Fast Segmentation of Kidneys in CT Images

Transcription

1 WDS'10 Proceedings of Contributed Papers, Part I, 70 75, ISBN MATFYZPRESS Fast Segmentation of Kidneys in CT Images J. Kolomazník Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Abstract. In this paper we present a fast implementation of segmentation algorithm suited for extraction of human kidneys in CT images. Kidneys have quite simple shape and that allows us to use faster algorithms than those which can adapt to changes in topology (e.g. level-sets). We based our 3D segmentation method on parametric snakes executed in volume slices to obtain contours, which define organs volume. This allows strong high-level parallelization of implemented method, which is scaling well with increasing number of processing units. We also decreased computational load by introducing more complex and descriptive statistical models, which substitute few expensive operations in segmentation process, reduce number of parameters and still give similar results. Introduction Precise segmentation of internal organs has various applications in medical practice visualization, diagnosis or medical treatment monitoring. We are going to present fast segmentation algorithm designed for segmentation of human kidneys from CT volume data. Segmentation of 3D objects from CT can be done in two ways. We can implement algorithm, which is three dimensional in its nature levelsets ((Adalsteinsson and Sethian, 1994)), subdivision surfaces ((Orderud and Rabben, 2008), 3D split/merge algorithms, watershed segmentation ((Straka et al., 2003)), etc. Or we can decide to use two dimensional segmentation algorithm executed in volume slices and coordinate these computations in some way. We chose second option for several reasons. 2D segmentation algorithms have longer tradition ((McInerney and Terzopoulos, 1996)) and are used more often, interaction with user is easier in two dimensions and our data have lower resolution in one dimension (CT images with thick slices). In the first section we are going to describe segmentation algorithm formally and in the second section we will talk about statistical models, which can be used for implementation of derived terms. 2D Segmentation Algorithm B-Spline Parametric Snakes As a base we chose parametric snakes from (Jacob et al., 2004), implemented in (Krajíček, 2007). Their version proved to be quite successful in precise segmentation but it was very slow and was controlled by large number of parameters we try to improve that. We used B-spline curve as contour representation. And we formulate energy term which for each contour shape computes amount of potential energy. This energy term is designed in such way that state with minimal energy defines optimal contour position. Optimization of this system is done by the iterative algorithm described in the end of this section. Energy terms A classical scheme of system energy is this: E snake = E image + E internal + E constraint (1) In following sections we describe these energy terms in detail. E image = E region + E edge (2) 70

2 Region energy term. Most important part of the image energy is region energy term. Its value is computed from region statistics and recent research on deformable models focuses on this type of term. Statistical approach gives us robust tool against noise, which is omnipresent in all real data computation doesn t depend on unstable image features, such as edges or corners. Main advantage of region based term is its large basin of attraction, so balloon force or similar terms, which are needed in other frameworks, can be omitted. If we need fast implementation we must prevent computation of region integrals, which is used in formulations of all region terms in this section. Mathematical analysis gives us tool called Green s Theorem, which allows us to convert region integrals to integrals along region contour. How to apply Green s Theorem can be found in (Jacob et al., 2004). As our first version of region energy term we use formulation from (Krajíček, 2007) and (Jacob et al., 2004): E region = log ( ) P(f(s) s Rin ) ds (3) P(f(s) s R out ) This integral gains its minimal value when represents region with higher probability P(f(s) s R in ) then P(f(s) s R out ). But this region term uses only distributions of intensity inside and outside of the segmented regions and ignores any information about position in space. To solve the previously mentioned problem we modify this term by using different probability function we use four dimensional distribution (intensity and spatial coordinates). ( ) P(f(s) s Rin ) E region2 = log ds (4) P(f(s) s R out ) There is one thing we can do with this complex model. We will need some basic formulas from conditional probability. If events A and B are independent following expressions are valid. P(A B) P(A B) = P(B) P(A B) = P(A B) P(B) (5) If we suppose that P(f(s)) and P(s R in ), respective P(f(s)) and P(s R out ) are independent, we can use 5. P(f(s) s R in ) = P(f(s) s R in ) P(s R in ) (6) From that we obtain: E region2 P(f(s) s R out ) = P(f(s) s R out ) P(s R out ) (7) ( ) ( ) P(f(s) s Rin ) P(s Rin ) = log + log ds (8) P(f(s) s R out ) P(s R out ) We can change this sum to weighted sum by introducing parameter α, which allows us to control influence of each sub-term: ( P(f(s) s Rin ) E region2 = α log P(f(s) s R out ) +(1 α) log ( P(s Rin ) P(s R out ) ) + ) ds (9) Notice special cases α = 1, we get the original 3 term, and α = 0, we get term whose value is controlled only by shape statistical model. Assumption of independence isn t really right. We try to segment homogeneous region, so independence of P(f(s)) and P(s R in ) is more or less kept. But outside of the region is situation different. Tissue around segmented organ is dependent on position, because organs are mostly at the same place and densities are correlated. Used simplification proofed sufficient, when segmenting kidneys of people with higher level of interorgan fat. Mathematically speaking, error caused by independence assumption isn t that big. But if we want to develop more general method we have to use the original formula 4 and not its simplifications 8 and 9. Later we will show how we can use these terms for fast initialization of parametric snakes. 71

