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Hough transform and uncertainty handling.Application to circular object detection in ultrasound medical imagesB. Solaiman, B. Burdsall and Ch. RouxDept. Image et Traitement de l'Information, ENST-Br, B.P. 832, 29285 Brest, FranceTel: (33 2 98 00 13 08 , Fax: (33) 2 98 00 10 98, email: Basel.Solaiman@enst-bretagne.frabstractThe Hough Transform (HT) has been shown tobe an effective method to detect parameterizedcurves in images by mapping image edge points intomanifolds in the parameter space and then finding"peaks" in the parameter space[1-2].In this paper, the application of the HT toultrasound images of the peri-esophageal region forthe localization of the aorta is discussed. Theproposed algorithm has the important feature ofintegrating a priori knowledge as well as two typesof uncertainties through the accumulation process.The first uncertainty type is related to the impreciseestimation of the gradient direction. The second,concerns the ambiguity due to the lake of a preciseaorta radius determination.1. The Hough Transform accumulationkernel conceptObject detection is an important first stage inmany recognition tasks. One of the main methods forachieving this objective is the Hough Transform(HT), first introduced by Hough (1962) for theidentification of straight lines in pictures [1].The HT converts the problem of extractingcollinear point sets in the image space into maximadetection in the Hough space. Duda and Hart (1972),[2], have extended the same method to extract otherparametric curves. As pointed out by manyresearchers, the principal limitations of the methodare its computational complexity requirements andmassive demand for storage (exponential in thedimensionality of the parameter space). In thispaper, the HT is revisited from a "knowledgeaccumulation" point of view in order to handledifferent forms of uncertainty in low-level vision. Infact, the object detection problem can be formulatedas follows :Assume an input NxN image {I(x,y), x, y = 0, 1,..., N-1}, where I(x,y) denotes the gray level at thepixel location (x,y), and that objects of an a prioriknown form (given as a parameterized curve) arecontained in the considered image. The objectdetection problem aims at the precise localization ofpotential objects contained in this image.From a knowledge-based system point of view,the HT starts by considering each individual pixel,by means of its gray level as well as its spatialcontextual information, as an elementary source ofknowledge (or an elementary expert).This elementary expert partially contributes tothe global object detection final solution.With no doubt, the most important aspect in theHT concerns the precise definition of the method ofhow each elementary expert contributes into the finalsolution according to the definition of the followingconcepts :1 - the Hough Space,2 - the Accumulation Kernel (AK),3 - the Accumulation Evidence (AE), and4 - the Knowledge Combining Strategy (KCS).The Hough Space (HS) denotes the physicallocus in which the contribution of all elementaryexperts are considered. The AK is defined as therestricted area in the HS delimiting the contributionof each elementary expert. The AE denotes thedegree of belief attributed to each individualelementary expert. Finally, the KCS aims at thedefinition of the approach to be used in order tocombine different contributions issued fromelementary experts.Let f(X,A 0 )=0, where X=(x,y) and A 0 is aparameter vector, be an analytic expression defininga parameterized curve. The first step in theapplication of the HT, aims at the definition of theparameter vector A 0 as well as the quantization ofthe parameter space into rectangular n-dimensionalcells called the Hough Space (HS).This analytic expression means that if we aregiven a parameter vector A 0 , then, the curve ofinterest is formed by pixels in the image planesatisfying the analytic curve expression.Now, let us inverse the situation, if we are givenan edge pixel E k = (x k ,y k ), then, the set parametervectors generating curves in the image plane thatpass through the edge pixel E k is given by theanalytic expression : f(E k ,A)=0.

