A Fast Reconstruction Algorithm of Coronary Artery

A Fast Reconstruction Algorithm of Coronary Artery

Based on Geometry Feature

PAN Jun-jun，ZHANG Yan-ning，ZHAO Rong-chun，CUI Chang-zong

(Department of Computer Science and Technology，Northwest Polytechnology University，Xi’an 710072)

Abstract

Inherent to the particularity of 3-D reconstruction of coronary artery in Angiograms, an algorithm based on geometry feature of coronary artery is presented. According to the geometry feature of coronary artery, the method of binocular stereo is used. In the sequence of from up to down, from left to right, from trunk to branch, the control points which have the feature of intersection, corner, tip are chosen and matched, and make the blood vessel between two control points can be simplified as a line with different diameter. Extract and generate the volume data from each control point. Then do the 3-D rendering by the way of “display the blood vessel with lines instead of curve, and approach the curve by adequate lines”. The experiment demonstrates that this algorithm is fast, simple, small in data, controllable in accuracy. It is suitable for the 3-D reconstruction of blood vessel, which belongs to the objects of line geometrical figure and the quality of angiographic images is not very high.

Keywords: Angiogram, Geometry feature, 3-D reconstruction, Volume data, Control point

1. Introduction

The coronary artery disease is one of the fatal diseases that threaten a lot to human’s life. Both of its occurrence rate and mortality rate are very high. To judge the function of coronary artery, the angiography is often used to display the figure and position of coronary artery in clinical medicine. To observe the coronary artery more accurately by omnifaceted methods, and aid in the diagnoses and operation, there are growing requests in 3-D blood vessel visualization from 90s and apply to the chest surgery operation initiatively. Therefore, there is great sense and value in analyzing and treatment of coronary artery disease by researching the 3-D reconstruction of coronary artery from angiograms.

As there are many complicated characteristics in angiograms, such as strong noise, low contrast, tree structure, the gray level is not well-distributed and so on. There are a great many difficulties in accurate reconstruction by detecting the contour of blood vessel automatically. At present, there are some useful methods in it, such as “double square box region of search” presented by Hoffman can be used to tracking the blood vessel automatically. And it can get the local size, the position of pipeline and other parameters of coronary artery at the same time. Another good method presented by Coatrieux and Collorec [6] is the extracting algorithm of blood vessel axis and contour, which is developed from the vector-tracking algorithm. It has the advantages of little manual manipulation, low time cost, and can be able to detect the artery iterated from thick vessels to vessels. However, all the methods above have a same disadvantage. That is the demand of image quality is very high. At present, the image quality of X-ray angiogram is limited and low, and the definition and resolution is not high enough by the means of common image pre-processing. What’s more, a great many of knowledge of anatomy is needed in detection and reconstruction. If assist it with the Expert System [12], there will be more complexity and expenditure to impact its popularization in clinical medicine. To overcome the difficulties above and make the reconstruction accuracy high enough, usually the way of manual detection and Computer Human Interaction are applied to detect the blood vessel and choose the corresponding points yet. As some subjective decisions being made in this semi-automatic reconstruction method, the reconstruction accuracy of blood vessel is limited. Hence when the accuracy is unchangeable, it’s the main aim to lower its complexity, reduce manual manipulation, accelerate the speed of reconstruction and lower the quantity of volume data. Pointing to this aim, an algorithm is presented based on geometry feature of coronary artery in this paper. It obtains the 3-D information and does the 3-D rendering of volume data according to the feature of coronary artery. The experiment demonstrates that this algorithm is simple, fast, small data, controllable in accuracy. Now this algorithm has been realized in the platform of microcomputer and tested by practical examples in application.

1. The Particularity in 3-D Reconstruction of Coronary Artery from Angiograms and the Algorithm Design

According to the difference of data description in reconstruction, the 3-D reconstruction can be divided into two types: One is describing the 3-D structure by connecting geometrical figure unions (such as triangle) and warping the object surface (such as “Matching Cubes Algorithm”), which is called surface rendering. Another is projecting the voxels to the display plane directly, which is called volume rendering (such as “Ray Tracing Algorithm”). The characteristic of former method is part reconstruction, suitable for reconstruction of tissue or organs of which the surface feature is clear, such as the bone in CT. The demand for the segmentation accuracy of this method is very high, so its reconstruction result is not good enough for tissue such as blood vessel, of which the contour is not clear and the brightness is changeable. Moreover the figure of blood vessel is of line type, which is not suitable for the curve surface warping. So reconstruct coronary artery by this method purely is not practical. The characteristic of latter method is that the information is abundant for reconstructed images, the result is realistic and of high accuracy. But the disadvantage of this method is that it needs plenty pictures of high accuracy, the calculation is large and the request of hardware level is high because each voxel need to search during rendering. In fact, the space of reconstruction in angiograms is small, just the little proportional places which contain the blood vessel. So the efficiency will be low if using the way of volume rending because the whole image will be searching and displaying including some voxels that is redundant, even bad for human vision. In view of the fact that the quality of present X-ray angiogram is limited, and it is low in speed and real time, Surface Rendering is not practical either.

