Vector Quantization for Feature-Preserving Volume Filtering

Transcription

1 Vector Quantization for Feature-Preserving Volume Filtering Balázs Csébfalvi, József Koloszár, Zsolt Tarján Budapest University of Technology and Economics Department of Control Engineering and Information Technology Magyar tudósok körútja 2., Budapest, Hungary, H Abstract In volume-rendering applications the input data is usually acquired by measuring some kind of physical property. The accuracy of the measurement strongly influences the quality of images produced by recent 3D visualization techniques. In practice, however, a high signal-to-noise ratio cannot always be ensured. For example, in virtual colonoscopy, CT scans are generated preferably at low radiation dose resulting in noisy volume data. In this paper a novel denoising method is presented, which is based on vector quantization. It is assumed that a highly accurate reference volume is available, like a CT scan acquired at high radiation dose. This reference data is used to calculate a generic codebook for filtering other noisy data sets. The major advantage of this approach is that noise can be efficiently reduced without removing important details. 1 Introduction Volumetric data usually contains measured values of a physical property, like the X-ray attenuation factor in case of a CT scan. Since the measurement is typically inaccurate, a denoising step has to be included into the volume-rendering pipeline. Traditional filtering techniques, which are based on a simple convolution or ordered statistics (like median filtering), calculate a filtered data value from a voxel neighborhood assuming that neighboring voxels represent likely the same material. Thus these methods exploit the spatial coherence of the data set, but do not utilize any other a priori knowledge. In this paper a special case of denoising is concerned, where it is assumed that such an a priori knowledge is available, and it is represented by a highly accurate reference data set. In a medical CT data, the signal-to-noise ratio can be improved by increasing the radiation dose used in the acquisition process. However, to set an appropriate radiation dose a compromise has to be made. On one hand the patient has to be saved from unnecessarily high radiation dose, and on the other hand the acquired volume data needs to be accurate enough in order to generate high quality images for the diagnosis. Using the same CT scanner the same tissue is represented by nearly the same density values in data sets of different patients. This is not valid for MRI data, where the data values belonging to the same tissue can vary even inside a volume. This phenomena is referred to as bias or global signal fluctuation in the literature. In this paper we mainly concentrate on filtering CT data for virtual colonoscopy, therefore we do not deal with the problem of global signal fluctuation. Assume that an accurate CT data has already been generated at high radiation dose. Small blocks of such a volume contain either characteristic transitions between different tissues or homogeneous regions, which are similar in different data sets. Therefore this previously acquired data can be reused as a reference volume for denoising data sets generated at low radiation dose. (a) 5 mas. (b) 40 mas. Figure 1: Slices of CT scans acquired at low (a) and high radiation dose (b). VMV 2004 Stanford, USA, November 16 18, 2004

