Simulated Wave Propagation and Traceback in Vascular Extraction

Transcription

1 Simulated Wave Propagation and Traceback in Vascular Extraction Francis K.H. Quek, Cemil Kirbas, and Xiayun ong, Vision Interfaces and Systems Laboratory (VISLab) Department of Computer Science and Engineering, Wright State University, Dayton, Ohio Abstract This paper presents an approach for the extraction of vasculature from angiography images by using a wave propagation and traceback mechanism. We discuss both the theory and the implementation of the approach. Using a dual-sigmoidal filter, we label each pixel in an angiogram with the likelihood that it is within a vessel. Representing the reciprocal of this likelihood image as an array of refractive indices, we propagate a digital wave through the image from the base of the vascular tree. This wave washes over the vasculature, ignoring local noise perturbations. The extraction of the vasculature becomes that of tracing the wave along the local normals to the waveform. While the approach is inherently SIMD, we present an efficient sequential algorithm for Key words Wave propagation, traceback, vascular extraction I. INTRODUCTION A variety of medical imaging techniques such as X-Ray Angiography, Computed Tomography and Magnetic Resonance Imaging/Angiography are capable of obtaining data on vasculature. This paper describes our approach to the extraction of this vasculature from the image data to obtain its connection morphology and the structure of each vessel. As with the processing of most medical images, signal noise, drift in image intensity and lack of image contrast pose significant challenges to the extraction of blood vessels. In X-Ray angiography, for example, the consistency of the pixel intensity is dependent on a number of factors. Angiograms are obtained by injecting the patient with a radio-opaque dye before imaging. The consistency of this dye in the vessel, the depth of the vessel, and noise in the imaging process result in images that are difficult, even for a human expert, to interpret. Furthermore, because X-Ray angiograms are projections of the three-dimensional reality into a two-dimensional representation, there is a fair amount of self occlusion among vessels. Hence, one cannot extract these vessels directly using the image intensities or gradients (edges) alone. Researchers have taken different approaches to this problem. These may be classified as: traditional pattern recognition techniques, trackingbased approaches, model-based approaches, artificial intelligence approaches, neural network-based approaches and front propagation methods. Representative of traditional pattern recognition approaches are Ritchings and Colchester [1] who apply a syntactic pattern recognition scheme, and Thackray and Nelson [2] who use morphological and adaptive thresholding. The former processes X-ray angiograms by applying an edge detector. The ribbon segments, formed by pairing opposing edges, are grouped to obtain extended vessel tracts. The system does not attempt to determine the structure of the vessel segments. The goal was to label for the diagnosis of vascular abnormality. Thakray and Nelson [2] describe an approach which extracts vascular segments using a set of 8 morphological operators, each of which represents an oriented vessel segment (in 8 orientations). The second set of approaches apply explicit vascular tracking. Liu and Sun [3] present an approach that extracts extended tracts of vasculature in X-ray angiograms by an adaptive tracking algorithm. iven an initial point within a vessel, they apply an extrapolation update scheme [4] that involves the estimation of local vessel trajectories. Once a segment has been tracked, it is deleted in the angiogram image. This procedure is performed recursively to extract the vascular tree. This algorithm requires the user to specify vessel starting points, and does not appear extensible to 3D extraction. Aylward et al [] describe an approach by which the medial axes of tubular objects such as vessels in an angiogram are approximated as directed intensity ridges. As with [3], these ridges are tracked by estimating the local vessel directions. The authors show results of a vascular tree extracted from a MR angiogram. This required a fair amount of user intervention (10 mouse clicks in all). The third class of approaches are model-based in that they apply explicit vessel models to extract the vasculature. Klein, Lee, and Amini [6] describe an approach to extract vessels from X-ray angiograms using deformable spline models or snakes. In their approach, the user provides an initial estimate of the location of the vascular entity, and the system refines the estimate by deforming snake to minimize some energy function. They use a B-spline model in their snake implementation. The energy function defines such constraints as the smoothness or coherence of the contour, the closeness the contour is to image edge pixels, and the compactness of the boundary. They use a abor filter to determine the image (or edge) energy term to attract

