Wavelet Based Edge Detection Method for Analysis of Coronary Angiograms

A. Bezerianos', A. Munteanul, D. Alexopoulos', G. Panayiotakis', P. Cristea2

1 Dept. of Medical Physics, University of Patras, Patras, Greece Dept. of D. S.P., Techical University of Bucharest, Bucharest, Romania 2

Abstract

The assessment of coronary anatomy is one of the prime determinants in choosing medical or intewen- tional therapy for patients with ischemic heart disease. We report a wavelet based method of coronary border identification which has the advantage of the detection of the edges at different scales (the image changes are computed in a variable neighborhood), unlike the con- ventional methods, where a fixed, heuristic neighborhood is used. Additionally, the conventional methods are more noise sensitive than a wavelet based method. We propose an algorithm to combine the information from the multiple scales. The computer determined diameters are compared to the actual diameters of the simulated vessels of three test objects. These comparisons show that our method allows accurate identijkation of the borders of phantom vessels (correlation coefficient in the range 0.90-0.99for different wavelets).

1. Introduction

Coronary luminal morphology is a key factor in the evaluation and treatment of patients with ischemic heart disease and plays a critical role in choosing medical ver- sus surgical therapy in this situation. The standard clini- cal method in the assessment of coronary artery disease is the visual estimation of the luminal diameter stenosis. Although this method is simple and rapid, it provides a poor correlation with direct measures of coronary stenosis and is associated with significant inter- and intraobserver variability [ 11.

Various methods for automated border identification of coronary arteries have been developed to overcome these shortcomings and to improve the accuracy of coronary morphologic analysis. Most of them consist in fitting of an edge model or in the application of an edge operator with constant size. However, as suggested by Marr and Hildreth [2] a good description of the image edges can be obtained by applying an edge operator at

different scales and then by combining the edge informa- tion. This method involves three problems: (1) to choose between a maxima edge detection equivalent to a Canny [3] edge detection and a zero-crossing edge detection equivalent to a Marr-Hildreth [2] edge detection, (2) the use of a continuous scale or a dyadic scale, (3) the ways in which the edges detected at different scales are efficiently combined to build up the final edge image

In this paper we analyze the information given by the Discrete Wavelet Transform (DWT) proposed by Mallat et al. [4]. The paper is organized as follows. Section 2 presents briefly the DWT and the evolution across the scales of the maxima locations for four types o f edges: step, ramp, pulse, and stair edges. In section 3 we present the algorithm that combines the information from the multiscale images. In the fourth section, using different wavelets, we apply the method to determine the diame- ters of the simulated vessels of three phantoms. and we compare them with their actual diameters. Section 5 ex- plains the main conclusions we have obtained.

2. Theoretical basis

0276-6547195 $4.00 0 1995 IEEE 573

Let be O(x) the smoothing function defined by

Mallat et al. in [4], the wavelet Y(x) the first-order de-

rivative of O(x), and Os(.) the dilation at the scale s of

@( x) : Os( x) = @( x / s) / s. The wavelet transform of f( x) at the scale s is computed by convolving the signal with the dilated wavelet:

Wsf(x) =f*Ys(x)= S - ( f " q . ) d dx

The local extrema of Wsf( x) correspond to the edges detected at the scale s To allow fast numerical imple- mentation, we imposs :A dyadic scale s = (2J)1a instead

of a continuous one. In two dimensions the wavelet trans- form contains the two componcnts of the gradient vector

( 2 )

Computers in Cardiology 1995

The edge points are the inflexion points of the surface (f*O,, )(x,y) and correspond to the local maxima points

of the modulus M,,f(x,y) of the gradient vector detected

in the direction of o(f*O,j )(x,y) with the horizontal

axis. In addition, at the maxima locations, M,,f(x,y)

measures the values of the first-derivative, therefore using the amplitude information one can easily differentiate small amplitude fluctuations from important discontinuities. This is an advantage over zero-crossings approach which gives only the position information.

For the fast DWT, in [4] is defined the smoothing operator S,, by: S,,f(x,y) = (f*(D2’)(x,y) where

cD(x,y) is a smoothing function. The wavelet transform

at the scale 2, provides the details that are available in S2,-’f(x,y) but have disappeared in S2,f(x,y). If we

denote by h(n) and g(n) the filters associated with @(x) respectively the wavelet Y(x) ? the fast algorithm for the 1D DWT is:

I

-2 S,,+,f(n)= Ch(k).S, ,f(n- 2-’k) , Vj j r0 (3)

W

Wz,+’f(n) = xg(k) .S2 , f (n-2Jk) , b’j’j20 (4) --w

where w is the length of the filter g. The coefficients of h are: h(O)=h(-1)=0.375 and h(l)=h(-2)=0.125.