2 Taking a look at a noisy 2D slice, the different tissues can still be recognized by a radiologist due to his/her a priori anatomical knowledge (see Figure 1). In 3D visualization, however, the noise level of the data is more critical. The quality of images is strongly influenced by the accuracy of the estimated gradients, which are rather sensitive to the signalto-noise ratio of the density values. Figure 2 shows images generated by direct volume rendering using CT data sets acquired at low (a) and high radiation dose (b). Although smooth isosurfaces can be reconstructed by low-pass filtering the input data, fine details that are important for the diagnosis might also be removed. (a) 5 mas. (b) 40 mas. Figure 2: Direct volume rendering of CT data sets acquired at low (a) and high radiation dose (b). Instead of calculating a filtered value directly from a voxel neighborhood, we treat denoising as a pattern matching problem. Therefore we apply vector quantization for filtering purpose. The codebook of the vector quantizer is generated from the reference volume, where the codewords correspond to small blocks of voxels. The codewords of an appropriate codebook are supposed to accurately represent the typical tissue transitions. In the noisy volumes, a voxel neighborhood of each voxel is compared to the patterns stored in the generic codebook, and the filtered density value of the given voxel is determined from the nearest codeword. Previously vector quantization was used for efficient volume compression. It has been shown that a usual coherent volume data can be well represented by indices to a codebook storing a relatively small number of typical voxel patterns. Our work has been motivated by the idea that the coherence could be exploited not just for creating compact representations but also for recognizing the characteristic tissue transitions in noisy volumetric data. In Section 2 previous work related to virtual colonoscopy and volume filtering is reviewed. The basic conception of vector quantization is revisited in Section 3, and its adaptation to volume denoising is discussed in Section 4. The results are presented in Section 5 and finally the contribution of this paper is summarized. 2 Previous Work Virtual colonoscopy is a method to detect pathological changes, like polyps in a human colon based on the CT scan of the abdominal region [2, 1, 10]. Its major goal is to avoid the insertion of a real endoscope into the colon, which is rather inconvenient for the patient. There are two fundamentally different approaches for virtual colonoscopy, which are surface shaded display (SSD) [8, 2] and direct volume rendering (DVR) [5, 10]. The boundary surface of a colon can be extracted by the well-known marching cubes algorithm [8]. The obtained triangular mesh can be interactively rendered by the conventional graphics cards using an appropriate occlusion culling technique [1]. The marching cubes algorithm, however, produces a piecewise linear approximation of the colonic surface. Therefore, the edges of the triangles become clearly visible when the user zooms into the fine details. Nevertheless, the boundary surface can also be directly rendered without generating an intermediate geometric representation. Applying this approach, the exact intersection points of the isosurface and the viewing rays are analytically evaluated [12]. Since it is rather expensive computationally, interactive frame rates can be achieved only on a multiprocessor architecture. Such a first-hit ray casting usually does not provide enough information for the diagnosis. For example, a polyp-like feature can be safely detected as a polyp, if its internal structure is visualized as well. Therefore physicians prefer direct volume rendering rather than pure surface rendering, which is capable to semitransparently visualize the colonic surface using an appropriate transfer function [10]. For virtual colonoscopy, CT data is preferably acquired at low dose in order to reduce the harmful influence of the ionizing radiation. The quality of the measured data depends on the product of X-ray tube current (ma) and exposure time (s), and it is expressed in mas. Unfortunately, low-dose CT acquisition (less then 20 mas) results in noisy data. If

