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those algorithms visually as well as objectively [8, 9, 10]. Among those different registration algorithms, the voxel similarity approaches to image registration have attracted significant attention since these full-volume-based registration algorithms don’t rely upon data reduction or segmentation, and involve little or no user interaction. They can also be fully automated and offer quantitative assessment. Maintz et al. [4] lists the reported paradigms and Studholme et al.

[11], Penney et al. [12], and Holden et al. [13] compare many of them. Among various different similarity measures, mutual information is the most prominent (see [14, 15, 16], and [17]). Many papers and reports have been published on this similarity measure since its first publication and advances in this area have been recently reviewed in Pluim et al. [17].

A cross-entropy optimization approach to image registration was reported recently in Zhu [18]. Cross-entropy minimization as a principle was formally established by Shore and Johnson [19, 20]. They also studied the properties of cross-entropy minimization [21]. In addition to image registration, this measure has been applied to the areas of spectral analysis in Shore [22], image reconstruction in Zhuang et al. [23], biochemistry in Yee [24], process control in Alwan et al.

[25], non-linear programming in Das et al. [26], and electron density estimation in Antolin et al. [27], among many others.

Cross-entropy, also known as Kullback-Leibler divergence, is an informationtheoretic measure that quantifies the difference between two probability density functions (pdf). It can be either maximized or minimized, depending on how a priori pdf is given. Cross-entropy maximization degenerates to mutual information maximization, conditional entropy minimization or joint entropy minimization under certain conditions. Cross-entropy has two close relatives known as reversed cross-entropy and symmetric divergence, which have been applied to spectral analysis (see [28, 29]) and neural networks [30]. It is reported that cross-entropy, reversed cross-entropy, and symmetric divergence spectral analyses have comparable performance. However, it is not clear how reversed cross-entropy and symmetric divergence perform as registration measures. This chapter explores their use as similarity measures for medical image registration and compares their performance.

Since 1999, the imaging vendors have been developing new imaging devices which combine two different imaging modalities into a single apparatus [31]. General Electric Medical Systems, Philips Medical Systems, and Siemens Medical Solutions/CTI all have released PET/CT devices (Discovery LS, Biograph, and

Gemini, respectively, [32]. Given this hardware approach to image registration, the question arises as to the continued need for software registration techniques. It is our opinion however, that software image registration will continue to play a vital role in many instances and that the development of registration algorithms shall remain an important research area for years to come. In many cases, hardware registration is impractical or impossible and one must rely on software-based registration techniques. For example, when monitoring treatment effectiveness over time, software image registration is necessary since the single or multimodality images are acquired at different times. In addition, applications involving intersubject or atlas comparisons require software registration since the images originate from different subjects. Other applications for software registration include the correction of motion that occurs between sequential transmission and emission scans in PET and SPECT as well as the positioning of patients with respect to previously determined treatment plans. The need to offer multiple different combinations of imaging modalities (i.e., PET/MR, SPECT/MR, PET/CT, etc.) would be impractical. As most researchers agree, the hybrid devices will likely play a major role primarily in radiation oncology.

The remainder of this chapter is organized as following: section 10.2 defines all three measures and discusses how they can be used in an image registration context. Section 10.3 addresses the implementation issues. Section 10.4 details the experimental setup to test cross-entropy, reversed cross-entropy, and symmetric divergence image registration by both maximization and minimization. Section 10.5 presents the registration results, along with a discussion. Section 10.6 concludes with a brief summary.

10.2Cross-Entropy, Reversed Cross-Entropy, and Symmetric Divergence

Cross-entropy, reversed cross-entropy, and symmetric divergence can be defined for pdfs of any-dimensional random variables. To make it relevant to the image registration context, in the following equations only a vector variable (u, v) is considered. In these equations, u and v are voxel gray values at corresponding points in two images, f (x, y, z) and g(x, y, z) that are known as the reference image and the floating image, respectively. This gray value pair is considered in

almost all similarity measure-based image registration. Note that, if there are more than two images to be registered (see [17, 33]), these definitions can be easily extended to the n-dimensional case [18].

