### Topological Data Analysis and Persistent Homology

Persistent homology is a new area of computational topology that aims to understand the underlying high-dimensional structures from low-dimensional local topological structures. Instead of analyzing shapes and images at a fixed scale, as usually done in traditional approaches, persistent homology observes the changes of topology over different scales and finds the most persistent topological features that are robust under noise perturbations. By 2009, there had been many theoretical advancements in the field, but large-scale real-world applications beyond toy examples to clearly demonstrate its usefulness and power were lacking. Thus, the concept had been a theoretical curiosity in medical imaging fields for many years. We wrote the first paper on the application of persistent homology to MRI using the cortical thickness of the human brain (Chung et al., 2009).MATLAB codes:

Exact Topological Inference for Network Differences

Persistent and Min-Max Diagrams for Cortical Thickness

Betti Plots for Brain Networks

Topological Network Distances

### Large-scale Brain Network Analysis

Diffusion tensor imaging (DTI) regularly produces up
to a half million white matter fiber tracts per
brain, which correspond to a half million edges and
one million nodes in a network. The resulting
networks defy many standard network methods due to
the sheer computational load. The connectivity
matrix corresponding to such data would contain a
hundred billion connections, which results in the
small-*n*large-

*p*problem (Chung et al. 2015). Specifically, when the number of voxels (

*p*) are substantially larger than the number of images (

*n*), it produces an under-determined model with infinitely many possible solutions. Our group was first to develop the compressed sensing (CS) technique to thin out the significant number of noise connections in the brain network (Lee et al., 2011).

Any CS or sparse brain network model is usually parameterized by a tuning parameter that controls the sparsity of the representation. Increasing the sparse parameter makes the solution more sparse. Thus, sparse models are inherently multiscale, where the scale of the model is determined by the sparsity. Many existing sparse network models use a fixed sparse parameter that may not be optimal in other datasets or studies. Depending on the choice of the sparse parameter, the final classification and statistical results will be different (Chung et al., 2015). Therefore, there is a need to develop a multiscale network model that provide a consistent analysis results and interpretation regardless of the choice of parameter. Persistent homology offers one possible solution to the multiscale problem. Instead of studying images and networks at a fixed scale, as usually done in traditional approaches, persistent homology summarizes the changes of topological features over different scales and finds the most persistent topological features that are robust under perturbations. This robust performance under different scales is needed for network models that are parameter and scale dependent. Instead of building networks at one fixed parameter that may not be optimal, I have developed a new persistent homological framework for analyzing the collection of sparse models over every possible sparse parameter (Chung et al., 2015; Chung et al., 2017). The method exploits the topological structure in a family of sparse models in drastically speeding up the computation for building large-scale brain networks with billions of connections.

MATLAB codes:

Transposition Test for Large-scale Network Data

Structural Brain Connectivity from Diffusion Tensor Images

Heritability of Large-scale Brain Networks

Epsilon Neighbor Structural Brain Network Construction for DTI

Betti Plots for Brain Networks

Sparse Hyper-Network Model

Topological Network Distances

### Heat Kernel Smoothing & Diffusion on
Manifolds

A lot of anatomical features that characterize brain
shape variability can only be measured relative to a
cortical surface. It is necessary to develop a
surface-based framework for increased sensitivity
and specificity of analysis. A significant hurdle in
this area had been caused by the lack of
surface-based techniques that can incorporate
the non-Euclidean geometric nature of the cortical
surface. The concept of smoothing anatomical
measurements along the brain tissue boundaries for
this purpose (Chung et al. 2001; Chung
et al. 2003). Diffusion smoothing, a concept
introduced in Chung et al. (2001), utilizes
isotropic heat flow as a way to smooth brain surface
measurements such as cortical thickness, curvatures
or functional MRI. The drawbacks of diffusion
smoothing are the complexity of numerically solving
the PDE and the numerical instability that depends
on the smoothness of anatomical boundary. To address
the issues, heat kernel smoothing was
introduced (Chung
et al., 2005). In heat kernel smoothing, the
analytically unknown nonlinear heat kernel is
approximated linearly and smoothing is performed as
the iterative convolution in a spatially adaptive
fashion. These two methods are the standard baseline
methods in building more complex cortical
surface models in the field. The methods can
easily detect surface specific tissue growth and
atrophy that are more difficult to detect using the
traditional 3D volume based methods. MATLAB codes:

Polynomial Approximation to Heat Diffusion on Manifolds

Heat Kernel Smoothing: Iterated Kernel Smoothing Version

Heat kernel smoothing via the Laplace-Beltrami eigenfunctions

Diffusion smoothing on manifolds

### Weighted Fourier Series Representation

Weighted Fourier Series (WFS) representation incorporates parameterization, image smoothing, surface registration and statistical inference in a single mathematical framework while reducing the Gibbs phenomenon (ringing artifacts) associated with the traditional Fourier analysis (Chung et al., 2007). In WFS, signals are represented as a weighted linear combination of eigenfunctions of the self-adjoint operator of the underlying manifold. The weights are exponentially decaying and related to the heat kernel of the manifold. On a sphere, the eigenfunctions are spherical harmonics. In the Euclidean space, they are the usual Fourier basis. We have recently shown that WFS is in fact mathematically equivalent to the diffusion wavelet transform (Chung et al., 2015). Thus, WFS inherits all the nice localization properties of wavelets. As a special case of WFS, we have developed weighted spherical harmonic (SPHARM) representation. The iterative residual fitting (IRF) algorithm was proposed for estimating more than 20000 spherical harmonic coefficients per brain (Chung et al., 2007) The algorithm sequentially breaks down one gigantic problem into smaller problems in an iterative fashion.MATLAB codes:

Weighted Spherical Harmonic Representation (Weighed-SPHARM)

Cosine Series Representation of 3D Curves and White Matter Fiber Tracts

Amygdala/hippocampus surface modeling using SPHARM

### Tensor-Based & Surface-Based Morphometry

In computational neuroanatomy,
region-of-interest (ROI) free MRI morphometric
techniques mainly referred to as tensor-based
morphometry (TBM) and deformation-based morphometry
(DBM) are popular. In trying to model shape
variations in medical images, it is not clear what
sort of image features should be analyzed. Building
on the framework of fluid dynamics, Chung
et al. (2001) modeled and analyzed the
distribution of the Jacobian determinant,
divergences and vorticities of brain deformation
systematically in a unified statistical framework.
These quantities are essential in measuring the
tissue growth and atrophy at each voxel. Chung
et al. (2001) demonstrated the Jacobian
determinant is the only meaningful biomarker in
tissue growth and atrophy. The Jacobian determinant
approach has been used in a wide variety of
applications including the localization of the
region of brain tissue growth before and after
cochlear implant in positron emission tomography
(PET) images. The later studies (Chung
et al. 2003) further extended the idea of the
Jacobian determinant in the Euclidean space to
manifolds in a more generalized tensor formulation
and were able to surface specific growth locally in
developing children. Currently, the Jacobian
determinant is the standard baseline feature for
characterizing tissue growth and atrophy in brain
imaging. MATLAB codes:

Amydala Surface Modeling Using Keith Worsley's SurfStat

Mandible Surface Modeling Using the Laplace-Beltrami Eigenfunctions

Voxel-Based Morphometry to Modeling Binary Stroke Lesions in DIffusion Weighted Images

### Other MATLAB packages

General linear models (GLM) in imagingHow to read ImageJ ROI file format into MATLAB?

Image format conversion Tips

Corrected p-value on cortex

Functional data analysis on a unit circle

Deformation field manipulation codes