In summary, pointing to the particularity above in 3-D reconstruction of angiographic coronary artery, the particular algorithm should be designed to solve this problem, not by the common methods mechanically. As follows, the particularity of coronary artery is analyzed in anatomy and the algorithm based on the geometry feature of coronary artery is designed.

In anatomy, there are three geometry features as follows in human’s coronary artery generally:

(1) The whole coronary artery is a tree structure, consisting of many branches.

(2) In part, the blood vessel is of line figure, which can be simplified to a smooth curve.

(3) There is a relatively constant diameter in each part of blood vessel section, nearly without texture.

By the theory of solid geometry, a spatial curve can be determined by two projections of different direction. Therefore, the method of binocular stereo is used to extract the 3-D information of coronary artery, i.e. extracting the 3-D information of coronary artery by the parallax from stereo pair of images. In this way, not only the complexity of this problem can be lowered, keeping the given accuracy of reconstruction, but also the radioactive harm can be reduced to the least. Because just two pictures of angiograms makes the time of X-ray clairvoyance reduced to the most, which conform to the clinical request.

The algorithm can be divided into two steps:

The first step: The 3-D volume data exaction of coronary artery

(1) First, register the two original angiographic images.

(2) After registration, according to the tree structure, Choose and match the control points that have the feature of intersection, corner, tip from the stereo pair of images in the sequence of from up to down, from left to right, from trunk to branch. And make the blood vessel between two control points can be simplified a line with different diameter. As the feature of these control points is obvious, this method is very suitable for manual detection.

(3) Match each pair of control points and obtain the 2-d information of each pair of control points from the pair of angiographic images, including the blood vessel diameter at the position of this control point.

(4) By camera calibration, transform the stereo information of control points to the 3-D data of coronary artery in the world coordinate. Create and store the file of volume data.

The second step: The 3-D rendering of coronary artery

In 3-D rendering, the blood vessel between two control points is nearly a line with a constant width. Hence the colorful line with different width can be used for the basic geometric figure union to connect these control points. This rendering method is called “display the blood vessel with lines instead of curve, and approach the curve by adequate lines”. The blood vessel between control points is nearly smooth, so this method will not reduce given accuracy too much if adequate control points are obtained. The remarkable advantage of this method is fast, simple, small in data, controllable in accuracy because it no need to search, match and display each voxel of coronary artery during the 3-D rendering. The whole algorithm will be stated detailedly in the following two section.

1. The Volume Data Extraction of Coronary Artery

3.1 Image Registration

As X-ray angiography belongs to the perspective projection, the projection size and position of coronary artery can’t be the same because of different focal length. So the image registration is needed first. The registration of angiograms belongs to the monomodality medical image registration. It can be resolved by the two-dimensional linear transform for one image:

（1）

is the coordinates of each pixel in original image. is the coordinates of each pixel in this image after registration. is the image zoom proportion factor. is a 21 image translation vector. , can be determined by comparing the vertical distance and the position of two pairs of corresponding control points from the pair of angiographic images.

3.2 Choosing and Matching the Control Points

Travel the angiograms in the sequence of from up to down, from left to right, from trunk to branch after registration. Choose and match the control points which have the feature of intersection, corner, and tip manually. This method belongs to Human Computer Interaction. In this way, not only avoid the difficulties of image restoration, detection and segmentation of coronary artery in angiograms, but also make the opportunity that the surgeon who is familiar with the structure of coronary artery can join in the operation and plays a active role, embody the advantage of expert experience, and enhance the reconstruction accuracy.

After registration, each pair of control points should be search in the same row, i.e. they are in the same epipole line. In fact, as the existence of practical error, the searching must be in a area not in a line, there is a restrain regulation in matching the control points: The corresponding points will be searching in a area which is of the same height from the pair of angiographic images. The formula is as follows:

（2）

3.3 Camera Calibration and 3-D Coordinates of Control Points Calculation

Calibrate and survey the depth of each pair of control points from the generated control points set by the method of binocular stereo, and generate the volume data in the three-dimensional coordinate system.

