All transform coefficients are coded using fast quadtree partitioning (FQP) scheme after the each subband was scaled. The coder only uses a simple quadtree partition- ing, which is similar to SPECK algorithm, instead of both quadtree partitioning and the octave band partitioning of SPECK. Fig.5 shows the basic scheme of FQP algo- rithm.
We adopt the integer powers of two as the threshold of coefficient quantization.
We say that a of coefficients is significant with respect to n if
Otherwise it is insignificant. We can write the significance of a as
Efficient Lossy to Lossless Medical Image Compression 91 When the outcome of the test is “1”, we say the set is significant for bitplane n, otherwise it is insignificant.
Fig. 5. Partitioning scheme of FQP algorithm
Starting with S as the root set from the whole map, If S is significant for threshold Set S is spitted into four quadrant sets, collectively denoted O(S). Four subsets are respectively corresponding to all subbands in the same level. We adopt the sig- nificance test for the same n to each of these sets and split four again only if signifi- cant. Significant sets continue to be recursively split until all there are four pixels, where upon, the significant ones are found and appended to a list of significant pixels, called the LSP. The sizes and coordinates of all insignificant sets and pixels can be appended to an array of insignificant sets and pixels (AISP). The progressing is not stop until all current sets of type S have been tested against n. Then, n is decremented by 1, and the FQP is continued for all Set S in AISP until n is equal to 1.
4 Experimental Results
We compare the EMSD-SQP algorithm with SPIHT and SPECK. Different medical images are selected as the test images.
Table 3 shows comparison of the PSNR performances among the EMSD-SQP al- gorithm, the based-IWT SPIHT and SPECK for Barbara, MRI and CT images. We adopt (3,3), which is the best IWT for compression. In table 4, the PSNR values using the EMSD-SQP algorithm based on (2,2), (4,2), (3,3) and (4,4) for MRI image are shown and compared with the results of DWT-based SPIHT and SPECK.
Table 5 shows lossless compression comparison of EMSD-SQP, SPIHT and SPECK on Barbara, MRI and CT image. Fig.6 gives the coding results of EMSD- SQP algorithm for the MRI image based on (3,3). Fig.7 shows the lossy reconstructed CT image using EMSD-SQP algorithm based on (3,3) at 0.25 bpp and 1.0bpp.
5 Conclusions
In this paper, we propose a so-called EMSD-SQP algorithm that has three primary advantages for medical image coding. Firstly, multiple subband decomposition (MSD) optimizes the transform coefficient distribution of each subband and improves energy compaction. Secondly, each subband is scaled according to its significance.
92 L.-b. Zhang and K. Wang
The scaling factors are the integer powers of two. Thirdly, fast quadtree partitioning (FQP) is presented to avoid octave band partitioning of SPECK, reduce the computa- tional complexity and increases the lossy and lossless compression efficiency. We expect this idea is valuable for future research in medical image coding and its appli- cations.
Fig. 6. Coding results of MRI image using EMSD-SQP algorithm based on (3,3) at 0.25 bpp (left), 1.0bpp (right).
Efficient Lossy to Lossless Medical Image Compression 93
Fig. 7. Coding results of CT image using EMSD-SQP algorithm based on (3,3) at 0.25 bpp (left), 1.0bpp (right).
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Geodesic Active Regions Using Non-parametric Statistical Regional Description and Their Application to Aneurysm Segmentation from
CTA
Monica Hernandez and Alejandro F. Frangi
Computer Vision Group, Division of Biomedical Engineering, Aragon Institute of Engineering Research, University of Zaragoza, Zaragoza, Spain
{mhg,afrangi}@unizar.es
Abstract. The inclusion of statistical region-based information in the Geodesic Active Contours introduces robustness in the segmentation of images with weak or inhomogeneous gradient at edges. The estimation of the Probability Density Function (PDF) for each region, involves the definition of the features that characterize the image inside the different regions. PDFs are usually modelled from the intensity values using Gaus- sian Mixture Models. However, we argue that the use of up to second order information could provide better discrimination of the different regions than based on intensity only, as the local intensity manifold is more accurately represented. In this paper, we present a non parametric estimation technique for the PDFs of the underlying tissues present in medical images with application for the segmentation of brain aneurysms in CTA data with the Geodesic Active Regions model.
