X-ray image distortion correction based on SVR

YUAN Ze-hui, LI Shi-zhong

(College of Mechatronic Engineering， North University of China， Taiyuan 030051， China)

袁澤慧， 李世中

(中北大學(xué) 機(jī)電工程學(xué)院， 山西 太原 030051)

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X-ray image distortion correction based on SVR

YUAN Ze-hui, LI Shi-zhong

(CollegeofMechatronicEngineering，NorthUniversityofChina，Taiyuan030051，China)

X-ray image has been widely used in many fields such as medical diagnosis, industrial inspection, and so on. Unfortunately, due to the physical characteristics of X-ray and imaging system, distortion of the projected image will happen, which restrict the application of X-ray image, especially in high accuracy fields. Distortion correction can be performed using algorithms that can be classified as global or local according to the method used, both having specific advantages and disadvantages. In this paper, a new global method based on support vector regression (SVR) machine for distortion correction is proposed. In order to test the presented method, a calibration phantom is specially designed for this purpose. A comparison of the proposed method with the traditional global distortion correction techniques is performed. The experimental results show that the proposed correction method performs better than the traditional global one.

X-ray image; distortion correction; support vector regression machine

X-ray technology has been widely used in a number of fields, especially industrial and medical fields[1]. In the last years, there has been an increasing interest in using projected images to generate three-dimensional (3D) tomographic reconstructions. However, because the the original projected images are distorted, the quality of the reconstructed images is deteriorated. Therefore, how to correct the distortions becomes necessary for using X-ray images.

Among various types of distortion, pincushion distortion and S-distortion are predominant[2]. The former is the most visually apparent due to photocathode curvature. The latter is introduced by the electron optics of image intensifier and interaction with external magnetic fields, mainly earth magnetic field.

Correction for spatial distortion in X-image is achieved by imaging a regular calibration grid, consisting of either transparent holes or small opaque objects, which represent the control points and are used to compute the correction coefficients. An interpolation function is to build the map between ideal and raw pixel locations of control points.

Several methods have been developed to correct the distortion in X-ray images. Normally these correction methods are classified into global methods[3-6]and local methods[7]. In global correction techniques, a single pair of equations in the form of a high-order polynomial (usually a 4th or 5th-order polynomial) are used to correct the image. The global methods are implemented easily and fast. However, for some local distortions that may appear in the acquired image, they can not be corrected and require an excessive calibration of the system. For overcoming the existing problems of global methods, local method is developed. In local correction methods, the raw image is divided into many small areas (usually rectangular or triangular) firstly, and the distortion of each local area is corrected respectively. Since different correction equations are used for each local area, for the neighboring regions, the discontinuities are introduced, which represent the major problem in local methods.

Both global methods and local methods have their advantages and disadvantages. In this paper, a new global method based on support vector machine (SVM) is proposed, which is based on the strong regression ability of SVM.

1 Material and method

The correction for two-dimensional X-ray image distortion is usually achieved by imaging a regular grid phantom, consisting of either transparent holes or small opaque objects, which are used as control points. In this paper, a thin plate with a series of small holes and a large hole, placed in a regular pattern, is specially designed for the correction work. The small holes are inside a circle with a diameter of 3.1 mm, while the large hole, displaced at the centre of the thin plate and used as the original control point, has a diameter of 4.2 mm. The step between holes (the distance between their centers) is 6 mm. The image of the designed thin plate is shown in Fig.1.

Fig.1 Thin plate attached to the face of XRLL

In this paper, a new global method based on support vector regression (SVR) for correcting distortion is proposed. As other correction methods, the presented approach mainly involves: (1) automatic control points detection (2) correction modeling (3) application of the correction model to individual images, i.e. image correction. In the following, a detailed description of these steps is given.

1.1 Automatic grid detection

As described in above, the centers of holes are used as control points for distortion correction. To detect the centers of holes precisely, image prepossessing and segmentation are carried out. After that, the segmented disjointed pixels are grouped into hole-clusters. Following the clustering operation, for each hole-cluster, mass center approach is used calculate its center coordinates. Next, all the center coordinates are put into a matrix according to their ideal grid orders.

Distortion correction is to build a mapping model, i.e. a correction model, between the control point coordinates (the holes’ mass centers in this case) of the raw distorted image and the ideal one. After the procedure described previously, the distorted control point coordinates are detected automatically. However, for the ideal image, it is not easy to be obtained since the ideal X-ray image is unknown. In our case, a large hole at the center of the thin plate is specially designed, which is based on the characteristics of pincushion distortion and S-distortion that the distortion in the central part is small. To obtain other small holes’ ideal mass centers, the step between two neighboring centers is calculated by averaging all the steps of each neighboring couple of holes. Then according to the mass center of the large hole and the calculated step, the other small holes’ ideal mass centers are obtained.

1.2 Support vector regression machine theory

The support vector machines (SVMs) were proposed by Vapnik[8]in 1998. In the last years, it has been widely used for classification[9-10]and regression[11-12]. When SVM is applied to regression by introducing an alternative loss function, it is termed as SVR.

The basic idea of SVR is that the data vectorxfrom the inputXis mapped into a high-dimensional feature spaceFby a nonlinear mappingΦ(x),andthenlinearregressionisconductedinFspace.