3 the signal-to-noise ratio of the input data is too low, recent visualization techniques produce low-quality images, that can hardly be used for polyp detection. The boundary surfaces can be made smoother by filtering the input data by a low-pass Gaussian filter. Such a filtering, however, results in a loss of information due to the removal of high-frequency details. In order to preserve such fine details, a nonlinear Gaussian filtering has been successfully applied for CT data acquired at radiation dose at 20 mas [14]. Another approach is to perform filtering and rendering simultaneously. Using ray casting, several samples along a viewing ray contribute to the corresponding pixel color, where the weighting of the sample contributions is defined by a transfer function. Applying an appropriate multidimensional transfer function, taking higher order derivative information into account, ray casting can be interpreted as a kind of filtering, which emphasizes the material boundaries [3, 4]. Recently a new method for filtering noisy CT and MRI data was published, which is based on firstand second-order derivative techniques [13]. The first order solution adapts the Canny edge detector to 3D in order to detect the boundary surface of a colon. Due to the robust normal estimation, this technique removes the noise but introduces a surface displacement. To avoid this problem, a second-order solution was proposed. It is exploited that second-order edge detectors LoG (Laplacian of Gaussians) and SDGD (second derivative component in the gradient direction) have opposite surface displacements. Therefore, by combining these operators, the displacement effect can be reduced. This approach, however, requires a specialized rendering tool adapted to the shaded display of the detected boundary surfaces. Therefore it does not support the semitransparent visualization of a polyplike feature. In contrast, our goal is to develop a general feature-preserving filtering technique, which is not restricted to the segmentation of a boundary surface, and does not require a specialized rendering method. 3 Vector Quantization A general vector quantizer maps a d-dimensional vector x onto an index, that references a codebook. The corresponding codebook entry contains a vector y of the same dimensionality, which is used to reproduce the original vector x. The accuracy of the reproduction is characterized by some metric δ. The most common distortion metric is the squared distance metric δ(x, y) = x y 2. An interesting aspect of vector quantization is how to find an optimal codebook for a particular input distribution, which minimizes the distortion of the reproduction. Vector quantization can be interpreted as a partitioning of d-dimensional Euclidean space into m Voronoi cells. An arbitrary d- dimensional vector falls into a particular cell if the reference point of the cell is closer to the given input vector than the reference point of any other cell. The m reference points are stored as codewords in the codebook of the vector quantizer. To find the optimal codebook for a certain set of input vectors is an NP-hard problem. However, using the classical LBG-algorithm, which is an extension of a method proposed for scalar quantization [9, 7], a reasonably good partitioning can be obtained [6]. The LBG-algorithm is an iterative technique, which successively minimizes the distortion of the codebook. First of all, an initial codebook is determined by randomly selecting reference vectors from the input set. In the second step, each input vector is classified into a Voronoi cell by finding the nearest reference vector. In the third step, the reference vectors are replaced by the centroids of the cells and the codebook is updated. The second and third steps are then iteratively repeated until the distortion falls below a predefined threshold. Previously vector quantization was applied for efficient volume compression [11, 15]. This approach benefits from the coherence of the volumetric data. For example, in a medical data set, the blocks of voxels can be well partitioned into clusters, since such blocks can typically represent only a couple of materials (air, water, fat, soft tissue, and bone) and the possible transitions between these materials. Therefore voxel blocks of size 2 3 can be well represented by a relatively small number of codewords, which have 8 vector components. It has been shown that using a codebook containing 256 entries, the original voxel densities can be more accurately reconstructed than using a downsampled volume as a compact representation [11]. Thus a block of voxels can be efficiently stored in one single byte, which is an index referencing the corre-

4 Figure 3: Noise filtering using vector quantization. sponding codeword. In this paper, vector quantization is not applied for volume compression but for noise filtering. In volume compression the voxel-to-voxel coherence is exploited. CT data sets, however, are coherent in an other sense as well. In different data sets, the same tissues are represented by nearly the same density values. Such a volume-to-volume coherence can be utilized for filtering noisy data, where a highly accurate reference data set is used as an a priori information about the distribution of typical voxel patterns. 4 Denoising by Vector Quantization Assume that the reference data is represented by a high-quality CT scan previously acquired at high radiation dose. A vector quantizer can be trained by using this accurate data set. Applying vector quantization for volume compression, the codewords represent 3D blocks of voxels. In contrast, our approach is slice-based, thus a codeword represents 2D blocks of pixels on the slices. Its main reason is that the slice thickness (the sampling rate along the axis perpendicular to the slices) can vary from volume to volume. The distribution of typical 3D block patterns would be strongly influenced by the slice thickness of the reference data. However, we want to create a generic codebook, which can be used then for filtering other noisy data sets. These noisy volumes might have a different slice thickness than that of the reference volume, and as a consequence the significantly different distribution of the 3D block patterns would result in a higher distortion. For training the vector quantizer, the LBG algorithm is applied. In the codebook, each codeword represents a block of pixels, which contains a typical pattern of a slice in a certain type of volume data. Depending on the signal-to-noise ratio of the input data, the size of the pixel blocks can be, for instance, 3 3 or 5 5. In the initial step, m reference vectors are obtained by randomly selecting m different pixel blocks from the slices of the reference data. Therefore, if all the density values in a randomly selected block are identical to the density values of a previously selected block then the given random sample is rejected. In the iterative steps of the LBG algorithm, the global distortion of the current codebook is measured by integrating the local distortions of all the overlapping pixel blocks in the reference data set. The main drawback of the classical LBG algorithm is that it is computationally expensive and requires many iterations to achieve a low distortion level. Furthermore the so called empty cell problem might arise. Empty cells are the result of collapsing codebook entries during refinement steps. Since these empty cells are not detected by the algorithm, many codebook entries can be wasted. Recently an optimized vector quantizer algorithm based on covariance analysis was published [15], which efficiently handles the empty cell problem. In our application, however, the efficiency of the codebook generation is less critical than in case of volume compression. Note that a generic codebook has to be created just once from a reference data, and then it can be used for filtering different noisy data sets. Here it is exploited that the characteristic pixel patterns and their distributions in different volumes containing the CT scan of the abdominal region, are pretty similar. Without loss of generality, assume that the size of the pixel blocks is 3 3. Therefore each codeword contains 9 components, which represent 3 3 density values in a typical noiseless pixel block. After having a generic codebook created, a noisy data set is filtered in the following way. A pixel neighborhood of each voxel on the corresponding slice is compared to the codewords, and the nearest codeword is determined. In this nearest codeword assignment, not necessarily the squared distance metric is applied. For example, the contributions of the neighboring pixels to the distortion can be weighted by the reciprocal squared distance to the current voxel. After having the nearest codeword found, its central component is considered to be a filtered density value of the current voxel (see Figure 3). This filtering process can be interpreted as a kind of