At coarse scales, the modulus maxima have different positions than at the fine scales. This is due to the smoothing of the image by @(x,y) . We estimate further the shifting of the wavelet maxima for four edge types: step, ramp, pulse and stair edges. The election of these classes of profiles is based on the work developed by Williams et al. [5]. We describe the 1D case and in section 3 we extend in two dimensions. Let Slf (n) be the discrete signal that models one of the above edges, and no the inflexion point. In a vicinity Vnn we v e r e :

Vk eVno , Slf(no + k ) + Slf(no - k ) = 2 . Slf(no) (5 )

For the pulse and the stair profiles we verlfy the same equation in two adjacent vicinities Vnn and V,,‘ which correspond to the two inflexion points “0 and n1 Generally, if nj is the inflexion point of S2]f(n) and w;

is the width of the edge at the scalej, we define the edge centered in nj as E,,. = [nj - w. n. -I wj] and the

vicinity of nj as VnJ = [nj - wj - 2J-’,nj + w, + ZJ]. For

VkEVn , S, , f (nj+k)+S2,f(nj-k)= 2*S2,f(nj) (6)

We show that the inflexion point of S,,+,f(n) is:

1’ 1

n, - 2’-1 . From (3) we derive

S2J+~ f (n, - 2j-l) = 2. S,, f (nj ) . (h(0) + h( 1)) (7)

Using(6)and(7)weprovethat: 2.S,J+lf(nj -2JP1)=

=S2J+~f (n , -2J -1+k)+S2 ,+ l f (n j -2’-’-k) (8)

therefore the shifting of the inflexion point of S,,+,f(n)

with respect to the inflexion point of S,,f (n) is 2J-l,

j 2 0 . It comes out that the shifting of the local maxima of W,,+,f(n) with respect to the inflexion point of

Slf(n) (actual position of the edge) is 2’ - 1 / 2 . This is true for all j in the cases of the step and the ramp profiles. At the coarse scales, for the pulse and the stair profiles, an error occurs between this theoretical shifting and the actual s W n g of the wavelet maxima because of the influence between the edges. If ny and ni are two

adjacent inflexion points, for great values of j , either V 0 nEn; f 0 , or Vn; nE,? f 0 or both. One can see

this phenomenon in figure 1, where we plot the evolution across the scales of this error for all the four profiles. For the step and the ramp profiles the error is 0, but it increases with the scale for the pulse and the stair models.

“1

A I A I I , I

0 0 0 0 0 0 0

1 0 0 0 0 0 O D

A , n A A I n I A e o 0 2 2 2 3 0 0

Figure 1. The evolution across the scales for the shifting error (absolute value) for a signal that contains the ramp, pulse, stair and step edge profiles; in the third and the fourth details the maxima of the stair profile are overlapping.

574

3. Method

In this section we describe the scheme that solves the problem of combining the information from the multiscale edges.

At every scalej we are looking for the local maxima of the discrete modulus M2,f(m, n) in four directions,

A = 0~,go~,45~,-4j~} and we create four different

gradient images: IJ(k) , k E A . We shift all the local

maxima w(m,n) EIj(k) in the new positions

w(m + 2 j - l - 1,n + 2J-l - 1) to compensate the theoreti- cal shifting described in section 2.

We cannot use the information from the first detail U = I ) because the edges in coronary angiograms are noisy and most of the maxima detected in the first detail are due to the noise.

In [6] the coronary images are divided into two groups: a “good image quality group which includes the images with good contrast and little noise and a “poor” image quality group with the images with very little con- trast or an extreme amount of noise.

For “good’ angiograms, the second detail gives the actual positions of the edges because in S,~f(m,n) the

edges are less noisy and no influence occurs between them. As was stated in section 2, due to the shifting error, if w(m,n) is a local maxima in 12(k), k E A , the

corresponding maxima in 13(k) is not necessarily in the same position (m,n); we find it along the gradient k, in one window centered in (m,n). For every k E Athe

{

windows are pr k= 0” ,

k= 90” ,

k= 49,

k= -45”, We start the searching procedure from the center

(m,n) of the window and we continue simultaneously to the lower and the upper limits of the window until a maxima is find.

We have found experimentally that for all the four edge profiles and for various edge widths, the error due to the influence between the edges is less or equal than +3in W23f(m,n) , therefore a proper value of w3 is:

w3 = 3 . We create further four final gradient images by

adding the maxima from 12(k), k t A and the corres-

ponding maxima found in 13(k). With this procedure,

creasing the amplitude of the local maxima found in the second detail with the amplitude of the local maxima found in the third detail, we improve the accuracy of the detected borders and we enhance the signal to noise ratio in the final edge image.