Given a set of training data {(x1,y1),(x2,y2),…,(xn,yn)}∈X×R,thegoalofSVRistofindafunctionf(x)thatisnotmorethanεfromthetargetsyiforallthetrainingdata,andatthesametime,itisasflataspossible.Thelinearfunctionf(x)isformulatedas

(1)

FlatnessinEq.(1)meansthatoneseekssmallw. One way to ensure this is to minimize the Euclidean norm, i.e. ‖w‖2.Thenthisproblemcanbeformulatedasaconvexoptimizationproblembyrequiring

(2)

(3)

TheconstantC>0determinesthetrade-offoferrormarginbetweentheflatnessoff(x)andtheamountofdeviationinexcessofεthatistolerated.ToenabletheSVRtopredictanonlinearsituation,SVRmapstheinputdataintothefeaturespacebyΦ(X),whichisdenotedby

ThedecisionfunctioncanbecomputedbytheinnerproductsofΦ(x)TΦ(xi)withoutexplicitlymappingxinto a higher dimension, which saves considerable computation efforts.Φ(x)TΦ(xi)iscalledkernelfunctionK(x,xi)≡Φ(x)TΦ(xi).ChoosingdifferenttypesofkernelfunctionscangeneratedifferentSVRmodels.

2 Experiment results

2.1 Determination of SVR parameters

The generalization ability of SVR relies entirely on a group of parameters. In this paper, a mixture kernel of polynomial kernel and radial basis kernel is used. And for the loss function,ε-insensitive loss function is selected. Therefore, it is needed to determine the following parameters: regularization factorC,εof theε-insensitive loss function,σof Gaussian radial basis function, the orderdof polynomial order Kernel, and the mixture coefficientλ.

The final SVR coefficients were selected as

2.2 Experiment results

In this section the presented SVR model is tested to correct the raw distorted X-ray image. Meanwhile it is compared with the most commonly used polynomial model, and the polynomial model’s order is set to 4. The raw distorted image is shown in Fig.2. And the corrected images are presented in Fig.3 and Fig.4, respectively.

Fig.2 Raw distorted X-ray image

Fig.3 Corrected image using polynomial model

Fig.4 Corrected image using SVR model

In order to compare the two controllers performance quantitatively, the following measures are taken.

Fig.5showstheresidualdisplacementofthecorrectedimagesusingpolynomialandSVRmodel,respectively.ItiseasytonoticethatforFig.5(b),alltheerrorsarelessthan0.6pixels.HoweverinFig.5(a),althoughmostoftheerrorsarelessthan0.6pixels,therearestillafewnumberofcontrolpoints’residualdisplacementswhichareevenlargerthan2.5pixels,whichcorrespondstothelocaldistortionsthattheglobalpolynomialmodelcannotcorrect.

Fig.5 Residual displacement histograms of corrected images

2) For the corrected image, two control points in a row are formed into a vector. Since the control points, i.e. the holes are displaced in a grid pattern, the angel between the formed vector with the horizontal vector is calculated. It is easy to know that the ideal angle is equal to 0. To explain conveniently, the vector formed by the control points in a row is named row-center-vector.

The histograms of calculated angles of images corrected by global pronominal model and SVR model are shown in Fig.6(a) and (b), respectively.

It is easy to observe that, compared with Fig.6(a), there are few angles larger than 1.2 degree in Fig.6(b), which means that by using SVR model, the distortion near the bounding is corrected mostly. Meanwhile, it is can be known that there are 462 angles which are less than 0.4°, which is much larger than 375 in Fig.6(a).

Fig.6 Histogram of angles between row-center-vector and horizontal vector

It can be concluded from the comparison results that the display of control points in image corrected by SVR model is much close to the ideal grid, i.e. for correcting the distorted images, the SVR model performs better than global polynomial model.

3 Conclusion

In this research, a correction algorithm based on SVR model has been proposed to correct the distortion of the acquired images by X-ray projector. Based on the extracted centers, the correcting SVR model is built and then applied to the raw distorted images. The experiments show that much of the distortion in raw image is found to be effectively corrected. And the developed SVR model performs better than the 4th order polynomial model in two aspects.

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基于支持向量回歸機(jī)的X射線圖像畸變校正研究

X射線圖像已經(jīng)被廣泛的應(yīng)用于各個(gè)領(lǐng)域， 如醫(yī)學(xué)診斷， 工業(yè)檢測(cè)等。 然而， 由于x射線及其成像系統(tǒng)的物理特性， 得到的X圖像存在嚴(yán)重畸變， 這極大限制了X射線在一些高精度場(chǎng)合的應(yīng)用， 因此需對(duì)原始的x射線圖像進(jìn)行校正。 所謂圖像畸變校正， 就是設(shè)法建立畸變圖像特征點(diǎn)坐標(biāo)與理想圖像特征點(diǎn)坐標(biāo)之間的映射模型， 即校正模型， 將畸變的點(diǎn)恢復(fù)到原來(lái)的位置。 目前常用的畸變校正方法主要分為兩類： 全局校正法和局部校正法， 這兩種方法都有各自的優(yōu)缺點(diǎn)。 本文提出了利用支持向量回歸機(jī)建立校正模型的一種全局校正方法， 并將其與全局多項(xiàng)式模型做比較， 通過(guò)比較兩種模型的校正圖像中特征點(diǎn)分布與理想分布的符合度， 檢驗(yàn)所提出算法的效果。 實(shí)驗(yàn)結(jié)果表明， 提出的算法效果明顯好于傳統(tǒng)的多項(xiàng)式方法。

X-射線圖像； 畸變校正； 支持向量回歸機(jī)

YUAN Ze-hui, LI Shi-zhong. X-ray image distortion correction based on SVR. Journal of Measurement Science and Instrumentation, 2015, 6(3)： 302-306. [

袁澤慧， 李世中

(中北大學(xué) 機(jī)電工程學(xué)院， 山西 太原 030051)

10.3969/j.issn.1674-8042.2015.03.018]

Received date： 2015-05-14 Foundation items： National Natural Science Foundation of China (No.61305118)

YUAN Ze-hui (yuanzehui1985@126.com)

1674-8042(2015)03-0302-05 doi： 10.3969/j.issn.1674-8042.2015.03.018

CLD number： TP391.41 Document code： A