3 Edge energy term. Region energy term is good for rough shape estimation. But for exact localization we must use some term, which is using image features. These terms have only small basin of attraction term has effect only when curve is close to edge. One of the most popular versions of edge energy term is based on integral of the square of the gradient magnitude along the curve. M E edgemag = f(v(t)) 2 dt (10) 0 Previously defined edge term was using only gradient magnitude and its direction was ignored. This flaw can be solved by different edge energy term: E grad = = C C k ( f(r) dr) (11) f(r) (dr k) (12) This one inceases energy if angle between contour normal and image gradient is too big. But this edge term is quite sensitive to noise. Internal and constrain energy term. In classic active contours algorithm ((Kass et al., 1988)), internal energy term was defined to maintain smoothness of the contour. But we are using parametric curves for shape representation so we have smooth derivatives by definition. But curves still can develop high curvature segments or collapse two control points into one - we need to prevent that. Derivation of suitable internal energy term can be found in (Jacob et al., 2004). Constrain energy term serves as tool for incorporating additional conditions, such as user input. In (Krajíček, 2007) was this term used for preventing curve divergence by shape restriction, generated from contours in previous slices. We did not use constraint energy term in our implementation. Optimization We used modified gradient steepest descent algorithm for locating point of minimum energy. But any other algorithm, which uses gradient of optimized function can be used. How to compute gradient of energy function can be found in (Kolomaznk, 2009; Krajíček, 2007; Jacob et al., 2004). Most of such numerical algorithms can end up in local extreme. So we need to initialize contour pretty close to the final shape. To prevent this we now effectively use weights of energy sub-terms. We can initialize our algorithm by arbitrary shape and run few iterations with boosted weight of shape energy sub-term (α in 9 is set to 0). We can also use spline with low number of control points to make it faster. In successive iterations we steadily increase influence of other region energy terms. This brings contour near the optimal shape. In the final phase we boost influence of edge energy term, which force spline to follow object contour. In our implementation one point initialization is enough we use it as center of circle with area similar to the final shape and then we execute optimization algorithm. See figure Figure 1. Segmentation steps in slice: 1 Init point computed from user input; 2 Snake initialized by circle with center in init point, radius computed from information stored in statistical model; 3 First phase of segmentation, controlled primarily by statistical shape model; 4 Second phase of segmentation, controlled mainly by image data. 72