For “poor” angiograms with this algorithm we obtain spurious edges because in S,1 f (m, n) the edges are still

noisy and consequently, W2z f (m, n) contains maxima

coming from the noise. In this case, in the first stage, we do not take the information about the positions of the inflexion points from the second detail, because edge localization errors can occur due to the noise. We per- form the same procedure, considering thz position infor- mation given by the third detail and the value w4 of the searching window in the fourth one: w4 = 4 . We correct further the positions of the edges with those detected in the second detail searching along the gradient in a win- dow with w2 = 3 ; from this detail we use also the am- plitude information. Taking the edges positions from the third detail we separate by searching with a small win- dow in the second detail, the maxima coming from the edges from the maxima due to the noise.

We create the final edge image by simply adding the four final gradient images. In figure 2 we present the result of the method using one coronary angiogram of “good’ quality.

Figure 2. Computer detected borders in an image of “good’ quality.

4. Results

We study in this section the behavior of the method for several types of wavelets using four simulated vessels

removing the information from the first detail and in- from three arterial phantoms.

575

We have derived three classes of smoothing functions ~ ( x ) that approximate well a Gaussian function of variance rs . The error between the smoothing functions and the Gaussian function was minimized in distinct ranges of o .We have derived the corresponding wavelets and the G filters using the method described in j4j. The coefficients of the G filters are presented in table I. We have tested also the G filter designed in 241 (filter 4).

filter \ n 1.

-2 -1 0 1 0.1069 I 2.6090 I -2.6090 -0.1069

2. 3. 4.

Table I. The G filter coefficients for different wavelets.

0.0077 2.2166 -2.2166 -0.0077 0.2196 2.1164 -2.1164 -0.2196

0.0 2.0 -2.0 0.0

We have performed the border detection algorithm using separately the above filters and in each phantom we have computed the average diameter for each of the four vessels. In order to express the diameters of the vessels in millimeters, we have employed a l-cm rectilinear grid which was recorded and digitized in the same time as the phantoms. The diameters of the simu- lated vessels were lmm, 1.8 nun, 2 mm and 2.33 mm.

For every G filter the computer-determined dlameters have been compared with the actual dlameters; the estimation parameters, the mean error e , the mean absolute error e,, and the correlation coefficient C, are presented in table 11.

- e

1. 2. 3. 4. 0.01” 0.02” 0.00“ 0.02”

I I I I I I - e, C C

0.07nim 0.08” 0.06” 0.08”

96% 91% 99% 90%

Table 11. The estimation parameters for all the four wavelet filters.

As we see in the above table the best results are ob- tained with the third filter. The second and the fourth filters are more sensitive to the noise and the localization errors are higher than for the other two filters. For the third filter we have computed also the average across all the vessels of the percentage of the absolute error from the actual diameters; the result is 4.4395% that is compa- rable with what was obtained with other methods [6]: 4.4%.

Running the algorithm with the third filter we have calculated the minimal lumen diameters in 10 coronary

angiograms of dfierent qualities: seven of them were graded as “good” quality images and the other three of them as “poor” quality images. The computer-determined diameters are highly correlated (c, =93%) with those detected manually by an experienced observer.

Using angiograms showing a tree structure of the vessels, our method detects the edges at one bifurcation level lower than the tested conventional methods.

For a 512 x 512, 256 gray-levels coronary angiogram the algorithm is performed on a PC486 computer in 130 seconds, therefore the method presented here has sub- stantial promise for real time implementation.

5. Conclusions

We have developed an algorithm of coronary border identfication which use the information given by the Discrete Wavelet Transform proposed by Mallat [4]. We have followed the evolution across the scales of four types of edge profiles, and we have extended the results in two dimensions looking for the maxima in the gradi- ent direction. We have proposed the methods to process the multiscale information, separately for “good and “poor” quality angiograms. Our phantom studies using several types of wavelets have revealed that the method allows accurate identification of the coronary borders.

References

[l] Zir LM, Miller SW, Dinsmore RE, Gilbert JP, Harthorne JW. “Interobserver variability in coronary an- giography”. Circulation 1976;53 :627-3 2. [2] Marr D, Hildreth E. “Theory of edge detection”. Proc. Royal Soc. London 1980;207:187-217. [3] Canny J. “A computational approach to edge detec- tion”. IEEE Trans. on P M 1986;8:679-98. [4] Mallat S, Zhong S. “Characterization of signals from multiscale edges” IEEE Trans. on PAM1 1992;14:710-32 [5 j Williams DJ, Shah M. “Normalized edge detector”. Proc. 10th. Int. Conf. on Pattern Recognition

[6] Steven RF, Mary1 RJ, Christopher JW, David JS, Robert FW, Carl WW, Melvin LM, Steve MC, “Automated analysis of coronary arterial morphology in cineangiograms: geometric and physiologic validation in humans” IEEE Trans. on Med. Imaging 1989;8:387-400.

1990; 11942-46.

Adress for correspondence.

Anastasios Bezerianos. Department of Medical Physics, University of Patras, Patras 26500. Greece.

576