2 2 the snake. The approach is most suitable for the accurate extraction of vascular segments. The amount of user interaction and computation required makes it impractical for extracting entire vascular structures. The fourth class of approaches may be described as artificial intelligence-based. Smets et al [7] present a knowledge-based system for the delineation of blood vessels on subtracted angiograms. The system encodes general knowledge about appearance of blood vessels in these images in the form of 11 rules (e.g. that vessels have high intensity center lines, comprise high intensity regions bordered by parallel edges etc.). These rules facilitate the formulation of a 4-level hierarchy (pixels, center lines, bars, segments) each of which is derived from the preceding level by a subset of the 11 rules. The show results of their system that indicate that the system is successful where the image contains high contrast between vessel and the background, and that the system has considerable problems at vessel bifurcations and self-occlusions. The fifth class of approaches in the research literature employ neural networks in image segmentation and vessel detection. Nekovei and Sun [8] describe their backpropagation network for the detection of blood vessels in X-ray angiography. This system applies the neural network directly to the angiogram pixels without prior feature detection. Since angiograms are typically very large, the network is applied to a small subwindow which slides across the angiogram. The pixels of this subwindow are directly fed as input to the network. Pre-labeled angiograms are used as the training set to set the network s weights. A modified version the common delta-rule is to obtain these weights. This system does not extract the vascular structure. Its purpose is to label the pixels as vessel or non-vessel. The last class is front propagation approaches. Caselles et al [9] and Malladi et al [10] use propagating interfaces under a curvature dependent speed function to model anatomical shapes. They used the Level Set Method approach developed by Osher and Sethian [11] and adapted it to shape recognition process. The main idea behind the Level Set Method is to represent propagating curves as the zero level set of a higher dimensional function which is given in the Eulerian co-ordinate system. Hence, a moving front is captured implicitly by the level set function. This approach has some advantages that make it attractive. Sethian developed another method, called the Fast Marching Method [12], which uses a wave propagation approach for specialized front problems. Fast Marching Methods are used in the problems where the front advances monotonically with a speed that does not change its sign. The Fast Marching Method s advantage over the Level Set Methods is that it is more computationally efficient. II. THE PROBLEM REFRAMED Figure 1 shows a section of an X-ray angiogram. Notice the variation in intensity within the darker vessel. The chal- Figure 1. A section of a vascular angiogram lenge is to ignore the local intensity variations while extracting the entire vasculature in a noisy image. While one cannot easily determine a clear set of thresholds to determine which pixels represent vessels, one can ascertain that certain pixels are more likely to belong to vessels. Hence, instead of applying firm thresholds, one might use a filter to assign a likelihood that pixels in an image are vessel pixels. To do this, we apply a dual sigmoid function to the data. This function assigns those intensity values that are definitely vessels to 1.0 and intensity values that are definitely not vessels to 0.0. Other intensity values are assigned a likelihood value according to the function. We can think of the reciprocal of the resulting image as a cost function array C (x y). iven a point of origin at the base of the vascular tree, the problem of detecting the vascular path to any extremal point in the tree may be thought of as that of minimizing the path integral: I `(x y) C (`(x y)) d` (1) where `(x y) is some path through the cost space, for all possible paths. Unfortunately, this is an intractable NPcomplete problem. We reframe the problem by modeling the cost function as a set of refractive indices, and the image as a medium through which a wave may be propagated. Pixels that are definitely not vessels are assigned an index of infinity (they are barriers to the wave), and pixels that are definitely vessels are given an index of 1. Other costs are assigned linearly between 1 and the highest refractive index (this is variable). Hence, if we propagate a wave at the base of the vascular tree, the wave would travel faster through pixels that are more likely to be vessel pixels and slower through less likely pixels. The problem of finding `(x y) that minimizes equation 1 from any extremal point in the vasculature becomes one of tracing the wave back along the direction of local normals to the wavefront. One may think of the approach as modeling the image as a pond where the lower indices are shallower than the larger indices. If a stone were dropped into the pond, the wave would propagate faster over the shallower regions and slower over the deeper regions. A trace following the local wavefronts from any point in the pond over which the wave propagates, will invariably take us back to the origin of the