Assume the joint pdf of random variable (u, v) is p(u, v). Also assume a prior estimation of p(u, v) is available and denoted as q(u, v). The cross-entropy is

thus defined on a compact support D = Du × Dv , as in Shore and Johnson [19]

ηCE( p, q) = (10.1)

D q(u, v)

where Du and Dv are supports of u and v, respectively.

If the roles of q(u, v) and p(u, v) are switched, one has the reversed cross-

To make the definition symmetric with regard to p(u, v) and q(u, v), one can

which is the definition of symmetric divergence.

In the case where cross-entropy optimization is utilized to perform image registration, if a favorable (also known as desirable or likely) priori pdf is given, an estimate of the true pdf can be found by minimizing the cross-entropy. On the other hand, if an unfavorable priori pdf is given, an estimate of the true pdf can be obtained by maximizing the cross-entropy. The same would be true when reversed cross-entropy and symmetric divergence are used as similarity measures for image registration.

A favorable pdf can be computed based on previous registration results [34]. Theoretical analysis can also provide information regarding a favorable priori. For example, the voxel values in images of the same modality and of the same patient are linearly related and it has been shown that MR image can be used to simulate a PET image [35]. Since the true pdf is expected to be close to a likely priori, the cross-entropy, reversed cross-entropy, and symmetric divergence will be minimized. It is worthy to note that the cross-entropy, reversed cross-entropy, and symmetric divergence are not the only frameworks to exploit the priori knowledge in registration. The recently proposed likelihood maximization approach can also systematically use this knowledge (see [34, 36]).

When an undesirable prior q(u, v) is given, the cross-entropy, reversed crossentropy, or symmetric divergence are maximized. When images are in registration, we would expect that the voxel values at the same coordinates in the two images are related. Therefore, unfavorable pdfs can be uniform, proportional to one of the marginal pdfs, or the product of two marginal pdfs. The case where the unfavorable pdf is the product of two marginal pdfs is worth noting. In this case, the voxel values are statistically independent since their joint pdf is the product of their marginal pdfs. Cross-entropy maximization based on these three priori estimates degenerate to entropy minimization, conditional entropy

minimization, and mutual information maximization, respectively, [18].

Let’s assume the marginal pdfs of u and v are h f (u) and hg (v), respectively, and the unfavorable priori pdf q(u, v) is equal to the product of two marginal

pdfs. Putting q = h f (u)hg (v) into Eqs. (10.1) (10.2) and (10.3), one arrives at

One may notice that Eq. (10.4) is the definition of mutual information. The measures defined in Eqs. (10.4–10.6) are maximzied when used in image registration.

10.3 Implementation

Figure 10.1 shows the overall flow chart of the registration process. It applies to all three similarity measures as well as the minimization and maximization cases. When the similarity measures are minimized, a favorable priori is used in the similarity calculation. When they are maximized, the priori estimate is calculated from the marginal pdfs on-the-fly. As can be seen, no matter which measure is used, the registration algorithm has the same structure and they can be implemented in the same manner. Since Eqs. (10.1)–(10.6) are in a continuous form, they must be discretized to be solved numerically.

Floating Image

Initial Transform Transform Update Transform

No

Compute

CE/RCE/SD

CE/RCE/SD

Optimal (min/max)?

Yes

Final Transform

Figure 10.1: Flow chart of image registration by the cross-entropy, reversed cross-entropy, and symmetric divergence optimization. CE: cross-entropy; RCE: reversed cross-entropy; SD: symmetric divergence.

The two images involved in the registration are identified as the reference image and the floating image. The floating image will undergo rotation and translation to match the reference image. Before the automatic registration starts, an initial set of registration parameters must be set and the floating image is appropriately transformed (i.e., rotated and translated). The similarity measures between the reference image and the transformed floating image are then computed. If the similarity number is not optimal, the registration parameters are updated, otherwise, the registration process stops and the optimal registration parameters are output. The scheme to update the registration parameters is determined by the optimization algorithm employed. In the following subsections, the key steps in the registration process are expanded and discussed in detail.