Abdolla25/MRI-Task_1

MRI Assignement_1

Name	Section	Bench
Abdullah Ahmed Zahran	2	5
Samar Ibrahim	1	40
Shymaa Gamal	2	1
Sondos Mohamed	1	42

This is a representation for Bloch Equation in 3-D space using Python IDE

The animations below shows the following:

• M vector with angular frequency 30deg
• Excitation Mode with angular frequency 30deg
• Relaxation Mode with angular frequency 30deg
• Excitation - Relaxation Mode
• Excitation - Relaxation Trajectory on X-Y plane
• Relaxation - Excitation Mode
• Relaxation - Excitation Trajectory on X-Y plane

Bloch Equation Code:

Main Equations for M vector:-

X = sin(z) * cos(t)

Y = sin(z) * sin(t)

Z = cos(z)

where t represents the angle of rotation, z is the angular frequency, by changing z value with respect to t time we have the following representations:

Excitation Model

z = pi/2 * sin(t/(4))

Relaxation Model

z = pi/2 * cos(t/(4))

Also you can set numberOfLoops = 1 to generate single mode [Excitaion or Relaxation] only or = 2 to generate dual modes.

import numpy as np
import imageio
from qutip import Bloch
def animate_bloch(vectors, duration=0.1, save_all=False):
   numberOfLoops = 1 #Enter '1' to apply only ONE mode [Excitation or Relaxation] or '2' to to apply only BOTH modes [Excitation and Relaxation] starting with the selected mode
   if numberOfLoops == 1:
       maxAngle = 2*np.pi
   elif numberOfLoops == 2:
       maxAngle = 4*np.pi
   mode = 1 #Enter '0' to activate Excitation Mode or '1' to activate Relaxation Mode 
   omega = np.pi/6 #Enter the angular frequency 
   z = 0
   sqAngle = np.pi/2
   a = 5
   vectorM = Bloch()
   # vectorM.view = [90,90] #Activate this line to view magnetization’s trajectory on x-y plane
   images=[]
   for t in np.arange(omega, maxAngle, 0.1):
       if mode == 0:
           z = np.pi/2 * np.sin(t/(4))
           if t == 2*np.pi:
               mode = 1
       elif mode == 1:
           z = np.pi/2 * np.cos(t/(4))
           if t == 2*np.pi:
               mode = 0
       else:
           pass
       vectorM.clear()
       vectorM.add_vectors([np.sin(omega)*np.cos(t), np.sin(omega)*np.sin(t), np.cos(omega)])
       vectorM.add_vectors([np.sin(z)*np.cos(a*t), np.sin(z)*np.sin(a*t), np.cos(z)])
       vectorM.add_vectors([np.sin(sqAngle)*np.cos(t), np.sin(sqAngle)*np.sin(t), np.cos(sqAngle)])
       filename = 'temp_file.png'
       vectorM.save(filename)
       images.append(imageio.imread(filename))
   imageio.mimsave('bloch_anim.gif', images, duration=duration)
vectors = []
animate_bloch(vectors, duration=0.1, save_all=False) 

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Animations in 3-D:

Tho following vectors represents different cases:

• Green: M vector on Relaxation with 0/30deg angular frequency.
• Orange: M vector on Excitation/Relaxation with B.
• Blue: M vector on Excitation with 90deg with Z vector.

M vector with angular frequency 30deg

Excitation Mode with angular frequency 30deg

Relaxation Mode with angular frequency 30deg

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Excitaion-Relaxation Mode

Excitaion-Relaxation Trajectory

Relaxation-Excitaion Mode

Relaxation-Excitaion Trajectory

Fourrier Code:

Here we use absoulute, angle ,real and imaginary to show each fourier component in Matplot show

import numpy as np
from cv2 import cv2 as cv
import matplotlib.pyplot as p
from numpy import random
def main ():
    img=cv.imread('1.jpg',0)
    spectrum =np.fft.fftshift(np.fft.fft2(img))
    p.subplot(321)
    p.imshow(img, cmap='gray')
    p.title('input image') 
    p.subplot(322)
    p.imshow( np.log(np.abs(spectrum)) , cmap='gray')
    p.title('Magnitude Spectrum') 
    p.subplot(323)
    p.imshow(np.angle(spectrum),cmap='gray')
    p.title('Phase Spectrum') 
    p.subplot(324)
    p.imshow(np.real(spectrum),cmap='gray')
    p.title('real Spectrum') 
    p.subplot(325)
    p.imshow(  np.imag(spectrum),cmap='gray')
    p.title('imaginary Spectrum') 
    p.show()
############################################   Plot Random Variable     ################################################################
    z = random.random(200)
    p.ylabel("Z axis")
    p.xlabel("Time")
    p.plot(z)
    p.show()

if __name__=="__main__":
    main()

Image And its Fourier Components

Original Image

Fourier transform components

Gray Beam

Plot Random Variable