3 3 wave. This approach is able to find the vessels through the intensity variations because the wave is able to wash over these variations while maintaining its general path through the vasculature. III. WAVE PROPAATION AND TRACEBACK We apply a general wave propagation technique that produces a digital saw-tooth waveform over a medium of any dimension. For simplicity, we shall begin with one dimensional wave propagation, and extend it through the third dimension. We shall also describe the algorithm to trace the wave back to the source. A. Digital Wave Propagation The basic premise in our wave propagation approach is to apply local neighborhood operations in such a way as to make the results globally robust (i.e. insensitive to local noise variations). We define two kinds of scalar quantities, medium states, m i 2 fm 1 :::m k g, and wave states, w i 2 fw 1 :::w n g. We define a medium point as any N ; dimensional data point that has a medium state value, and a wave point as a data point that has a wave state. The general wave propagation algorithm may then be described as follows: Set Current Wave State to w n Tag the desired wave origin with w n while 9 medium points that are neighbors of wave points /* propagate the wave by one step */ Decrement Current Wave State if Current Wave State is less than w 1 Reset Current Wave State to w n 8 medium point that is neighbor of wave points /* Decrement once within one wave step even if it is has several wave point neighbors */ Decrement the medium point state by 1 if the medium point state is less than m 1 Set the medium point to a wave point with the current wave state Figure 2 shows an one dimensional wave propagation sequence with 4 medium states 1 ::: 4 and 8 wave states W 1 :::W 8. The wave origin was set to the center of the 1D data comprising 11 points. After the first wave step, the first neighbors were set to 0 which is less than 1. Hence the data points are replaced with the current wave state which was W 7. This process repeated for 11 wave steps until the wave had propagated over the entire medium. Figure 2 shows the result of this propagation as a step waveform. Notice the repetitive saw-tooth form of the wave. B. Two-Dimensional Wave Propagation In extending the wave propagation to 2D, the kind of neighborhood in the pixelated representation needs to be defined. Obviously, the kind of neighborhood used affects the shape of the waveform generated. Figure 3a and 3b show a comparison between 4 and 8 neighborhood propagation respectively. In this example, the original medium W 8 W W Current Wave State = W W 7 W W7 W8 W W7 W8 W W W7 W8 W7 W W 4 Current Wave State = W4 W7 W8 W7 W W 3 W 3 Current Wave State = W4 W7 W8 W7 W 0 3 W2 2 1 W4 W7 W8 W7 W W2 2 Current Wave State = W4 W7 W8 W7 W W2 1 W 8 1 W8 W4 W7 W8 W7 W 0 W 6 W8 W4 Propagated Wave W 7 W8 W7 W Current Wave State = 8 Current Wave State = 6 Current Wave State = Current Wave State = 4 Current Wave State = 1 Current Wave State = 7 W2 W2 0 W 7 Current Wave State = 6 Propagated Wave Figure 2. 1D wave propagation with 4 medium states and 8 wave states Figure 3. a. 4-neighborhood, b. 8-neighborhood, and c. alternating 4 and 8 neighborhood propagation and their saw-tooth form over a homogeneous medium states were all set to M 1 and the number of wave states was 16. Neither propagation morphology adequately approximates isotropic radial propagation. In our implementation, we alternated between 4 and 8 neighborhood propagation to yield the result shown in figure 3c. Depending on hardware configuration the 2D algorithm must be modified for computational efficiency. If a SIMD image neighborhood processor were available, our general algorithm needs very little modification to run efficiently. In a general purpose CPU, however, testing every medium pixel to see if it has a wave state neighbor for every wave step would be prohibitively costly. In our implementation, we employ a wavefront list whose neighbors are inspected to see if they need to be decremented or propagated over. For each wave step, we test and update the medium pixel neighbors of each wavefront point in this list according to our wave propagation rules. If a wavefront point has no more remaining medium point neighbors, it is removed to the wavefront list. Also, new wave points generated during any wave state iteration is added to the wavefront list in preparation for the next wave state iteration. We perform the wave propagation until the wavefront list is empty. To implement the alternating 4 and 8 neighborhood propagation shown in figure 3c, we add another weak neighbor tag bit to each pixel. For the central point C shown in figure 4, points [ ] are all strong neighbors in W7