5 (a) (b) (c) (d) (e) (f) Figure 4: (a, d): CT slice acquired at low radiation dose. (b, e): Reconstruction by Gaussian filtering. (c, f): Reconstruction by vector quantization. pattern matching. A pixel block is taken from the noisy data as a pattern, and the most similar noiseless pattern is searched for in the codebook. In case of very low signal-to-noise ratio the pixel neighborhood taken into account can be extended to size 5 5. However, due to the higher variation of the 25-dimensional input vectors, more codewords are necessary to reduce the distortion below a tolerable level, and as a consequence nearest neighbor search gets slower. The efficiency of filtering is further reduced by the increased number of vector components, which slows down the distance calculation. In order to remedy such problems, several methods have been published, which restrict the nearest neighbor search to a subset of the codebook. Nevertheless, for vector spaces of higher dimensionality, most of these methods are just slightly faster than the brute-force exhaustive search. Therefore we use a greedy algorithm optimized only for fast distance calculation [15]. This so called partial search technique terminates the calculation of the distance to a certain codebook entry if the partially evaluated distance is already higher than the current minimal distance. 5 Implementation We implemented our filtering technique in C++ and tested it on real-world medical data sets on a 2GHz AMD Athlon XP 2600 PC with 1GB of RAM. In order to have a reference for a quantitative distortion analysis, simulated noisy data sets were generated from original CT scans acquired at high radiation dose. We added a Gaussian noise to the projections of the slices instead of adding noise directly to the voxel densities, and the simulated noisy slices were produced by using the classical filtered back projection algorithm. In such a way a CT acquisition process at low radiation dose can be correctly simulated. First we tried to reconstruct the original data set with a simple Gaussian filtering, and then we used vector quantization for the reconstruction. The RMS errors are shown for CT scans of two

6 (a) (b) (c) (d) (e) (f) Figure 5: Virtual colonoscopy using data sets of different noise levels. (a, d): CT data acquired at high radiation dose (40 mas). (b, e): CT data acquired at low radiation dose (5 mas). (c, f): Data set reconstructed from the noisy CT scan by using vector quantization. data set number of slices slice thickness RMS of simulated noisy data RMS of Gaussian filtering RMS of vector quantization different patients in Table 1. These data sets have the same slice resolution ( ) but they differ in the number of slices and in slice thickness as well. The codebook of the vector quantizer was optimized for data set CT#1. Since CT#2 contains also the abdominal region of a patient, the typical pixel patterns and their distribution are similar to that of CT#1. Therefore the original slices of CT#2 can be reconstructed by vector quantization at nearly the same error level as in case of CT#1. Note that, in both cases vector quantization results in lower RMS errors than that of a Gaussian reconstruction. CT# CT# Table 1: RMS errors of data sets reconstructed by Gaussian filtering and vector quantization. e, f), it is clearly visible, that a low-pass Gaussian filtering results in blurring. In contrast, vector quantization removes the noise from the homogeneous regions as well, but preserves sharp edges. Figure 5 shows the direct volume rendering of data sets CT#1 (a) and CT#2 (d). Both data sets were acquired at high radiation dose (40 mas). We also visualized the low-dose (5 mas) CT data sets of the same patients (b, e). Due to the lower signalto-noise ratio of density values, the estimated gradients are perturbed, resulting in jagged boundary We also tested our denoising method on real noisy data sets acquired at low radiation dose (5 mas). Since in this case, there is no reference data available, the distortions cannot be quantitatively analyzed. Therefore we visually compared the results of different reconstruction techniques. Figure 4 shows an original slice of a noisy data set (a, d), and its denoising by Gaussian filtering (b, e) and vector quantization (c, f). In the zoomed images (d,

