Summary How MRI Works - Part 3 - Fourier Transform and K-Space (Youtube) www.youtube.com
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The Fourier transform is a powerful mathematical tool that decomposes functions into sinusoidal waves, enabling the analysis of complex signals and the transition between different frames of reference.
Slides
Slide Presentation (12 slides)
Key Points
- Fourier transforms represent any function as a sum of sinusoidal waves of different frequencies, where each wave is specified by frequency, amplitude, and phase
- The Fourier transform decomposes a function f(t) into its constituent sinusoidal frequencies, resulting in a new function F(ω) that specifies the complex phasor (amplitude and phase) associated with each frequency ω
- Important Fourier transform pairs include: sine wave, decaying exponential (Lorentzian distribution), and square pulse (sinc function)
- The Fourier shift theorem allows transitioning between the laboratory frame and the rotating frame of reference used in MRI
- K-space is the Fourier domain in MRI, where the location in K-space corresponds to a particular 2D sinusoidal frequency, and the relationship between K-space and image space is crucial for understanding MRI
- K-space is often displayed as a 2D grayscale image, where the brightness of a voxel corresponds to the magnitude of the complex Fourier transform, and due to the Hermitian property, only half of K-space needs to be acquired
- The application of magnetic field gradients in the x and y directions creates a 2D sinusoidal pattern in the spin sample, and the vector sum of these encoded spin magnetic moments corresponds to the complex phasor at a particular location in K-space, which is the foundation for the gradient echo pulse sequence in MRI
Summaries
38 word summary
Fourier transforms decompose functions into sinusoidal waves. Sine waves have opposing phasors, while exponentials yield Lorentzian distributions. The Fourier transform of a pulse is a sinc function. Fourier shift theorem enables transitioning between frames, crucial for understanding k-space.
50 word summary
Fourier transforms decompose functions into sinusoidal waves. Sine waves have opposing phasors, while exponentials yield Lorentzian distributions. The Fourier transform of a pulse is a sinc function. The Fourier shift theorem enables transitioning between frames, crucial for understanding k-space, which encodes spatial frequencies and determines field of view and resolution.
129 word summary
Fourier transforms are crucial in MRI imaging, decomposing functions into sinusoidal waves. The Fourier transform of a sine wave has two complex phasors with opposite phases, while a decaying exponential yields a Lorentzian distribution. The width of the Lorentzian is inversely proportional to the T2 relaxation time. The Fourier transform of a square pulse is a sinc function, representing the excitation pulse's frequency profile.
The Fourier shift theorem enables transitioning between the laboratory and rotating frames, crucial for understanding k-space - the Fourier domain encoding spatial information. K-space represents spatial frequencies, with low frequencies in the center and high frequencies at the edges. The relationship between k-space and image space determines the field of view and resolution. Understanding Fourier transforms and k-space is essential for comprehending MRI imaging principles.
353 word summary
Fourier Transforms and K-Space in MRI
Fourier transforms are a powerful mathematical tool that play a crucial role in MRI imaging. The key concept is that any function can be represented as a sum of sinusoidal waves with different frequencies, amplitudes, and phases. The Fourier transform decomposes a function into its constituent sinusoidal frequencies, resulting in a new function that specifies the complex phasor (amplitude and phase) associated with each frequency.
Several important functions and their Fourier transforms are discussed. The Fourier transform of a sine wave consists of two complex phasors at frequencies with opposite phases. The Fourier transform of a decaying exponential is the Lorentzian distribution, which has a real "absorption" component and an imaginary "dispersion" component. The width of the Lorentzian is inversely proportional to the decay rate, corresponding to the T2 relaxation time in MRI. The Fourier transform of a square pulse is a sinc function, representing the frequency profile of the excitation pulse.
The Fourier shift theorem allows transitioning between the laboratory frame and the rotating frame of reference used in MRI, which is crucial for understanding k-space, the Fourier domain used to encode spatial information. When dealing with digitized signals, considerations arise regarding the Nyquist frequency and aliasing, as well as the efficient Fast Fourier Transform (FFT) algorithm.
K-space is the Fourier domain in MRI, where the location corresponds to a particular 2D sinusoidal frequency. Low spatial frequencies in the center represent gradual changes in intensity, while high spatial frequencies near the edges represent sharper details. The relationship between K-space and image space is crucial, as the step size in K-space dictates the field of view, and the size of the K-space array determines the image resolution.