4 C Figure 4. a.two-dimensional neighborhood index and Traceback in homogeneous medium b. biased and c. improved Wave Direction Wave Direction W 6 1 W Figure 6. Wave propagation through a. an angiogram segment and b. an angiogram segment with noise W 6 W 1 W 8 W 8 W 1 W 6 Current traceback point Figure. Aliasing example:1d wave with medium and 8 wave states neighborhood, while in 4 neighborhood, points [ ] are strong neighbors and points [1 3 7] are weak neighbors. We want to perform 4 and 8 neighborhood propagations on odd and even current wave state iterations respectively. The weak neighbor tag bit is initially set to 0. On odd current wave state iterations, when a medium pixel neighbor to the wavefront is inspected for the first time, this bit is set to 1 if it is a weak-neighbor (i.e. a diagonal neighbor), and its medium state is left unchanged while the first-visited and iteration-toggle tag bits are set the same way as before. If this medium pixel is visited again in the same wave state iteration because it is a strong neighbor of another wave pixel, we know that it had been visited, but not yet decremented in the current iteration if both the iteration toggle and the weak neighbor tag are 1. In this case, we clear the weak neighbor tag and decrement the medium state. Hence, we know that a medium pixel had been decremented in an odd wave state iteration if the iteration toggle and weak neighbor tag are 1 and 0 respectively. On even wave state iterations, all neighbors need to be decremented. Thus, we simply set all weak neighbor tags to 0 on even iterations. C. Traceback Algorithm As with the wave propagation, our traceback algorithm is a local neighborhood process. As described earlier, we trace the wave back along the direction of local normals to the wavefront. Let the current traceback point be centered at C. The normal to the wavefront at C would be in the direction of the steepest ascent in the wave from C. However, more than one pixel having the same wave state may satisfy this steepest ascent criterion. Hence, a good scanning rule for finding the steepest ascent path is necessary. Consider the labeled 2D neighborhood of a center pixel in figure 4a. If we simply took the sweep order [ ],we would get a biased traceback direction toward the northwest direction as shown in figure 4a which is undesirable. Figure 4b and c shows the result of improved traceback sweep sequence of [ ] followed by the reverse order: [ ]. This will remove the directional bias since any direction will not be permitted to dominate. Figure 7. a. Wave propagation and b. wave count images through an angiogram segment A key characteristic in our wave propagation paradigm is evidenced in our traceback algorithm. Since the waveform is cyclic, one would not know what the steepest descent wave pixel is if there are more medium states than half the number of wave states. Take the situation with 4 medium states and 8 wave states. If a wave pixel has wave state w 1, the steepest ascent pixel must be within the set fw 2 :::w g. Any wave neighbor in the set fw 1 w 6 :::w 8 g would actually belong to the previous wave cycle. If, however, we had medium states, there would be a directional ambiguity during traceback. Consider the two examples shown in figure. In both cases, the wave was propagated from left to right, and the correct traceback direction is in the reverse direction. As can be seen in our examples, an ambiguity arises as to whether a w 6 neighbor to the current pixel resulted from a propagation from w 6 down to w 1 through a medium or from a w 1 to a w 6 through a 3 medium. This is an example of aliasing that occurs when the wave sampling frequency is less than twice the maximum frequency of change in the medium. Hence, our system conforms the the classical Nyquist sampling theorem. IV. WAVE PROPAATION IN VESSELS Figure 6a shows our 2D wave propagation in an angiogram segment. The wave origin is marked by the cross and the traceback can be seen as the light colored line through the wave. Notice that the traceback does indeed approximate the lowest cost path rather than the medial axis. It snakes through the vessel, cutting the corners to find the shortest path. Figure 6b shows the wave propagation through the same vessel with 0% aussian noise added. The robustness of the algorithm is evident as the traceback is almost identical to that in figure 6a. Our wave propagation approach yields another side effect that is beneficial to the extraction of the vascular structure.

