C O N T E N T S:

- The MR image is then reconstructed by performing the inverse Fourier Transform of the acquired raw data (k-space).(More…)

- The user can further explore this option (Half Fourier) by varying the Half Fourier parameters e.g. selecting some extra (more than half of the kspace data lines) PE lines and its effects on the reconstructed image.(More…)

Image Courtesy:

link: http://www.geogebra.org/m/802709

author: geogebra.org

description: fourier – GeoGebraBook

**KEY TOPICS **

**[1] The inverse Fourier transform of the kspace provides the Magnetic Resonance (MR) image. [1] Once the full k-space data is acquired, its inverse Fourier transform (iFFT) provides an MR image. [1]**

*The MR image is then reconstructed by performing the inverse Fourier Transform of the acquired raw data (k-space).*The image below shows the ideal Fourier transform, FFT without windowing, and FFT after applying a hamming window. [2] The Fast Fourier Transform (FFT) is a method to calculate the fourier transform efficiently for discrete sets of points. [3] Fast Fourier Transform (FFT) is a very powerful tool for revealing the useful frequency components of a signal, even when the signal is influenced by noise. [2] Fast Fourier Transform (FFT) cannot be applied directly on the acquired NC trajectory based k-space data. [1] FFT originates from the Fourier Transform (FT), a mathematical way to extract the frequency information by decomposing any periodic signal into a combination of sine and cosine waves. [2] To adapt to digital signals, a discrete version of Fourier Transform is proposed and named Discrete Fourier Transform (DFT). [2] Spectral methods work by using the Fourier transform (or some varient of it) to calculate the derivative. [3]

You clearly define your goals of wanting to work out the physics/ technical issues of shifting pitches, such as the nyquist frequency limitation and the means of computing the frequencies of a sound file using the Fourier Transform. [4] The Fourier transform data has the amplitude graphed against its frequency, so I can look at the different cycles of the unemployment rate. [4] With discrete data set of N ~ 1000, I can perform the fast Fourier transform or the FFT to efficiently compute the discrete Fourier transform. [4] Discrete Fourier transform can be expressed into the next set of equations from the previous ones. [4] Discrete Fourier transforms take a very long time to compute. [4] Given the time, I believe that the simple low-pass filter was sufficient to show how a filter affects the Fourier transform data and its inverse. [4] I inverse Fourier transform the low-pass filtered data to compare it to the original data. [4] The bottom equation is used to get the inverse of the Fourier transform or the original function from the Fourier transform equation. [4] Fourier transform and spherical harmonics are mathematical tools that can be used to represent a function as a combination of periodic functions (functions that repeat themselves, like sine waves) of different frequencies. [5] Fourier transform is often used in audio processing to post-process signals as combination of sine waves of different frequencies, instead of single streams of sound waves. [5] A function x(t) (representing, perhaps, an audio signal) has a Fourier Transform X( ? ), given below (note the functions were drawn arbitrarily – don’t try to find mathematical representations). [6] Find the Fourier Transform, Y(?), for the function, y(t), shown below. [6] The Fourier transform decomposes a function of time into the frequency components that make up the original function. [4] Assuming that the trend will continue, I can look at the lower frequency end of the most significant peak fluctuation in the Fourier transform graph of figure 1.1. [4] The first part of the project was to Fourier transform the unemployment data. [4] This post is part 2 of a series ( part 1 ) leading up to a geometric interpretation of Fourier transform and spherical harmonics. [5] This handy analogy can help us take another step closer to a geometric interpretation of Fourier transform and spherical harmonics later. [5]

The latter subtracts the Fourier-transformed incident fields from the Fourier transforms of the scattered fields; logically, we might subtract these after the run, but it turns out to be more convenient to subtract the incident fields first and then accumulate the Fourier transform. [7] Basically, we’ll tell Meep to keep track of the fields and their Fourier transforms in a certain region, and from this compute the flux of electromagnetic energy as a function of ω. [7] According to Fourier’s theorem, every function can be written in terms of a sum of sines and cosines to arbitrary precision; given function f(t), the Fourier transform F(f(t)) is obtained by taking the integral of f(t)’s sines and cosines function. [4] It keeps running for an additional 50 time units until the square amplitude has decayed by 1/1000 from its peak: this should be sufficient to ensure that the Fourier transforms have converged. [7] The top equation in Eq. 2 is the discrete Fourier transform. [4]

**POSSIBLY USEFUL **

**[1] The user can visualize the Half Fourier k-space data and its effects on the reconstructed image. [1]**

*The user can further explore this option (Half Fourier) by varying the Half Fourier parameters e.g. selecting some extra (more than half of the kspace data lines) PE lines and its effects on the reconstructed image.*Another important property of the k-space is the existence of symmetrical data; according to this property only half of the k-space data will need to be collected and the remaining half of the k-space can be estimated mathematically by using complex conjugate synthesis; this process is known as Half Fourier. [1]

This document takes a look at different ways of representing real periodic signals using the Fourier series. [8] This article explains how we can model the human heartbeat with a mathematical expression, using Fourier Series. [9] It will provide translation tables among the different representations as well as example problems using Fourier series to solve a mechanical system and an electrical system, respectively. [8]

The Fourier series describes a given function as infinite linear combinations of stationary plane waves, each of which may be characterized by an amplitude and a wave number. [1] The table below summarizes how to get one set of Fourier Series coefficients from any other representation. [8] Here’s an interactive graph that allows you to explore the concepts behind the Fourier Series. [9]

The blue graph in the animation is called an amplitude spectrum. [2] [xyz-ihs snippet=”Amazon-Affiliate-Native-Ads”] Animation: The user can visualize the animated effect on the reconstructed image by gradually increasing or decreasing the size of the central k-space window. [1] The “Visualize Trajectory” MATLAB window ( Figure 3a ) also loads the gradient waveforms showing the animation of traversing only 16 lines (Rectilinear trajectory) in k-space (far fewer than the actual experiment). [1]

The user is also able to visualize different kinds of trajectories (Cartesian and Non-Cartesian) using animations. [1] Beside that the user can visualize the effect of filtering schemes with the help of the animations. [1] Different animation effects are “low pass filter to high pass filter” and similarly from “high pass filter to low pass filter.” [1]

The main purpose of showing this animation is to familiarize the user for the direction of the phase encoding and frequency encoding. [1] The initial animation moves slowly because we selected Calculate on Demand when calculating Vorticity Magnitude. [10] Toggling off Calculate on Demand will skip this process during the first animation, and instead load all the variable values upon the initial calculation. [10]

Figure 5b provides a sketch of the animation of spiral trajectory. [1] Animation (a) for Cartesian Trajectory and (b) shows the animation for Spiral trajectory. [1]

Starting from FT, below is an animation (from wikipedia) that illustrates how it works. [2]

**RANKED SELECTED SOURCES **(10 source documents arranged by frequency of occurrence in the above report)

1. (16) Journey through k-space: an interactive educational tool

3. (6) OpenBCI

4. (4) Blog | Ming-Lun “Allen” Chou

5. (3) Meep Tutorial – AbInitio

6. (3) Fourier Series – PrattWiki

7. (2) Calculating Variables and Using Contour Color Cutoff

8. (2) Numerical Solution of the KdV – WikiWaves

9. (2) E12 Exam 2 Review | Online Homework System

10. (2) Fourier Series Graph Interactive