To compare the properties of our saliency measure with the SN measure in a quantitative manner, we analyze the behavior of them by applying to three simple curves: a straight line, a circle and a catenary. First we compute the saliency of these curves using the SN measure. We consider only curves with no gaps such that and for all Because a straight line has a constant curvature the saliency calculated using SN measure is
For a circle of perimeter the curvature is a constant
Therefore, the saliency of this circle is as derived in [12]. For
a catenary its cesàro equation is Using a
continuous formulation, the curvature term of the SN measure can be derived as
Then the saliency of this curve is computed by The
calculated saliency values of these curves at different lengths are shown in Fig.
3(a) from which we can observe that when the lengths of these curves are short the straight line is found more salient than the other two because its sum of curvature is zero and its saliency grows linearly with the length of the line. This is consistent with the analysis that the SN measure favors straight lines more than other curves. However, when at longer lengths the preference of the SN measure changes and the circle is calculated as the most salient curve among these three ones. This shows that the SN measure lacks scale invariance. The above observations indicate that the SN method treats these curves differently even if they are scaled uniformly.
When using our measure, we set to 0.9. For simplicity, the intensity gradient term and view-dependent term are both supposed to be 1 for all these
An Adaptive Enhancement Method for Ultrasound Images 43
Fig. 4. An experiment on a liver US image. ( a ) The test image. ( b ) Edge map obtained via canny operator. ( c ) The salient map of SN method. ( d ) The result of the proposed method.
experiments. Therefore, our saliency equation Eq. (1) becomes
For the straight line, the curvature gradient as the curvature is 0 for all arcs. Hence the saliency of a straight line of length is
Because a circle has a constant curvature too, the curvature gradient is also equal to 1. Thus, the saliency of the circle with perimeter is similar to that of the straight line, i.e. Then we consider the catenary. Its curvature gradient is calculated as
Therefore the saliency is
From the calculated saliency values shown in Fig. 3(b), we see that, as long as they are at a same length (short or long), the straight line has the same saliency as the circle. This satisfies our purpose that all salient structures with the same length and smoothness should obtain the same saliency. Meanwhile, it is noticed that the catenary is always less salient than the straight line and the circle. The longer their lengths are, the more different are their saliency values. The reason for this is that the catenary has higher curvature gradients than the other two curves and our proposed measure favors long curves with low curvature change rate. This novel saliency measure is computed via a proposed local search algorithm developed in [8].
44 J. Xie, Y. Jiang, and H.-t. Tsui
Fig. 5. An experiment on a brain US image. ( a ) The original US image. ( b) The edge map obtained via the canny edge operator. ( c ) The salient map of the SN method. ( d ) The result of the proposed method.
4 Experimental Results
Before applying the proposed method to real US data, the characteristics of the algorithm was first studied on several synthetic data as shown in [8]. Experimen- tal results on real US data are shown in Fig. 4 and 5. The original test images are shown in Fig. 4(a) and 5(a). The edge images derived by the canny operator are shown in Fig. 4(b) and 5(b). They are undesirable for there are too many erroneous edges due to the speckle noise in the test US images. Fig. 4(c) and 5(c) are the salient maps of the SN method. They are better than canny edge maps for only long, smooth and strong boundaries were enhanced. The results of our method are shown in Fig. 4(d) and 5(d). Compared to results of SN method, the results of the proposed method had cleaner and thinner boundaries. This is because our local curvature gradient measure avoided the influence of noise more effectively than the curve accumulation measure. The local noise cannot be spread to the whole curve. Fig. 6 shows two experiments on the US image segmentation. We can observe that the preprocessing procedure has improved the segmentation results effectively.
5 Conclusions
In this paper, based on a novel saliency measure, we have presented an adaptive enhancement method for US images. One advantage of this method is that it can reduce the speckle noise on US images adaptively by analyzing the local speckle structures. Meanwhile, perceptual salient boundaries of organs are enhanced via computing the proposed measure of perceptual saliency. Because of this curva- ture gradient based saliency measure, our method can extract long, smooth and unclosed boundaries and enhance more types of salient structures equally than the SN method. Experiments show the proposed approach works very well on the given image set. This is useful for image guided surgery and other computer vi- sion applications such as image segmentation and registration. Further research is needed to increase the searching speed of the proposed algorithm for real-time applications.