4 Statistical Model KOLOMAZNÍK: FAST SEGMENTATION OF KIDNEYS IN CT IMAGES We formally deduced equations which control our segmentation process. Now we must find a way to represent our model in program. It is complicated to represent the model explicitly and precisely so we need to approximate probabilities somehow. First we want probabilities used in term 3. If we compute estimation of distribution parameters (e.g. estimated value, variation), we can use some known distributions to represent probability functions. Common choice could be normal distribution lots of phenomena have distribution similar to normal in practice. In our equations will be situations easy, because logarithm eliminate exponential function from Gaussian definition. For inner probability this works quite well in case of non-contrast CT data, because we have more or less homogeneous region. But for outer region or for contrast images is probability described by normal distribution insufficient. According to (Krajíček, 2007) we decided to represent probability in more precise way by normalized density histogram. This solution is more flexible, because we don t need to guess the type of a distribution, which best fit to our problem. In (Krajíček, 2007) were histograms computed in the beginning of segmentation and during algorithm were recalculated according to actual contour position in a slice. This is important for keeping histograms updated real distribution of density will not be so different. We chose different approach, we built histograms from training set. This mean that they don t reflect distribution in segmented image precisely, but instead show distribution in average case. Precision we lost by not incorporating histogram recalculation we try to compensate by different mechanism shape representation. Other probability functions are defined over 3D or 4D domain. We use 3D grid, where every element contains information needed to represent defined probability functions. Model Training We formally described principles of the statistical model used in our segmentation algorithm, but this model must be trained at first. We need precisely segmented training set to do that. Every sample in training set is formed from the binary mask and the original data. These data should be denoised or preprocessed by other filters, which will be applied on the input data before segmentation. This should assure that model will be trained for same type of data as we will process. Histogram Computing. Training of average normalized histograms is simple. For every sample we get two histograms the first is for region selected by mask and the second is for its supplement. So now we have two sets of histograms for inner regions and outer regions. We need normalized average histogram, so we simply do a summation of all histograms in each set. We obtain two super-histograms and we now normalize them. Normalized histogram sum to one over its domain we need this feature, because we want to use the result as probability density function. If we have small training set, we would consider normalized histogram smoothing to decrease influence of outliers. Train Data Alignment. We can t simply blend samples, because we will lost model precision. If we want combine samples with bigger precision, we need to align them at first. Complex registration is not needed, because we combine samples from different subjects and exact match is impossible anyway. We should use the same apparatus for aligning samples as we use for model alignment at the beginning of the segmentation process. We did few assumptions before we chose a set of transformations, which we use for alignment of kidney samples. The first thing is that kidneys from same body side are oriented similarly over the subjects. It means that we need to rotate samples only a little. For segmentation we obtain from user two points (kidney poles), so we can use the same points for setting transformation parameters. Training data can either contain coordinates of the organ poles or we can easily emulate user input by computing centers of gravity in two outer slices of training sample (kidneys have quite simple shape especially around poles). Another thing we should keep in mind is that we will use the model all the time during segmentation process. So we need to have all operations on it fast and simple. That is why we restricted transformation only to translation and skew. Kidneys are all oriented similarly in slices, so we omitted rotation around an axis perpendicular to slices. To alter the rest of the rotations we used skew transformation (parameters are easily computed from coordinates of the poles). Shape Model Training. When we have all training data aligned, we can proceed to shape model training. We will simply make average image of all aligned bit masks. If we assume that in every bit 73

