform, wherea grayscale image is converted to a representation of radiusand orientation as shown in Fig. 1. This inverse problem issolved using the alternating direction method of multipliers(ADMM) [6], offering fast convergence rate. We employ `1regularisation since it gives good results for sparse data, suchas lines in our work. We also propose an automatic robustmethod to detect B-lines, where the strong lines presented cansimply be detected from local peaks in the Radon transformeddomain.The remainder of this paper is organised as follows. Thebackground and related work in this field is stated in Section2. The proposed line detection method is described in Section3. The performance of the method is evaluated in Section 4.Finally, Section 5 presents the conclusions of this work.2. BACKGROUNDWhen lung density increases due to the presence in the lungof transudate, the acoustic mismatch between the lung and thesurrounding tissues is lowered, and the ultrasound beam canbe partly reflected at deeper zones and repeatedly. This phe-nomenon creates some vertical reverberation artefacts knownas B-lines [7]. Similarly appearing artefacts that should notbe confused with B-lines are Z-lines. They are short, broad,ill defined, vertical comet tail artefacts arising from the pleu-ral line but not reaching the distal edge of the screen. Theyare less echogenic than the pleural line and do not erase A-lines. The A-lines are repetitive horizontal echoic lines withequidistant intervals, which are also equal to the distance be-tween skin and pleural line.Two automatic B-line detection techniques have beenproposed in the literature. The first method was proposedby Brattain et al. [4] using angular features and thresholding(AFT). Five features are employed and the B-line is de-tected in a particular image column if each feature exceedsa predefined threshold. This method is not robust to thedifferent machine settings, as noise and intensity of the im-ages can be significantly different thereby requiring differentvalues of the threshold. The second method was proposedby Moshavegh et al. [5] using alternate sequential filtering(ASF). Firstly, the method detects the pleural line using arandom walk technique. The binary mask for the area un-derneath the pleural line is then generated from the absolutevalues of the Hilbert transform of the axial gradient com-ponents. A repeated sequential morphological opening andclosing approach is applied to the mask until possible B-linesare separated. The drawback of this method is that the re-sults always overestimate the number of B-lines because theZ-lines are not discarded. A semi-automatic approach wasproposed by Weitzel et al. [8]. The users define the area be-low the pleural line. The moving average filter is employedand a pre-defined thresholding technique is used to identifythe B-line.3. PROPOSED METHODThis section describes the proposed method of line restorationvia an inverse problem, presented in Section 3.1, and the pro-posed automatic B-line detection method, presented in Sec-tion 3.2.3.1. Line Detection in speckle imagesLines in the noisy ultrasound image are described using themodely = R−1x+ n, (1)where y is the observed ultrasound image, x is the line repre-sented by the orientation θ and distance ρ from the centre ofy. R and R−1 are a Radon transform and an inverse Radontransform, respectively. n is Gaussian noise.In a formulation without a noise term, a Radon transformis described as in Eq. 2, where δ(•) is the delta function. Tooperate with an image, the R and R−1 are discrete versions,which can be implemented as proposed in [9].x =∫R2y(i, j)δ(ρ− i cos θ − j sin θ)didj. (2)Fig. 1. Example of lung ultrasound image y (left column) andits Radon transform Ry (right column), where the horizontalaxis is θ varying from -45◦ to 135◦, the vertical axis is ρ vary-ing from −ρmax to ρmax, and the brighter intensity indicateshigher magnitude of the Radon transform.Lines x can be estimated using `1 regularisation approach asx = arg minx{12||y−R−1x||22 +α||x||+β||∇R−1x||}, (3)where α and β are regularisation constants. The first twoterms are similar to the method proposed in [10]. The thirdterm is included in the estimation because the output of theRadon transform is