K-space is often displayed as a 2D grayscale image, where the brightness corresponds to the magnitude of the complex Fourier transform. Due to the symmetry of the Fourier transform of real functions, K-space is Hermitian, allowing for faster acquisition by only acquiring half or more of K-space.
The understanding of Fourier transforms and K-space is essential for comprehending the fundamental principles of MRI imaging, from signal generation to image reconstruction.
568 word summary
Fourier Transforms and K-Space in MRI
Fourier transforms are a powerful mathematical tool that play a crucial role in MRI imaging. The key concept is that any function can be represented as a sum of sinusoidal waves with different frequencies, amplitudes, and phases. The Fourier transform takes a function f(t) and decomposes it into its constituent sinusoidal frequencies, resulting in a new function F(ω) that specifies the complex phasor (amplitude and phase) associated with each frequency ω.
Several important functions and their Fourier transforms are discussed. The Fourier transform of a sine wave consists of two complex phasors at frequencies ±ω, with opposite phases. The Fourier transform of a decaying exponential is the Lorentzian distribution, which has a real "absorption" component and an imaginary "dispersion" component. The width of the Lorentzian is inversely proportional to the decay rate α, which corresponds to the T2 relaxation time in MRI. The Fourier transform of a square pulse is a sinc function, which represents the frequency profile of the excitation pulse in MRI.
The Fourier shift theorem is an important concept, as it allows transitioning between the laboratory frame and the rotating frame of reference used in MRI. This is crucial for understanding k-space, the Fourier domain used to encode spatial information in MRI.
When dealing with digitized signals, several considerations arise. The sample rate determines the maximum frequency that can be accurately recorded, known as the Nyquist frequency. Sampling at less than twice the highest frequency leads to aliasing, where higher frequencies appear as lower frequencies in the sampled data. The Fast Fourier Transform (FFT) is an efficient algorithm for computing the discrete Fourier transform, with the output following a specific order.
K-space is the Fourier domain in MRI, where the location in K-space corresponds to a particular 2D sinusoidal frequency. Low spatial frequencies in the center of K-space represent gradual changes in intensity, while high spatial frequencies near the edges represent sharper details. The relationship between K-space and image space is crucial: the step size in K-space dictates the field of view, and the size of the K-space array determines the image resolution.
K-space is often displayed as a 2D grayscale image, where the brightness of a voxel corresponds to the magnitude of the complex Fourier transform. Due to the symmetry of the Fourier transform of real functions, K-space is Hermitian, meaning that half of the information is redundant. This allows for faster acquisition by only acquiring half or more of K-space.
The organization of the 2D discrete Fourier transform output follows a specific layout, with the DC offset at the center and the quadrants arranged in a particular order. This layout can be visualized and understood in terms of the relationship between the K-space coordinates and the matrix rows and columns.
Finally, the concept of K-space encoding is introduced, where the application of magnetic field gradients in the x and y directions creates a 2D sinusoidal pattern in the spin sample. The vector sum of these encoded spin magnetic moments corresponds to the complex phasor at a particular location in K-space, which is the foundation for the gradient echo pulse sequence in MRI.
In summary, this overview provides a comprehensive introduction to Fourier theory and its applications in MRI. Understanding Fourier transforms and their properties, as well as the concept of K-space, is crucial for comprehending the fundamental principles of MRI imaging, from signal generation to image reconstruction.
987 word summary
Fourier transforms are a powerful mathematical tool used extensively in science, engineering, and math. This video discusses how Fourier transforms relate to MRI imaging.
The key concept is that any function can be represented as a sum of sinusoidal waves of different frequencies. A sinusoid is fully specified by three parameters: frequency (ω), amplitude (a), and phase (φ). These three parameters can be combined into a single complex number called a phasor, which represents the amplitude and phase.
The Fourier transform takes a function f(t) and decomposes it into its constituent sinusoidal frequencies. The result is a new function F(ω), which specifies the complex phasor (amplitude and phase) associated with each frequency ω. Adding up all the scaled and shifted sinusoids described by F(ω) reconstructs the original function f(t).
This Fourier transform relationship can be visualized in 3D, with the real and imaginary parts of F(ω) plotted against frequency ω. For real functions f(t), the Fourier transform F(ω) has a Hermitian property, where the positive and negative frequency components are complex conjugates.