Two rigid body transformations will be used during this process: One is the camera calibration from left image to the three-dimensional coordinate system, including a rotation matrix and a translation vector . Another is the camera calibration from right image to the three-dimensional coordinate system, including a rotation matrix and a translation vector . Supposing the coordinates of corresponding control points in each pair are and in the three-dimensional space. The transform formula of the left point in each pair from the left camera coordinate system to the world coordinate system is as follows:

=+ （3）

The transform formula is the same for the right angiographic image:

=+ （4）

Find the minimum of the formula as follows by calculating which point is closed to the pair of sight ray most. Then find the answer of and according to the condition that this point is closed to the pair of sight ray most.

（5）

In the formula above, is the focal length of the left angiographic image, is the focal length of the right angiographic image. is the base of the world coordinate system. is the left sight ray rotated into the world coordinate system, is the right sight ray rotated into the world coordinate system. Solve this intersect sight rays problem by derivation of and , i.e. solve the equation in condition that both and is 0. Put the results of and into the sight ray equation of formula (3) and (4), then get a pair of three-dimensional coordinates of control points. The average of this pair of three-dimensional coordinates is the final result of this corresponding control point in the world coordinate system.

The parameters of , and ,,,can be get by calculation from the projection angles and other original data including the focal length of this pair of angiographic images after chosen the world coordinate system.

3.4 The Generation of Volume Data of Coronary Artery

The diameter of blood vessel in each control point is needed after getting its three-dimensional coordinates. By the reason of perspective projection, the diameter of blood vessel in each control point may not be same in the pair of angiographic images. However, the diameter of blood vessel is so small in contrast to the distance from the X-ray source to the focal plane, hence it can be simplified to the orthodox projection. The diameter of blood vessel in the world coordinate system can be estimated to the average of diameters from two angiographic images simply, i.e. . Finally, the volume data of coronary artery is generated by storage of the three-dimensional coordinates and the blood vessel diameter in each control point to a file with a structure array in structure data.

1. The 3-D Rendering of Coronary Artery

After getting the volume data of coronary artery, do the 3-D rendering in the same sequence of from up to down, from left to right, from trunk to branch. Connect these control points with lines of which the color parameter is （1，0，0）in OpenGL. One thing need to pay attention to is that the width of each line is equal to the diameter of each blood vessel section, and the diameter of each blood vessel section can be estimated to the average of diameters of control points in the end of this blood vessel section, i.e. （ is the diameter of the th control point， is the diameter of the th blood vessel section， which is equal to the average of diameters of control points in the end of this blood vessel section: and ）

1. The Result and Analysis of Experiment

According to the algorithm above, program with Visual C++ and OpenGL in the platform of microcomputer (CPU:Pentium4，Memory:128M). Take a test from a pair of angiographic images from a practical patient (fig.1). The result is as follows: (fig 2, fig. 3). In the result, the meaning for four angles of projection direction is: CRA stands for the projection angle toward the head; CRA stands for the projection angle toward the bottom; RAO stands for the projection angle toward the right; LAO stands for the projection angle toward the left.

(a) 15 o CAU 45 o RAO (b) 10 o CAU 55 o LAO

fig.1 The Original Angiograms

(a) 15 o CAU 45 o RAO (b) 10 o CAU 55 o LAO

fig.2 The Angiograms after Choosing and Matching the Control Points

fig.3 The 3-D Display of Reconstructed Coronary Artery

Analyze the reconstruction result from the time cost and data quantity, and make a comparison with the result analysis between this algorithm and volume rendering. The analysis result can be seen in table 1. The result shows that in this algorithm, the volume data is small, time of rendering is short, the 3-D model display is realizable. Another advantage of this algorithm is that the reconstruction accuracy is controllable. If the demand of accuracy is high, more control points will be chosen and matched to show more fine 3-D structure and information of object. If the demand of accuracy is low, less control points may be chosen and matched. So it can deal with different accuracy request in different situation.

Table.1 The Quantity of Volume Data and Time Cost in this Test

The Reconstruction Method / The number of control points / The number of blood vessel sections / The size of volume data / The cost time in calculation / The cost time
In display
This algorithm / 35 / 34 / 1.32K / 0.02s / 0.06s
Volume rendering / About 2000 / 0 / 120K / 1.3s / 4s

1. Conclusion

An algorithm based on geometry feature of coronary artery is presented in this paper, inherent to the 3-D reconstruction of coronary artery in angiograms. The way of binocular stereo is used, according to the geometry feature of coronary artery in anatomy. The control points are chosen and matched by a certain principle and sequence. Then doing the 3-D rendering by the way of “display the blood vessel with lines instead of curve, approach the curve by adequate lines” to connect these control points. The experiment realized by programming in the microcomputer demonstrates that this algorithm is fast, simple, small data and controllable in accuracy.