A Note on the Maximum Genus of Graphs with Diameter 4

Let G be a simple graph with diameter four, if G does not contain complete subgraph K3 of order three

One‐Dimensional Fourier Imaging and k‐Space

Magnetic resonance imaging offers the possibility to obtain spatially resolved anatomical information. This is accomplished by taking advantage of the Larmor relationship, which dictates that the frequency of the spins depends on the local magnetic field. This unit discusses the one dimensional Fourier imaging based on this relation. The one‐to‐one mapping of the signal from a given frequency to a given spatial location is explained. The image reconstruction based on well‐known Fourier transform reconstruction method is described in detail. The Fourier transform takes the MR signal as acquired in the time domain (usually referred to as the k‐space domain) and converts it to the frequency domain where 1‐D spatially resolved information can be obtained.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145275/1/cpmib0402.pd

Spin and Gradient Echoes

This unit discusses the basic concept of an echo. The phase of the spins plays the fundamental role in this process. A general description as to how an echo occurs is followed by a discussion on both spin echoes and gradient echoes.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145391/1/cpmib0401.pd

Signal, Noise, and Contrast

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145413/1/cpmib0600.pd

Basic Spin Properties and the Bloch Equations

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145203/1/cpmib0300.pd

Signal‐to‐Noise Ratio as a Function of Imaging Parameters

Signal‐to‐Noise Ratio as a Function of Imaging Parameters (Azim Celik, General Electric Company, Milwaukee, Wisconsin and Weili Lin, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina). The degree to which noise affects a measurement is generally characterized by the signal‐to‐noise ratio (SNR, as measured by the ratio of the voxel signal to the noise standard deviation). This unit describes the importance of SNR in describing image quality. SNR is the key parameter for determining the quality of any given imaging experiment. If the SNR is not high enough, it becomes impossible to differentiate tissues from one another or the background. The dependence of SNR on imaging parameters such as the number of repetitions, the number of k‐space samples (Nx, Ny, and Nz), the readout bandwidth, and voxel dimensions (Dx, Dy, and Dz) is explained in detail.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145331/1/cpmib0602.pd

Contrast

Contrast (Azim Celik, General Electric Company, Milwaukee, Wisconsin and Weili Lin, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina). SNR (signal‐to‐noise ratio) determines the effectiveness of an imaging experiment. However, even the highest SNR does not guarantee the usefulness of an image. An important aim of imaging for diagnostic purposes is to be able to distinguish between diseased and neighboring normal tissues. If the imaging method used does not have a signal‐manipulating mechanism which produces different signals for the diseased and normal tissues, then distinguishing the two tissues is not possible. This unit provides a detailed discussion of the contrast‐producing mechanisms that arise from the signal dependence on a wide variety of tissue parameters.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145270/1/cpmib0603.pd

Multiscale adaptive smoothing models for the hemodynamic response function in fMRI

In the event-related functional magnetic resonance imaging (fMRI) data analysis, there is an extensive interest in accurately and robustly estimating the hemodynamic response function (HRF) and its associated statistics (e.g., the magnitude and duration of the activation). Most methods to date are developed in the time domain and they have utilized almost exclusively the temporal information of fMRI data without accounting for the spatial information. The aim of this paper is to develop a multiscale adaptive smoothing model (MASM) in the frequency domain by integrating the spatial and frequency information to adaptively and accurately estimate HRFs pertaining to each stimulus sequence across all voxels in a three-dimensional (3D) volume. We use two sets of simulation studies and a real data set to examine the finite sample performance of MASM in estimating HRFs. Our real and simulated data analyses confirm that MASM outperforms several other state-of-the-art methods, such as the smooth finite impulse response (sFIR) model.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS609 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Basic Spin Properties and the Bloch Equations

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145288/1/cpmib0300.pd

Contrast Agents

Contrast agents in general are exogeneous substances employed to alter natural tissue contrast. The motivation behind the use of contrast agents in MR imaging is to further enhance contrast between normal and diseased tissue types and indicate functionality of an organ. The focus of this unit is to give to the basic mechanisms of contrast agents in MR without going into their clinical applications.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/145201/1/cpmib0604.pd