Several important functions and their Fourier transforms are discussed:
1. Sine wave: The Fourier transform of sine(ω₀t) consists of two complex phasors at frequencies ±ω₀, with opposite phases.
2. Decaying exponential: The Fourier transform of e^(-αt) is the Lorentzian distribution, which has a real "absorption" component and an imaginary "dispersion" component. The width of the Lorentzian is inversely proportional to the decay rate α, which corresponds to the T2 relaxation time in MRI.
3. Square pulse: The Fourier transform of a square pulse of duration τ is a sinc function (sin(ωτ/2)/(ωτ/2)). This sinc function represents the frequency profile of the excitation pulse in MRI.
These Fourier transform pairs are fundamental to understanding MRI physics. The decaying exponential models the free induction decay (FID) signal, while the sinc function relates to slice selection in MRI.
Additionally, the Fourier shift theorem is introduced, which allows transitioning between the laboratory frame and the rotating frame of reference used in MRI. This is an important concept for understanding k-space, the Fourier domain used to encode spatial information in MRI.
In summary, this video provides a thorough introduction to Fourier theory and its applications in MRI. Understanding Fourier transforms is crucial for comprehending how MRI works, from signal generation to image reconstruction. The key ideas covered include sinusoids, phasors, the Fourier transform, Hermitian properties, and important function pairs like the decaying exponential and Lorentzian, as well as the square pulse and sinc function. These fundamental concepts lay the groundwork for understanding k-space and the Fourier-based image formation process in MRI.
Fourier Transform and K-Space in MRI
The Fourier transform is a fundamental concept in MRI, allowing us to excite spins within well-defined slices of the body. The relationship between the duration of a square pulse and the frequency band of the sinc pulse is crucial: longer pulses excite a narrower band of frequencies, while shorter pulses excite a larger band. This relationship works both ways, as a sinc pulse in the time domain will excite a square-shaped band of frequencies.
Another important Fourier pair is the Gaussian function, where a broad Gaussian in the time domain yields a narrow Gaussian in the frequency domain, and vice versa. Multiplying a function f by a sinusoid of frequency ω₀ shifts the entire Fourier transform of f by ω₀. This is important in the context of the laboratory frame and the rotating frame of reference in NMR.
When dealing with digitized signals, several considerations arise. The signal is now discretized, with a specific sample rate and bit depth. The sample rate determines the maximum frequency that can be accurately recorded, known as the Nyquist frequency. Sampling at less than twice the highest frequency leads to aliasing, where higher frequencies appear as lower frequencies in the sampled data.
The Fast Fourier Transform (FFT) is an efficient algorithm for computing the discrete Fourier transform. The output of the FFT follows a specific order, with the DC offset, positive frequencies, the Nyquist frequency, and negative frequencies. The units of frequency in the FFT are cycles per field of view, where the field of view represents the full range of the input function.
K-space is the Fourier domain in MRI, where the location in K-space corresponds to a particular 2D sinusoidal frequency. Low spatial frequencies in the center of K-space represent gradual changes in intensity, while high spatial frequencies near the edges represent sharper details. The relationship between K-space and image space is crucial: the step size in K-space dictates the field of view, and the size of the K-space array determines the image resolution.
K-space is often displayed as a 2D grayscale image, where the brightness of a voxel corresponds to the magnitude of the complex Fourier transform. Due to the symmetry of the Fourier transform of real functions, K-space is Hermitian, meaning that half of the information is redundant. This allows for faster acquisition by only acquiring half or more of K-space.
The units of K-space frequencies are cycles per field of view, and the Nyquist frequency is determined by the number of pixels in the image. If the pixel dimensions are different in the x and y directions, the Nyquist frequencies will also be different, indicating that the pixels are not isotropic.
The organization of the 2D discrete Fourier transform output follows a specific layout, with the DC offset at the center and the quadrants arranged in a particular order. This layout can be visualized and understood in terms of the relationship between the K-space coordinates and the matrix rows and columns.
Finally, the concept of K-space encoding is introduced, where the application of magnetic field gradients in the x and y directions creates a 2D sinusoidal pattern in the spin sample. The vector sum of these encoded spin magnetic moments corresponds to the complex phasor at a particular location in K-space, which is the foundation for the gradient echo pulse sequence in MRI.
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Source: https://www.youtube.com/watch?v=R_4GuyJTzMo
Page title: How MRI Works - Part 3 - Fourier Transform and K-Space - YouTube
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