5 Figure 8. a. Wave propagation and b. wave count images through an angiogram segment Figure 7a shows a 32 wave state propagation through a homogeneous medium. Figure 7b is labeled with the wave count that tracks the number of the wave state cycles across the propagated wave. Figure 8 shows the concomitant wave propagation and wave count images for an angiogram segment. The wave count image effectively segments the vasculature into axial vessel segments. The connectivity of segments reflect the connectivity and self-occlusion of the vascular tree in the angiogram. This is especially useful in 3D wave propagation through MR or CT images since these do not exhibit self-occlusion. Hence all wave count segments with 3 or more adjacent segments are necessarily bifurcations in such 3D wave propagations. V. RESULTS We ran our wave propagation algorithm on a set of neurovascular angiogram images. The results obtained are very promising. Figure 9 shows an original and wave propagated angiogram images with measured vessel segments. We used our AIM system [13] developed in our lab to measure the vessels widths. We took 110 measurement over the vessel segments with widths between 3 pixels (0.3mm) and 3 pixels (6.24mm), from a set of 6 angiogram images, using the AIM system. Then we ran our wave propagation algorithm on these images and checked all these measurement points on the propagated vessels. The system could not propagate and extract vessel segments only on 4 of these 110 points. The rest of the vessel segments were successfully propagated and segmented by the system. The image sections in which 4 unsegmented vessel points lay were regions of high noise and low contrast. VI. DISCUSSION AND CONCLUSIONS This paper describes an approach for extracting vasculature from two dimensional medical images. Wave propagation and traceback allows us to extract not only the individual vessels, but the vascular connection morphology as well. Our system is robust to noise and is able to obtain the global network effectively. Our wave propagation and traceback algorithms can be easily extends to operate in three dimensional data. As with 2D wave propagation, 3D propagation will deal with weak and strong neighborhoods, with 8 and 4 neighborhoods corresponding to 26 and 6 neighborhoods respectively. Extension to traceback sweeping sequence to Figure 9. a. Original and b. Wave propagated angiograms with measured vessel segments 3D operations will require greater care to prevent a directional bias, but all the basic operations and ideas are the same with the 2D traceback. Our future work will focus on the 3D wave propagation and automated 3D traceback to extract the entire vascular tree in 3D data set. ACKNOWLEDEMENT This work has been supported by the Whitaker Foundation under Biomedical Engineering Research rant REFERENCES 1 R.T. Ritchings and A.C.F. Colchester, Detection of abnomalities on carotid angiograms, Pat. Rec. Let., vol. 4, pp , Oct B.D. Thackray and A.C. Nelson, Semi-automatic segmentation of vascular network images using a rotating structuring element (rose) with mathematical morphology and dual feature thresholding, IEEE Trans. on Med. Img., vol. 12, pp , Sept I. Liu and Y. Sun, Recursive tracking of vascular networks in angiograms based on the detection-deletion scheme, IEEE Trans. on Med. Img., vol. 12, pp , June Y. Sun, Automated identification of vessel contours in coronary arteriograms by an adaptive tracking algorithm, IEEE Trans. on Med. Img., vol. 8, pp , S. Aylward, S. Pizer, E. Bullitt, and D. Eberl, Intensity ridge and widths for tabular object segmentation and description, in Wksp on Math. Methods in Biomed. Image Analysis, pp , A.K. Klein, F. Lee, and A.A. Amini, Quantitative coronary angiography with deformable spline models, IEEE Trans. on Med. Img., vol. 16, pp , October C. Smets,. Verbeeck, P. Suetens, and A. Oosterlinck, A knowledge-basedsystem for the delineation of blood vessels on subtraction angiograms, Pattern Rec. Let., vol. 8, pp , R. Nekovei and Y. Sun, Back-propagation network and its configuration for blood vessel detection in angiograms, IEEE Trans. on Neural Nets, vol. 6, pp , January V. Caselles, F. Catte, T. Coll, and F. Dibos, A geometric model for active contours in image processing, NM, vol. 66, pp. 1 32, R. Malladi, J. A. Sethian, and B. C. Vemuri, Shape modeling with front propagation: A level set approach, PAMI, vol. 17, pp , February S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on hamilton-jacobi formulation, JCP, vol. 79, pp , J. A. Sethian, A fast marching level set method for monotinically advancing fronts, in NAS, vol. 93, pp , F. Quek, C. Kirbas, and F. Charbel, Aim:attentionally-based interaction model for the interpretation of vascular angiograph, IEEE Tran. on Inf. Tech. in Biomed., vol. 3, pp , June 1999.