quantised by a discrete pre-defined rangeof orientation Θ. ∇g is the gradient of g, defined as ∇g =(gxxg2y − 2gxgygxy + gyyg2x)/(g2x + g2y), where gx and gy are∂g∂x and∂g∂y in horizontal and vertical directions, respectively.This regularisation term is similar to that in a super-resolutionapproach for compressed video [11], which is used to sup-press quantisation noise.3.1.1. Alternating direction method of multipliersThe alternating direction method of multipliers (ADMM) [6]is employed to solve the problem in Eq. 3. It is a variant ofthe augmented Lagrangian scheme that uses partial updatesfor the dual variables. It is simple to implement by splitting alarge problem into a series of subproblems as follows.minimize f(u) + g(x),subject to u− x = 0.(4)wheref(u) =12||y −R−1u||22, (5a)g(x) = α||x||+ β||∇R−1x||. (5b)Then, the Augmented Lagrangian for (4) isLρ(u, x, z) =12||y −R−1u||22 + β||∇R−1x||+ α||x||+ zT (u− x) + γ2||u− x||22,(6)where z is the dual variable or Lagrange multiplier. γ > 0 isa penalty parameter and zT indicates the transport of z.The ADMM technique allows this problem to be solvedapproximately by first solving for u with x fixed, and thensolving for x with u fixed. The computation process con-sists of three-step iterations, namely i) u-minimisation, ii) x-minimisation, and iii) dual update, as follows.uk+1 := arg minuLγ(u, xk, zk), (7a)xk+1 := arg minxLγ(uk+1, x, zk), (7b)zk+1 := zk + γ(uk+1 − xk+1). (7c)where k is an internal iteration counter.3.1.2. ImplementationThe problem in (7a) is a quadratic function about u, whichcan be solved as in Eq. 8, where I is the identity matrix.uk+1 = arg minu12||y −R−1u||22 + (zk)T (u− xk)+γ2||u− xk||22,= ((R−1)TR−1 + γI)−1((R−1)T y + γxk − zk).(8)To solve Eq. (7b), we first ignore β||∇R−1x|| for sim-plicity, then we havexk+1 = arg minxαγ||x||+ 12||uk+1 − x+ zkγ||22,= Sα/γ(uk+1 +zkγ).(9)where Sλ(•) is a soft thresholding described asSλ(a) = sign(a) max(|a| − λ, 0). (10)Subsequently, we apply a smooth function to suppress thequantisation noise, occurring due to a discrete pre-definedrange of orientation Θ. Denoting ỹk+1 as the line image ofxk+1, ỹk+1 = R−1xk+1. Then, xk+1 is updated as describedin Eq. 11.xk+1 = R(ỹk+1 + β∇ỹk+1)(11)Note that if the step size for θ is very small, e.g. ∆θ < 0.25◦,this smooth term can be removed.The last step in each iteration is for updating z, which iszk+1 = zk + γ(uk+1 − xk+1). (12)3.2. B-line DetectionThe exact number of B-lines must be sought on the pleuralline, i.e. it is equivalent to the number of points from whichlines originate. Therefore, the procedure starts with detectingthe pleural line in the restored x, using the fact that it is alwaysthe brightest line. The search area of the maximum value islimited to Θp ∈ [70◦, 110◦] (0◦ starts from the x axis), andlimiting the range of ρ to ρ > 0 since the pleural line is al-ways located on the top half of the image. Then the A-lines,physiological horizontal lines below the pleural line, are de-tected using ΘA = Θp and the fact that they are equidistant.The lines are detected using local maxima technique proposedin [12]. The B-lines are detected at ΘB ∈ [−20◦, 20◦]. TheA-lines are erased by the B-lines, so any vertical artefacts withthe presence of the A-lines are removed – these artefacts arecalled Z-lines which are not used for diagnosis. Any detectedvertical lines that merge at the pleural line are counted as asingle B-line.4. RESULTS AND DISCUSSIONWe tested our methods with 50 paediatric lung ultrasoundimages. They were acquired using linear array transducersbecause they were best suited to small body size. If con-vex transducers are used, the images have to be transformedto rectangular images, which can be done using affine trans-form. We set α, β and γ equal to 1 throughout the experi-ment. The algorithm stopped with the convergence criterion||xk+1 − xk||/||xk|| < 10−3, resulting in the process endingin <25 iterations.The subjective results in Fig. 2 show the detected pleural,A-, B- and Z-lines (red = pleural line, blue = A-lines, yellow= B-lines, green = Z-lines). Fig. 3 shows the comparisonof our method with two existing approaches for automatic B-line detection, which are i) angular features and thresholding(AFT) [4] and ii) alternate sequential filtering (ASF) [5]. Thex-axis is the number of B-lines identified by the expert andy-axis is the results marked by automated approaches. Ourmethod achieves the best mean square error (MSE) which is0.45, whilst the MSE of the AFT and ASF are 1.44 and 1.02,respectively. Both existing techniques overestimate the num-ber of B-lines when it should be 0 or 1, since A-lines are nottaken into account, resulting in Z-lines being misclassified.They underestimate when the number of B-lines is between 2-4, since the shadow of the ribs in children causes unclear/fadelaser-like artefacts, unlike those in the adult cases.5. CONCLUSIONThis paper presents two novel algorithms for automated de-tection of B-lines in lung ultrasound images. The first methodrestores the lines in the speckle images by solving an inverseproblem based on the Radon transform. The proposed methodoffers a simple and fast implementation via alternating direc-tion method of multipliers. With the accurate line detectionresults from the first method, the second method identifies B-lines automatically from the local maxima in the Radon trans-(a) (b) (c) (d) (e)Fig. 2. Lung ultrasound images with (a) three B-lines and two Z-lines, (b) one B-line and one Z-line, (c) two B-lines and threeZ-lines, (d) one B-line and several A-lines, and (e) Z-line and several A-lines. Note that the pleural line is red, the B-lines areyellow, the Z-lines are green, and the A-lines are blue.Fig. 3. Estimation of the number of B lines. MSE of AFT,ASF and the proposed method are 1.44, 1.02 and 0.45, re-spectivelyform domain. The results show that our proposed method out-performs two existing automatic B-line detection up to 69 %.Future work will combine despeckling [13, 14] and PSF esti-mation as inverse problems for the case when the input is araw radiofrequency image.6. REFERENCES[1] M. Allinovi, M.A. Saleem, O. Burgess, C. Armstrong, and W. Hayes,“Lung ultrasound: a novel technique for detecting fluid overload inchildren on dialysis,” Nephrol. Dial. Transplant., 2016.[2] M. Allinovi, M.A. Saleem, O. Burgess, C. Armstrong, and W. Hayes,“Finding covert fluid: methods for detecting volume overload in chil-dren on dialysis,” Pediatr Nephrol., 2016.[3] G. Soldati, R. Copetti, and S. Sher, “Sonographic interstitial syndrome:The sound of lung water,” J. Ultrasound Med., vol. 28, pp. 163–174,2009.[4] L. J. Brattain, B. A. Telfer, A. S. Liteplo, and V. E. Noble, “AutomatedB-line scoring on thoracic sonography,” J. Ultrasound Med., vol. 32,no. 12, pp. 2185–2190, Dec. 2013.[5] R. Moshavegh, K. L. Hansen, H. Møller Sørensen, M. C. Hemmsen,C. Ewertsen, M. B. Nielsen, and J. A. Jensen, “Novel automatic de-tection of pleura and B-lines (comet-tail artifacts) on in vivo lung ul-trasound scans,” in SPIE Medical Imaging: Ultrasonic Imaging andTomography, 2016, pp. 1–7.[6] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Dis-tributed optimization and statistical learning via the alternating direc-tion method of multipliers,” Foundations and Trends in Machine Learn-ing, vol. 3, no. 1, pp. 1–122, 2011.[7] D. A. Lichtenstein, G. A. Mezière, J.-F. Lagoueyte, P. Biderman,I. Goldstein, and A. Gepner, “A-lines and b-lines: Lung ultrasoundas a bedside tool for predicting pulmonary artery occlusion pressure inthe critically ill,” Chest, vol. 136, no. 4, pp. 1014–1020, 2009.[8] W.F.Weitzel, J.Hamilton, X.Wang, J.L.Bull, A.Vollmer, A.Bowman,J.Rubin, G.H.Kruger, J.Gao, M.Heung, and P.Rao, “Quantitative lungultrasound comet measurement: Method and initial clinicalresults,”Blood Purif, vol. 39, pp. 37–44, 2015.[9] B. T. Kelley and V. K. Madisetti, “The fast discrete Radon transform:I. Theory,” IEEE Transactions on Image Processing, vol. 2, no. 3, pp.382–400, Jul 1993.[10] N. Aggarwal and W. C. Karl, “Line detection in images through reg-ularized Hough transform,” IEEE Transactions on Image Processing,vol. 15, no. 3, pp. 582–591, March 2006.[11] N. Anantrasirichai and C. N. Canagarajah, “Spatiotemporal super-resolution for low bitrate H.264 video,” in IEEE International Con-ference on Image Processing, 2010, pp. 2809–2812.[12] D. G. 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Automatic B-line detection in paediatric lung ultrasound

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