various institutes and research centers. this increase in data size calls for new algorithms that can mine and detect subtle changes in the images to address clinical questions. such clinical questions are very diverse and include group analysis, imaging biomarkers, disease phenotyping and longitudinal studies. group analysis in the group analysis, the objective is to detect and quantize abnormalities induced by a disease by comparing the images of two or more cohorts. usually one of these cohorts consist of normal (control) subjects, and the other one consists of abnormal patients. variation caused by the disease can manifest itself as abnormal deformation of anatomy (see voxel-based morphometry). for example, shrinkage of sub-cortical tissues such as the hippocampus in brain may be linked to alzheimer's disease. additionally, changes in biochemical (functional) activity can be observed using imaging modalities such as positron emission tomography. the comparison between groups is usually conducted on the voxel level. hence, the most popular pre-processing pipeline, particularly in neuroimaging, transforms all of the images in a dataset to a common coordinate frame via (medical image registration) in order to maintain correspondence between voxels. given this voxel-wise correspondence, the most common frequentist method is to extract a statistic for each voxel (for example, the mean voxel intensity for each group) and perform statistical hypothesis testing to evaluate whether a null hypothesis is or is not supported. the null hypothesis typically assumes that the two cohorts are drawn from the same distribution, and hence, should have the same statistical properties (for example, the mean values of two groups are equal for a particular voxel). since medical images contain large numbers of voxels, the issue of multiple comparison needs to be addressed,. there are also bayesian approaches to tackle group analysis problem. classification although group analysis can quantify the general effects of a pathology on an anatomy and function, it does not provide subject level measures, and hence cannot be used as biomarkers for diagnosis (see imaging biomarkers). clinicians, on the other hand, are often interested in early diagnosis of the pathology (i.e. classification,) and in learning the progression of a disease (i.e. regression ). from methodological point of view, current techniques varies from applying standard machine learning algorithms to medical imaging datasets (e.g. support vector machine), to developing new approaches adapted for the needs of the field. the main difficulties are as follows: small sample size (curse of dimensionality): a large medical imaging dataset contains hundreds to thousands of images, whereas the number of voxels in a typical volumetric image can easily go beyond millions. a remedy to this problem is to reduce the number of features in an informative sense (see dimensionality reduction). several unsupervised and semi-/supervised, approaches have been proposed to address this issue. interpretability: a good generalization accuracy is not always the primary objective, as clinicians would like to understand which parts of anatomy are affected by the disease. therefore, interpretability of the results is very important; methods that ignore the image structure are not favored. alternative methods based on feature selection have been proposed,. clustering image-based pattern classification methods typically assume that the neurological effects of a disease are distinct and well defined. this may not always be the case. for a number of medical conditions, the patient populations are highly heterogeneous, and further categorization into sub-conditions has not been established. additionally, some diseases (e.g., autism spectrum disorder (asd), schizophrenia, mild cognitive impairment (mci)) can be characterized by a continuous or nearly-continuous spectra from mild cognitive impairment to very pronounced pathological changes. to facilitate image-based analysis of heterogeneous disorders, methodological alternatives to pattern classification have been developed. these techniques borrow ideas from high-dimensional clustering and high-dimensional pattern-regression to cluster a given population into homogeneous sub-populations. the goal is to provide a better quantitative understanding of the disease within each sub-population. shape analysis shape analysis is the field of medical image computing that studies geometrical properties of structures obtained from different imaging modalities. shape analysis recently become of increasing interest to the medical community due to its potential to precisely locate morphological changes between different populations of structures, i.e. healthy vs pathological, female vs male, young vs elderly. shape analysis includes two main steps: shape correspondence and statistical analysis. shape correspondence is the methodology that computes correspondent locations between geometric shapes represented by triangle meshes, contours, point sets or volumetric images. obviously definition of correspondence will influence directly the analysis. among the different options for correspondence frameworks we can find: anatomical correspondence, manual landmarks, functional correspondence (i.e. in brain morphometry locus responsible for same neuronal functionality), geometry correspondence, (for image volumes) intensity similarity, etc. some approaches, e.g. spectral shape analysis, do not require correspondence but compare shape descriptors directly. statistical analysis will provide measurements of structural change at correspondent locations. longitudinal studies in longitudinal studies the same person is imaged repeatedly. this information can be incorporated both into the image analysis, as well as into the statistical modeling. in longitudinal image processing, segmentation and analysis methods of individual time points are informed and regularized with common information usually from a within-subject template. this regularization is designed to reduce measurement noise and thus helps increase sensitivity and statistical power. at the same time over-regularization needs to be avoided, so that effect sizes remain stable. intense regularization, for example, can lead to excellent test-retest reliability, but limits the ability to detect any true changes and differences across groups. often a trade-off needs to be aimed for, that optimizes noise reduction at the cost of limited effect size loss. another common challenge in longitudinal image processing is the, often unintentional, introduction of processing bias. when, for example, follow-up images get registered and resampled to the baseline image, interpolation artifacts get introduced to only the follow-up images and not the baseline. these artifact can cause spurious effects (usually a bias towards overestimating longitudinal change and thus underestimating required sample size). it is therefore essential that all-time points get treated exactly the same to avoid any processing bias. post-processing and statistical analysis of longitudinal data usually requires dedicated statistical tools such as repeated measure anova or the more powerful