The locus of these vectors in the HS is called theAccumulation Kernel (AK). For instance, if the slopand intercept parameterization of straight lines isadopted (i.e; y=a 0 x+b 0 ) then the AK associated tothe edge pixel E k =(x k ,y k ), referred to as AK k , is astraight line in the HS (i.e; b= - ax k +y k ).Also, in the case of a fixed radius circle detection(i.e; (x-a) 2 +(y-b) 2 =R 2 ), the AK k is a circle of radiusR centred on the edge pixel E k (i.e; (a-x k ) 2 + (b-y k ) 2= R 2 ).In this study, the concept of the AK is shown tobe of great interest when dealing with different typesof uncertainties. The fixed radius circular objectdetection and localization is particularlyinvestigated.The second step in the application of the HT,concerns the edge pixels detection. This is generallyconducted by applying an edge detector to theimage. The set of potential edge pixels (E) is formedby those having a gradient magnitude exceedingsome threshold "α":E ={X i,j such that g i,j >α}={E k = (x k , y k , g k , ϕ k ), k=1, 2, ..., K}.Therefore, each edge point E k , is associated withfour information elements, the pixel position (x k ,y k ),the gradient magnitude g k , and the gradient directionϕ k . After these preliminary steps, the accumulationprocedure is the most important step in theapplication of the HT.In the standard HT and for each edge pixel E k ,all the cells situated on the AK k are incremented byunity. This value is called the accumulationevidence.In this study, we refer to this incrementationprocedure as the Total Ignorance Accumulation,meaning that if the edge pixel position is the onlyinformation used E k =(x k ,y k ), then no incrementationpreference is given to any accumulation cell on theAK k .Ballard has proposed [3] to use the gradientmagnitude, g k , as a heuristic in the incrementationprocess. Instead of incrementing the AK k cells byunity, they are incremented by an accumulationevidence which is a function of the gradientmagnitude.Now, if we assume that the gradient directioninformation, ϕ k , is precise and reliable, then onlyone cell on the AK k needs to be incremented. Thissituation is referred to as the Total KnowledgeAccumulation.Kimme et al. have proposed [4] an intermediateaccumulation procedure by considering theimprecise nature of the gradient directiondetermination. In the case of a circle detection, onlythe cells situated on an arc in the direction of ϕ k andcentred on (x k ,y k ) are incremented.In this study, the uncertainty concerning thecircle radius value is integrated to the imprecisedirection knowledge. This uncertainty is due eitherto the lake of the precise radius value knowledge, or,to the fact that the searched shape is "a nearlycircular shape" where the radius is defined as arange of values instead of a single value.In this case, the accumulation kernel becomes asurface and not a linear curve, figure 1.In the case of Imprecise Direction-Radiusaccumulation, the accumulation evidence is adecreasing function of the parameter |ϕ−ϕ k | where ϕ∈[ϕ k -∆(r), ϕ k +∆(r)], figure 2.As the uncertainty in the direction increases, withan increasing radius, the accumulation kernel spreadsout in a radial fashion.Notice that the accumulation evidence is equal tounity for ϕ=ϕ k and for all r ∈[R 1 ,R 2 ]. This situationreflects the total imprecision concerning the realradius value.Otherwise, this accumulation evidence can beselected as a bell shaped function of the parameter rand centred on the radius R. The optimal radius,R Opt , of the circular form can be accuratelydetermined by a use of a simple 1D accumulator orhistogram of possible radii.For each edge pixel having a normal vectorpointing in the direction of the circle centre (x 0 ,y 0 ),the distance to (x 0 ,y 0 ) is measured and is stored inthe histogram (subject to the distance being within[R 1 ,R 2 ]). At the end of this accumulation, the modeor peak, of the histogram represents the most likelyradius, figure 3.This "selective" radius determination aims atreducing noise effects as well as at optimaloverlapped circles "discrimination".2. The Aorta detectionIn this study, the proposed approach is applied inthe aorta detection using echoendoscopic imagingsystems. Echoendoscopic imaging using ultrasoundis a technique encountered more and more often dueto its passive, unobtrusive, nature and subsequentlow risk to the patient. The visualization of theesophagus and peri-esophageal region providesimportant diagnostic information about the onset andevolution of esophageal tumors, figure 4-a. Thetumors form in the esophageal wall, and spread

outwards to the surrounding organs (e.g. the aortaand ganglions).Echoendoscopic image interpretation iscompounded by two types of noise, or interference,during the data acquisition which degrade theensuing visualization: Speckle noise results from thediffraction of the ultrasound by micro-anatomicstructures which have a size comparable to that ofthe wavelength.Bright harmonics are superimposed around thecentre of the esophagus. All these factors contributeto make the interpretation of ultrasound images verydifficult, even for trained medical specialists.3. Simulation results and furthercommentsThe aim of this study is the application of theproposed HT in order to have a precise detection andlocalization of the aorta. The use of a priori anatomicknowledge of the approximate radius of the aorta hasalready been discussed in the previous section.A priori information concerning the approximateaorta position in the image plane is simply integratedthrough the use of a weight function, in the Houghspace, assuming the value zero in areas where theaorta is not allowed to be detected (central part ofthe image), and unity in the rest of the image.Finally, the Canny-Deriche edge detector is usedbecause it allows the accurate estimation of thenormal vectors, and thus, increasing the efficiency ofthe accumulation.Obtained results in terms of detection andlocalization of the aorta are extremely promising andclearly show the robustness of the proposedapproach on even quite noisy ultrasound images.Two examples of the obtained results are given infigure4.Figure 4-a shows an echoendoscopic image of a"normal" patient. On the other hand, Figure 4-bshows the aorta region concerning a patient havingan extended esophageal tumor leading to animportant degradation of the esophagus and the aortawalls.In both cases, the resulting circularapproximation of the aorta is of high precision.Tested against the traditional Hough transform, theproposed approach leads to an important false alarmsreduction. Future work will concentrate on extendingthe use this approach to “elliptical” Houghtransforms in order to detect such structures like theganglions. The integration of anatomical knowledgewill be integrated to reduce the search space. Theuse of snake algorithms is also studied to make themodels deformable for generalized shape detectionand for individual modelling to the patient.References[1] P.V.C. Hough, "Methods and means forrecognizing complex patterns," U.S. Patent 3,069,654, 1962.[2] R.O.Duda and P.E.Hart,"Using the Houghtransforms to detect lines and curves in pictures,"Comm. of the ACM, 15(1), pp. 11-15, 1972.[3] D.H.Ballard,"Generalizing the Hough transformto detect arbitrary shapes", Pattern Recognition13(2), pp. 111-122, 1981.[4] C.Kimme, D.Ballard and J.Sklanski,"Findingcircles by an array of accumulators," Comm. ACM18, pp. 120-122, 1975.ϕ k + ∆ϕ kE kRϕ k + ∆Rϕ kϕ k − ∆REdge pixel EkE kRϕ kϕ k− ∆EkRR maxmin(a) (b) (c) (d)a)Total Ignorance Accumulation, b) Total Knowledge Accumulation, c) ImpreciseDirection, accumulation and d) Imprecise Direction-Radius Accumulation.Figure 1. Circular Accumulation kernels associated with an edge point

1r = constantϕ − ∆kϕ kϕ + ∆kFigure 2. Accumulation evidence associated with E k . r∈[R 1 , R 2 ] and ϕ∈[ϕk -∆,ϕk +∆]ϕyR2R1y 0rx 0xR 1R OptR 2Figure 3, Selective radius accumulation process(a)Figure 4; Aorta localization using the modified Hough transform(b)