Resources and Help A pyramid approach to subpixel registration based on intensity Abstract: We present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set, which can be either images two-dimensional or volumes three-dimensional. It uses an explicit spline representation of the images in conjunction with spline processing, and is based on a coarse-to-fine iterative strategy pyramid approach. The geometric deformation model is a global three-dimensional 3-D affine transformation that can be optionally restricted to rigid-body motion rotation and translation , combined with isometric scaling. It also includes an optional adjustment of image contrast differences. We obtain excellent results for the registration of intramodality positron emission tomography PET and functional magnetic resonance imaging fMRI data.

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Ruttimann, and Michael Unser, Senior Member, IEEE e Abstract— We present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set, which can be either images two-dimensional or volumes three-dimensional. It uses an explicit spline representation of the images in conjunction with spline processing, and is based on a coarse-to-? The geometric deformation model is a global three-dimensional 3-D af?

It also includes an optional adjustment of image contrast differences. We obtain excellent results for the registration of intramodality positron emission tomography PET and functional magnetic resonance imaging fMRI data. We conclude that the multiresolution re? In addition, our improved version of the Marquardt—Levenberg algorithm is faster.

Index Terms— Af? Of interest may be the detection of change, and the consolidation of data or image fusion , where different images of the same object need to be brought into correspondence. Remote sensing and biomedical imaging are typical application areas.

Due to the recurrent problem of registration, many solutions have been proposed [1]—[3]. Our speci? In particular, this requires the three-dimensional 3-D intramodal registration of positron emission tomography PET or functional magnetic resonance imaging fMRI images of the Manuscript received February 7, ; revised March 4, The associate editor coordinating the review of this manuscript and approving it for publication was Prof.

Janusz Konrad. Publisher Item Identi? T brain. In both applications, the changes to be detected are very small less than a tenth of the dynamic range for PET, and even less for fMRI , demanding accurate registration. This requirement rules out many of the methods involving? Furthermore, the dif? Therefore, we prefer to explore methods that use the unaltered intensity of all image pixels, thus exploiting effectively all available information. Another aspect is the interpolation needed in applying the transformation.

While the majority of published methods limit themselves to linear interpolation, we prefer a higher interpolation order to minimize image blurring and to achieve consistency in computing the spatial derivatives required for the registration process.

Our deformation model considers the combination of a full 3-D af? We derive simpler models by restricting the af? The Marquardt—Levenberg nonlinear optimization algorithm [4] is well suited for performing registration based on a least-squares criterion [5], [6]. In this paper, we introduce two extensions of this algorithm. First, we speed up its execution by taking advantage of the particular structure of our deformation model.

As a second extension, we cast Marquardt—Levenberg into a multiresolution framework, using a coarse-to-? Most iterations are carried out at the coarsest level, where the amount of data is so greatly reduced that the computational cost of one iteration is negligible.

Once convergence has been reached at any particular level, a switch to a? Government work not protected by U. In Section IV, we?

We also propose a heuristic for deciding when convergence has been reached. In Section V, we discuss the choice of the cubic spline interpolation model. In Section VI, we discuss the use of a multiresolution pyramid and its bene? Here, we adopt the classi? Following this scheme, we give below a short discussion of the most important recent registration methods. Image Features The image features used in a given algorithm have important practical signi?

For example, spatial coordinates landmarks are well adapted to intermodal registration, where the purpose is to register two volumes measuring different properties of an object. However, the selection of landmarks is recognized to be a dif? For many images, this is a serious drawback because registration accuracy can be no better than what is achieved by the initial selection of landmarks. For practical reasons, the number and precision of landmark locations is usually limited.

Hence, spatial coordinates and geometric primitives often oversimplify the data by being too sparse and imprecise. By contrast, registration methods based on initial intensity values can make effective use of all data available [13]; if necessary, some binary masking or other weighting process may be introduced to emphasize special features.

Robustness is controlled through the use of appropriate similarity measures [14]—[16]. With the use of theoretical models, intensity-based methods can produce continuous deformation? Search Space Geometric transformations can be divided into three categories: global, local, and displacement? The second category, sometimes called elastic mapping, allows the transformation parameters to exhibit Fig.

Slice of a typical PET image. For many types of optimizers, this strategy for convergence is signi? In addition, a multiresolution strategy improves robustness, in the sense that it decreases the likelihood of being trapped at a false local optimum. A distinctive feature of our approach is that we consider spline interpolation models that are superior to those typically used for image registration e.

By increasing the order of the spline, we can get arbitrarily close to the sinc interpolation model [7]. In practice, we use cubic splines because they are already remarkably close to this ideal, at a cost that is less than truncating and apodizing a sinc kernel [5].

Our spline model is well suited for computing image pyramids and for performing geometric transformations at various resolutions. It also allows for an easy computation of exact derivatives. By using the same model at each step, we ensure that the overall algorithm is internally consistent. The algorithm that we propose is entirely automatic.