7 surfaces (b, e). Furthermore, even the topology is changed by the noise, which makes also the automatic center line calculation more difficult. In contrast, using vector quantization for noise filtering (c, f), the original topology can be preserved and the smooth boundary surface of the colon can be more accurately approximated. The computational cost of our filtering method depends on the number of codewords in the codebook, and the size of the pixel blocks, which determines the number of components in a codeword. In our tests the size of the blocks was 3 3 and the codebook contained 256 entries. The noisy data sets of resolution and were filtered in 71 and 45 minutes respectively. Compared to Persoon s method [13], denoising based on vector quantization provides similar performance. Nevertheless, our method is not restricted to the detection of a boundary surface in a noisy data set. Therefore it does not rely on a specialized rendering tool. The filtered data sets can be visualized by a general volume-rendering implementation. Even direct volume rendering is supported by our method, since vector quantization reduces the distortion of all the voxels in the data. 6 Conclusion and Future Work In this paper a novel feature-preserving volumefiltering method has been presented, which is based on vector quantization. Although the efficiency of the technique was illustrated on filtering low-dose CT data for virtual colonoscopy, it is considered to be a general denoising method. However, it is assumed that a highly accurate reference volume is available as an a priori information. Furthermore, it is also assumed that the reference data set is coherent with the data to be filtered, thus they contain similar structures. For example, the density values and the distribution of typical voxel blocks are nearly the same in CT scans of different patients which contain the same region of the human body. As a consequence, a generic codebook can be used for filtering different data sets of the same type. The main advantage of our approach is that noise can be reduced without removing fine details and sharp edges between different tissues can be well preserved. In our future research, we would like to investigate other possible application fields of our filtering technique. For instance, in industrial neutron tomography, slices have to be reconstructed from only a couple of projections, since the measurement of one projection is time and cost demanding. The obtained noisy slices could be filtered by vector quantization. In this case the generic codebook is calculated from a reference data set reconstructed from several projections. Although scanning such a reference volume is expensive, it has to be done only once, and afterwards other noisy data sets can be efficiently filtered using the generic codebook. Acknowledgements This work has been supported by OTKA and IKTA 00159/2002. The CT data sets were provided by the Semmelweis University in Budapest ( References [1] Hietala, R., and Oikarinen, J A visibility determination algorithm for interactive virtual endoscopy. In Proceedings of IEEE Visualization 2000, [2] Hong, L., Muraki, S., Kaufman, A., Bartz, D., and He, T Virtual voyage: Interactive navigation in the human colon. In Computer Graphics (Proceedings of SIGGRAPH 97), [3] Kindlmann, G., and Durkin, J. W Semi-automatic generation of transfer functions for direct volume rendering. In Proceedings of IEEE Symposium on Volume Visualization, [4] Kniss, J., Kindlmann, G., and Hansen, C Multi-dimensional transfer functions for interactive volume rendering. IEEE Transactions on Visualization and Computer Graphics, 8, [5] Levoy, M Display of surfaces from volume data. IEEE Computer Graphics and Applications, Vol.8, No.3, [6] Linde, Y., Buzo, A., and Gray, R An algorithm for vector quantizer design. IEEE Transactions on Communications COM-28, January, [7] Lloyd, S Least squares quantization in pcm. IEEE Transactions on Information Theory, 28,

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