At the core of most image registration problems is determining a spatial transformation that relates the physical coordinates of two or more images. Registration methods have become ubiquitous in many quantitative imaging applications. They represent an essential step for many biomedical and bioengineering applications. For example, image registration is a necessary step for removing motion and distortion related artifacts in serial images, for studying the variation of biological tissue properties, such as shape and composition, across different populations, and many other applications. Here fully automatic intensity based methods for image registration are reviewed within a global energy minimization framework. A linear, shift-invariant, stochastic model for the image formation process is used to describe several important aspects of typical implementations of image registration methods. In particular, we show that due to the stochastic nature of the image formation process, most methods for automatic image registration produce answers biased towards ‘blurred’ images. In addition, we show how image approximation and interpolation procedures necessary to compute the registered images can have undesirable effects on subsequent quantitative image analysis methods. We describe the exact sources of such artifacts and propose methods through which these can be mitigated. The newly proposed methodology is tested using both simulated and real image data. Case studies using three-dimensional diffusion weighted magnetic resonance images, diffusion tensor images, and two-dimensional optical images are presented. Though the specific examples shown relate exclusively to the fields of biomedical imaging and biomedical engineering, the methods described are general and should be applicable to a wide variety of imaging problems.

Introduction and motivation

Recent advances in digital imaging technology have had profound impact on a variety of technical fields including communications, medicine, surveillance, military, entertainment, as well as many experimental sciences. The availability of charged coupled devices (CCDs), for example, has encouraged widespread use of digital cameras for a variety of purposes ranging from personal entertainment to automated surveillance systems. Microwave-based imaging technologies such as synthetic aperture radar (SAR) are widely used by militaries around the world for the purposes of intelligence gathering. In the biomedical fields, imaging modalities such as magnetic resonance imaging (MRI), computer assisted x-ray tomography (CT), and ultrasound (US), to name a few, are becoming increasingly used for diagnosing, treating, and monitoring pathologies. In addition, many scientific fields such as experimental biology, chemistry, and materials sciences are also becoming increasingly dependent on imaging technologies such as digital atomic force microscopes and high-field magnetic resonance resonance spectroscopy and imaging techniques, to name a few, for acquiring data to be used in validating and even generating scientific hypothesis.

Problem statement

As explained above, post-acquisition image alignment (registration) is routinely performed in biomedical research and clinical practice. Applications using image registration techniques include motion and distortion correction in fMRI, DT-MRI, and MR relaxometry experiments. In addition, image registration procedures are increasingly being used in computational based studies of neuroanatomy. This involves understanding the variability of tissue properties, including shape, across specific populations. An example is voxel-based morphometry, described in.

In general, many of the current post-processing methodologies can be summarized within a pipeline framework, as depicted in Figure (1.3). At first, a set of medical images is acquired and reconstructed using standard tomographic technologies. The tomographic reconstruction step in MRI typically involves a Fourier transformation of the data (though filtered back-projection methods are sometimes used) while reconstruction procedures in CT often involve Radon transformation methods. Regardless of the tomographic reconstruction method in use, in most quantitative imaging experiments the output of this step is a series of digital images to be stored in computer memory. Each image in this series can be though of as a function (real or complex) of discrete input coordinates. Mathematically, the nth image in this series is written as Sn(i), with i Zd, where d represents the dimensionality of the images. For example, a common digital image processing technique is to view the indexes of the image array i as i = [i,j,k]T where each coordinate i,j,k belongs to the set {0,··· ,255}, for example. Naturally, the coordinates i are associated with the spatial coordinate system of the laboratory. The specifics of this association are determined by the image acquisition and reconstruction procedure, but simplifications can be made such that a coordinate in the laboratory frame of view x Ω := [0,1]d Rd can be written as x = [cxi,cyj,czk]T . The constants cx,cy,cz represent the resolution (size of each sample) in the x,y and z dimensions, respectively, of the imaging system.

Because of patient motion, or device dependent geometric distortions which may not remain constant through acquisition of the entire image series, the series of images Sn(i), 1 6 n 6 N, may be misaligned with respect to each other. This means that a fixed image coordinate i may not represent the same structure or anatomical region in all images of the series. To ensure that the same coordinate i corresponds, as much as possible, to the same structure image registration is performed to bring the series of images into alignment (see figure (1.3)). This entails in finding functions fn : Ω → Ω, 1 6 n 6 N, so that the variability due to subject motion or geometric distortions in the series Sn(fn(x)), 1 6 n 6 N, is removed. In addition to removing artifacts related to motion and distortion the entire image series may also be spatially transformed onto a standardized coordinate system so that the data may be more conveniently interpreted. More on how these tasks are actually accomplished is to follow.