Since it is pixel based, no landmarks are required. We have used it successfully without modi? In Section III, we describe our registration procedure and include a rationale for the choice of our data-space, for the choice of its corresponding objective criterion, and for the? These parameters are often de? Finally, true displacement? Search Strategy Given a set of features and a parametric deformation, both the criterion to optimize and the optimization algorithm itself de?

The use of a least-squares criterion jointly with geometric primitives is popular [24], although it is sometimes replaced by more robust statistics, as those generated while using a distance map chamfer transform [25].

The least-squares criterion is also widely used with intensity values as image features [5], although researchers sometimes prefer more robust statistics, giving up maximum likelihood parameter estimation in the presence of Gaussian white noise in favor of assuring insensitivity to outliers [15]. The optimization algorithm re? If the data are regularly spaced, such as in the case of pixel intensity values, both Fourier [26], [27] and wavelet approaches [17] are applicable.

Several researchers have explored the possibilities of stochastic, or? Exhaustive search has also been investigated. With strong conditioning of the data, near real-time registration [29] can be achieved. Problem Addressed An alternative way to classify registration is to look at the type of problem it addresses. Again, three categories emerge: data fusion, motion estimation, and the detection of signi? In the? In these instances, registration allows the extraction of common features, for example by averaging, or by more re?

Data fusion also arises when one needs to? In the context of motion estimation second category , the problem is to estimate the displacement of a rigid object imaged on some background, with the added challenge of potential changes of pose [35]. Since typical applications are video coding, target tracking and autonomous vehicles, computational ef?

The third and last category encompasses the detection of signi? A new problem then appears, due to the fact that the registration process tries to align data that may be intrinsically dissimilar. This last consideration has sometimes led to robust registration methods using an internal criterion insensitive to outliers [14]—[16]. After registration, the task usually proceeds to detect dissimilar regions, given statistical decision criteria with respect to type I and type II errors [36].

Data Space In this paper, we consider pixel intensity values as our image features. This choice is appealing because it bypasses the segmentation of data into geometric primitives, a notoriously dif? As mentioned in Section II-A, the use of pixel intensity values facilitates the inclusion of the entire informational content of the data.

Moreover, the role of the reference and of the test volume can be exchanged at will. This symmetry property is not necessarily taken for granted with some algorithms based on geometric primitives, for example, in [37]. Although raw intensity values are well suited to tasks like the detection of change or intramodality registration, their associated drawback is their lack of universality: They are not well suited to the problem of intermodality registration, a task in which one usually must resort to an intermediate feature representation, for example their gradient [38] or their histogram [33], [34], [39].

Criterion Any automatic registration method requires the choice of an objective criterion that measures the similarity of the test data to the reference. As the optimization criterion, we select "2, the integrated square difference of the intensity values, sometimes named the residue.

Let fR be the reference data and fT the test data. Such a criterion lends itself well to minimization with respect to , and is well understood. In particular, this Euclidean dissimilarity measure is known to be maximum likelihood if the noise is additive, white, and Gaussian. Its drawback is a lack of robustness in the presence of severe outliers e. In the worst case, outliers predominate and the parameter for which "2 reaches its minimum can be quite different from 0 , the minimum in the noiseless case.

In medical images, outliers are always present, for example in the case of two PET images of the same brain at different functional states certain brain areas are expected to display different activity levels. However, we do not expect these outliers to be dominant. We include no gray-scale shift because it is virtually always true in medical images that some level typically the background with value zero has a physical interpretation, and should not be changed.

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## A Pyramid Approach to Subpixel Registration Based on Intensity

Ruttimann, and Michael Unser, Senior Member, IEEE e Abstract— We present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set, which can be either images two-dimensional or volumes three-dimensional. It uses an explicit spline representation of the images in conjunction with spline processing, and is based on a coarse-to-? The geometric deformation model is a global three-dimensional 3-D af? It also includes an optional adjustment of image contrast differences. We obtain excellent results for the registration of intramodality positron emission tomography PET and functional magnetic resonance imaging fMRI data.

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## A pyramid approach to subpixel registration based on intensity.

Abstract Abstract — We present an automatic subpixel registration algorithm that minimizes the mean square intensity difference between a reference and a test data set, which can be either images two-dimensional or volumes three-dimensional. It uses an explicit spline representation of the images in conjunction with spline processing, and is based on a coarse-to-fine iterative strategy pyramid approach. The geometric deformation model is a global three-dimensional 3-D affine transformation that can be optionally restricted to rigid-body motion rotation and translation , combined with isometric scaling. It also includes an optional adjustment of image contrast differences. We obtain excellent results for the registration of intramodality positron emission tomography PET and functional magnetic resonance imaging fMRI data. We conclude that the multiresolution refinement strategy is more robust than a comparable single-stage method, being less likely to be trapped into a false local optimum. In addition, our improved version of the Marquardt—Levenberg algorithm is faster.