5 mask meant 1 object and 0 background, value in averaged image give us probability of point being inside of the region. Supplement to 1 is probability of the point being outside of the region. We use these values and precompute logarithm of their ratio, which is used in region energy term. General Model Training. So we know how to train special parts of our statistical organ model, now we have to train general shape and density model. We use same procedure as for shape model, but computations in each element will be different. Every cell of model grid contains histograms for in probability, out probability and logarithm of their ratio. For shape model we averaged all aligned voxels from masks into one value, but for general shape/intensity model we add density values from aligned data voxels into cell histograms and we decide to which one by mask image. At the end, we normalize all cell histograms and compute their ratio logarithm. If we don t have enough training samples, we should consider histogram smoothing. Model Instances. For proper function of the model we need it to reflect real objects (kidneys in our case) very closely. To achieve that, we need to lower variation of the training data without losing representativity. First base is data alignment, which we already described. We can t proceed further without removing the most distinctive samples from training set but selective sample removal will certainly influence representativity. But there is a solution by reduction of the training set we defined new subclass of objects, so we use model trained from this reduced training set only for segmentation of the objects falling in this subclass. If we want to be able to successfully run segmentation on objects from the whole object class, and not only from its subset, we need to cover it by different subclasses and train separate model for each of them. For our case (kidney segmentation), we should have at least two models one for each side. But we will achieve better results by splitting problem to more categories. Not only for contrast and noncontrast but also for each distinguishable type of kidney shape (small, large, etc.). Trained user should easily decide the right category and choose appropriate model for segmentation. Number of categories depends on amount of the possible training data (danger of model undertraining for some small shape category), user friendliness (we don t want to flood user by large list of types) and precision of the models (if we have models without details, we don t need accurate categorization). For design of the object categories can be used some method from cluster analysis. Other Parts of the Model. We showed ways to model training, but during that process we can compute other statistics, which can be used for different purposes, other than application in energy terms. One possibility is computation of organ s point of gravity and weight in each layer. These values can be used in separate slice initialization version of sliced algorithm for preparation of curve s initial shape. In model can be possibly present whole average parametric initialization for snakes, which can be used instead of the initialization by circle. Similar statistics can be added to model for utilization in other segmentation methods, gradient image preprocessing, etc. 3D Sliced Segmentation Now we have finished 2D segmentation algorithm. For segmentation of 3D data we must execute algorithm in volume slices. We let user to specify two organ poles, which we use as region of interest definition and hint for alignment of model with actual dataset. Now we can proceed to execution of segmentation algorithm. In (Krajíček, 2007) algorithm started on those poles and used results obtained in each slice as initialization for the next one. We showed that we do not need precise initialization so we can start computations in each slice separately. Using initialization by neighboring slice, we can execute segmentation in two separate threads, each going from one pole ((Krajíček, 2007)). Separate initialization allows us completely independent computations in each slice. This means we can parallelize segmentation process on high level each slice can have its own thread. 74

6 a b c Figure 2. Separate slice initialization: 1 Poles set by user; 2 Computed initialization points for each slice; 3 Execute slice segmentation algorithm for separate initialization. Conclusion We successfully moved to statistical model most of the information needed by algorithm during its computation. This allowed us to use easy initialization of contour shape, reduce number of parameters controlling computations and most importantly we achieved to speed up original algorithm times (when executed in one thread). This makes segmentation of kidneys from thick slice CT series almost interactive when running in one thread (around 0.5s on common PC). On the other side we have to train precise representations of body organs, so we should have model for each kind of measurement (with or without contrast agent), demography group (kids, seniors, etc.) and pathology type to improve quality of results. References Adalsteinsson, D. and Sethian, J. A., A fast level set method for propagating interfaces, Journal of Computational Physics, 118, , Jacob, M., Blu, T., and Unser, M., Efficient Energies and Algorithms for Parametric Snakes, IEEE Transactions on Image Processing, 13, , URL Kass, M., Witkin, A., and Terzopoulos, D., Snakes: Active contour models, INTERNATIONAL JOUR- NAL OF COMPUTER VISION, 1, , Kolomaznk, J., Organ Segmentation, Master s thesis, Faculty of Mathematics and Physics, Charles University in Prague, Krajíček, V., Měření objemu v 3D datech, Master s thesis, Faculty of Mathematics and Physics, Charles University in Prague, McInerney, T. and Terzopoulos, D., Deformable models in medical image analysis: A survey, Medical Image Analysis, 1, , Orderud, F. and Rabben, S. I., Real-time 3d segmentation of the left ventricle using deformable subdivision surfaces, Computer Vision and Pattern Recognition, IEEE Computer Society Conference on, 0, 1 8, Straka, M., Cruz, A. L., Köchl, A., Srámek, M., Gröller, M. E., and Fleischmann, D., 3d watershed transform combined with a probabilistic atlas for medical image segmentation, in MIT 2003, pp. URL