An Adaptive Enhancement Method for Ultrasound Images 45
Fig. 6. Two experiments on US images using the classical Level Set Method.
( a ) The segmentation on an original US image. ( b ) The segmentation result using the enhanced US image. ( c ) The segmentation on another original US image. ( d ) The segmentation result using the corresponding enhanced US image.
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State Space Strategies for Estimation of Activity Map in PET Imaging
Yi Tian1, Huafeng Liu1, and Pengcheng Shi2
1 State Key Laboratory of Modern Optical Instrumentation Zhejiang University, Hangzhou, China
2 Department of Electrical and Electronic Engineering Hong Kong University of Science and Technology, Hong Kong
{eeliuhf, eeship}@ust.hk
Abstract. In this paper, we explore the usage of state space principles for the estimation of activity map in tomographic PET imaging. The proposed strategy formulates the dynamic changes of the organ activity distribution through state space evolution equations and the photon- counting measurements through observation equations, thus makes it possible to unify the dynamic reconstruction problem and static recon- struction problem into a general framework. Further, it coherently treats the uncertainties of the statistical model of the imaging system and the noisy nature of measurement data. The state-space reconstruction prob- lem is solved by both the popular but suboptimal Kalman filter (KF) and the robust estimator. Since the filter seeks the minimum- maximum-error estimates without any assumptions on the system and data noise statistics, it is particular suited for PET imaging where the measurement data is known to be Poisson distributed. The proposed framework is evaluated using Shepp-Logan simulated phantom data and compared to standard methods with favorable results.
1 Introduction
Accurate and fast image reconstruction is the ultimate goal for many medi- cal imaging modalities such as positron emission tomography (PET). Accord- ingly, there have been abundant efforts devoted to tomographic image recon- struction for the past thirty years. In PET imaging, the traditional approach is based on the deterministic filtered backprojection (FBP) method [3]. Typi- cal FBP algorithms do not, however, produce high quality reconstructed images because of their inability to handle the Poisson statistics of the measurements, i.e. the counts of the detected photons. More recently, iterative statistical meth- ods have been proposed and adopted with various objective functions, including notable examples such as maximum likelihood (ML) [12], expectation maximiza- tion (EM) [9], ordered-subset EM [2], maximum a posteriori (MAP) [10], and penalized weighted lease-squares (PWLS) [4].
For any statistical image reconstruction framework, one must consider two important aspects of the problem: the statistical model of the imaging system
G.-Z. Yang and T. Jiang (Eds.): MIAR 2004, LNCS 3150, pp. 46–53, 2004.
© Springer-Verlag Berlin Heidelberg 2004
State Space Strategies for Estimation of Activity Map in PET Imaging 47
Fig. 1. Digital Shepp-Logan phantom used in the experiments (left) and scale map (right)
and the noisy nature of measurement data. It is clear that an accurate statistical model is a prerequisite for a good reconstruction [1]. However, all aforementioned existing works do not consider the uncertainties of the statistical model, while in practical situations it is almost impossible to have the exact model information.
In addition, these existing methods assume that the properties of the organs being imaged do not change over time. In the dynamic PET, however, the activity distribution is time-varying and the goal is actually to obtain dynamic changes of the tissue activities [8].
In this paper, we present a general PET reconstruction paradigm which is based on the state space principles. Compared to earlier works, our effort has three significant novel aspects. First, our approach undertakes the uncertainties of both the statistical model of the imaging system and the measurement data.
Secondly, our method formulates the dynamic changes of the organ activity dis- tribution as state space variable evolution, thus makes it possible to unify the dynamic reconstruction problem and static reconstruction problem into a general framework. Finally, two solutions are proposed for the state space framework:
the Kalman filtering (KF) solution which adopts the minimum-mean-square- error criteria, and the filter which seeks the minimum-maximum-error es- timates. Since the principle makes no assumptions on the noise statistics, it is particular suited for PET imaging where the measurement data is known to be Poisson distributed. An evaluation study by using Shepp-Logan simulated phantom data of our proposed strategy is described. Experimental results, con- clusions and future work are also presented.
2 Methodology