neuroimaging Neuroimaging is the use of quantitative (computational) techniques to study the structure and function of the central nervous system, developed as an objective way of scientifically studying the healthy human brain in a non-invasive manner. Incre ...

, spatial normalization is an

image processing An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

step, more specifically an

image registration Image registration is the process of transforming different sets of data into one coordinate system. Data may be multiple photographs, data from different sensors, times, depths, or viewpoints. It is used in computer vision, medical imaging, milit ...

method. Human brains differ in size and shape, and one goal of spatial normalization is to deform human brain scans so one location in one subject's brain scan corresponds to the same location in another subject's brain scan. It is often performed in research-based

functional neuroimaging Functional neuroimaging is the use of neuroimaging technology to measure an aspect of brain function, often with a view to understanding the relationship between activity in certain brain areas and specific mental functions. It is primarily used a ...

where one wants to find common brain activation across multiple human subjects. The brain scan can be obtained from

magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...

(MRI) or

positron emission tomography Positron emission tomography (PET) is a functional imaging technique that uses radioactive substances known as radiotracers to visualize and measure changes in Metabolism, metabolic processes, and in other physiological activities including bl ...

(PET) scanners. There are two steps in the spatial normalization process: * Specification/estimation of warp-field * Application of warp-field with resampling The estimation of the warp-field can be performed in one modality, e.g., MRI, and be applied in another modality, e.g., PET, if MRI and PET scans exist for the same subject and they are coregistered. Spatial normalization typically employs a 3-dimensional nonrigid transformation model (a "warp-field") for warping a brain scan to a template. The warp-field might be parametrized by

basis function In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represen ...

s such as cosine and polynomia.

Diffeomorphisms as compositional transformations of coordinates

Alternatively, many advanced methods for spatial normalization are building on structure preserving transformations

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

s and

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...

s since they carry smooth submanifolds smoothly during transformation. Diffeomorphisms are generated in the modern field of

Computational Anatomy Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical metho ...

based on diffeomorphic flows, also called diffeomorphic mapping. However, such transformations via diffeomorphisms are not additive, although they form a group with function composition and acting non-linearly on the images via

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

. For this reason, flows which generalize the ideas of additive groups allow for generating large deformations that preserve topology, providing 1-1 and onto transformations. Computational methods for generating such transformation are often called LDDMM which provide flows of diffeomorphisms as the main computational tool for connecting coordinate systems corresponding to the geodesic flows of Computational Anatomy. There is a number of programs that implement both estimation and application of a warp-field. It is a part of the SPM and

AIR The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing f ...

programs as well as MRI Studio and MRI Cloud.org

References

Neuroimaging {{Tech-stub

Diffeomorphisms as compositional transformations of coordinates

See also

References