1 Introduction
Brain aneurysms are pathological dilatations of cerebral arteries developed on weakened vessel walls due to blood pressure. Two dimensional Digital substrac- tion angiography (DSA) is considered the gold standard technique for the de- tection and quantification of brain aneurysms. However, other less invasive ac- quisition techniques like Computed Tomography Angiography (CTA), Magnetic Resonance Angiography (MRA) or 3D Rotational Angiography (3DRA) are also used as complementary methods for these aims [15, 7]. In clinical practise, quan- tification is usually performed from Maximum Intensity Projections (MIP) of the original volumetric scan, which introduces a high degree of subjectivity to the quantification of the aneurysm. The use of computerized 3D segmentation techniques can play a crucial role in improving quantification of the aneurysm dimensions as well as for a correct interpretation of the 3D morphology.
The adoption of deformable models for segmentation in vascular cerebral structures has become very popular over the last years [9,16,1]. In particular, the ability to handle with topological changes in complex structures makes implicit
G.-Z. Yang and T. Jiang (Eds.): MIAR 2004, LNCS 3150, pp. 94–102, 2004.
© Springer-Verlag Berlin Heidelberg 2004
Geodesic Active Regions 95 deformable models a very suitable technique for modelling the shape of vascular structures and brain aneurysms [11].
Traditionally, implicit deformable models based on the Geodesic Active Con- tours approach depend on the gradient of the image as edge integration criteria.
Due to the low quality of the medical data, the front in evolution usually suf- fers from leakage in places with weak or inhomogeneous image gradient, and usually does not provide good results in brain vessels. There have been several efforts to include statistical region-based information in the process of segmen- tation [17,12]. In places with weak gradient, the region-based information drives the evolution of the active contour providing more robust segmentation. Previ- ous attempts for the inclusion of statistical region-based information into the deformable model has shown promising results in the segmentation of brain aneurysms in 3DRA and CTA data [3, 8].
The inclusion of statistical region-based information into the deformable model is done using region descriptors, which are defined in terms of the negative logarithm of a Probability Density Function (PDF) associated to the region [3].
The estimation of the PDF for each region, involves the definition of the features that characterize the image inside the different regions. In fact, the estimated PDF can be considered as a conditional PDF wherex is the point in the image domain andf is the vector of features used to describe the image in the estimation process.
In most previous attempts the estimation of the PDF for region descriptors are based on two main assumptions: image intensity is the most discriminant regional descriptor, and the statistics of image intensity can be described using parametric estimators of the PDF. In particular, PDFs are usually modelled from the intensity values using a Gaussian Mixture Model (GMM) [17, 12, 3]
with parameters estimated via the Maximum Likelihood principle. However, we argue that the use of up to second order information could provide better approximations for the different regions as, besides intensity, local geometrical information is introduced for region characterization and therefore, improve the segmentation. As the use of a Gaussian model with a higher dimension feature set could generalize poorly, the use of non parametric estimation techniques becomes necessary for PDF estimation.
This article proposes a method of introducing high order information for the estimation of PDF of the different tissues present in medical data. The technique is here applied to the segmentation of brain aneurysms in CTA data. The novelty of our method stems in the use of differential image descriptors of up to second order for the definition of non-parametric region descriptors of the main tissue types that are present in the CTA images. The underlying PDFs are estimated using adaptive Parzen windows based on the k-Nearest Neighbor (kNN) rule. The result is an algorithm that improves region PDF estimation in an optimal way, as feature selection is included in the method, providing accurate segmentations.
The paper is organized as follows. Section 2 explains the devised PDF esti- mation method used in the segmentation with Geodesic Active Regions model.
The results of the method and conclusions are reported in Section 3.
96 M. Hernandez and A.F. Frangi
2 Probability Density Function Estimation
For the estimation of the PDF for vessel, background and bone tissues in CTA, a method based on a non parametric estimation technique is proposed. First, the K-means algorithm is used for unsupervised construction of the train set.
Then, the kNN rule is used for PDF estimation in a multidimensional feature space defined by the multiscale derivatives of the image up to second order. The result is an estimation technique that takes into account not only the intensity distribution in the image but also approximates to a higher degree the local image structure.