linear mixed effects models. additionally, it is advantageous to consider the spatial distribution of the signal. for example, cortical thickness measurements will show a correlation within-subject across time and also within a neighborhood on the cortical surface - a fact that can be used to increase statistical power. furthermore, time-to-event (aka survival) analysis is frequently employed to analyze longitudinal data and determine significant predictors. image-based physiological modelling traditionally, medical image computing has seen to address the quantification and fusion of structural or functional information available at the point and time of image acquisition. in this regard, it can be seen as quantitative sensing of the underlying anatomical, physical or physiological processes. however, over the last few years, there has been a growing interest in the predictive assessment of disease or therapy course. image-based modelling, be it of biomechanical or physiological nature, can therefore extend the possibilities of image computing from a descriptive to a predictive angle. according to the step research roadmap, the virtual physiological human (vph) is a methodological and technological framework that, once established, will enable the investigation of the human body as a single complex system. underlying the vph concept, the international union for physiological sciences (iups) has been sponsoring the iups physiome project for more than a decade,. this is a worldwide public domain effort to provide a computational framework for understanding human physiology. it aims at developing integrative models at all levels of biological organization, from genes to the whole organisms via gene regulatory networks, protein pathways, integrative cell functions, and tissue and whole organ structure/function relations. such an approach aims at transforming current practice in medicine and underpins a new era of computational medicine. in this context, medical imaging and image computing play an increasingly important role as they provide systems and methods to image, quantify and fuse both structural and functional information about the human being in vivo. these two broad research areas include the transformation of generic computational models to represent specific subjects, thus paving the way for personalized computational models. individualization of generic computational models through imaging can be realized in three complementary directions: definition of the subject-specific computational domain (anatomy) and related subdomains (tissue types); definition of boundary and initial conditions from (dynamic and/or functional) imaging; and characterization of structural and functional tissue properties. in addition, imaging also plays a pivotal role in the evaluation and validation of such models both in humans and in animal models, and in the translation of models to the clinical setting with both diagnostic and therapeutic applications. in this specific context, molecular, biological, and pre-clinical imaging render additional data and understanding of basic structure and function in molecules, cells, tissues and animal models that may be transferred to human physiology where appropriate. the applications of image-based vph/physiome models in basic and clinical domains are vast. broadly speaking, they promise to become new virtual imaging techniques. effectively more, often non-observable, parameters will be imaged in silico based on the integration of observable but sometimes sparse and inconsistent multimodal images and physiological measurements. computational models will serve to engender interpretation of the measurements in a way compliant with the underlying biophysical, biochemical or biological laws of the physiological or pathophysiological processes under investigation. ultimately, such investigative tools and systems will help our understanding of disease processes, the natural history of disease evolution, and the influence on the course of a disease of pharmacological and/or interventional therapeutic procedures. cross-fertilization between imaging and modelling goes beyond interpretation of measurements in a way consistent with physiology. image-based patient-specific modelling, combined with models of medical devices and pharmacological therapies, opens the way to predictive imaging whereby one will be able to understand, plan and optimize such interventions in silico. mathematical methods in medical imaging a number of sophisticated mathematical methods have entered medical imaging, and have already been implemented in various software packages. these include approaches based on partial differential equations (pdes) and curvature driven flows for enhancement, segmentation, and registration. since they employ pdes, the methods are amenable to parallelization and implementation on gpgpus. a number of these techniques have been inspired from ideas in optimal control. accordingly, very recently ideas from control have recently made their way into interactive methods, especially segmentation. moreover, because of noise and the need for statistical estimation techniques for more dynamically changing imagery, the kalman filter and particle filter have come into use. a survey of these methods with an extensive list of references may be found in. modality specific computing some imaging modalities provide very specialized information. the resulting images cannot be treated as regular scalar images and give rise to new sub-areas of medical image computing. examples include diffusion mri, functional mri and others. diffusion mri a mid-axial slice of the icbm diffusion tensor image template. each voxel's value is a tensor represented here by an ellipsoid. color denotes principal orientation: red = left-right, blue=inferior-superior, green = posterior-anterior diffusion mri is a structural magnetic resonance imaging modality that allows measurement of the diffusion process of molecules. diffusion is measured by applying a gradient pulse to a magnetic field along a particular direction. in a typical acquisition, a set of uniformly distributed gradient directions is used to create a set of diffusion weighted volumes. in addition, an unweighted volume is acquired under the same magnetic field without application of a gradient pulse. as each acquisition is associated with multiple volumes, diffusion mri has created a variety of unique challenges in medical image computing. in medicine, there are two major computational goals in diffusion mri: estimation of local tissue properties, such as diffusivity; estimation of local directions and global pathways of diffusion. the diffusion tensor, a 3 × 3 symmetric positive-definite matrix, offers a straightforward solution to both of these goals. it is proportional to the covariance matrix of a normally distributed local diffusion profile and, thus, the dominant eigenvector of this matrix is the principal direction of local diffusion. due to the simplicity of this model, a maximum likelihood estimate of the diffusion tensor can be found by simply solving a system of linear equations at each location independently. however, as the volume is assumed to contain contiguous tissue fibers, it may be