Merge branch 'master' into 'master'

Made MATLAB consistent with python, added plotting to partial fourier

See merge request fsl/pytreat-practicals-2020!27

talks/matlab_vs_python/bloch/bloch.m

@@ -18,7 +18,7 @@ subplot(2,1,2);
 plot(pulseq(:,3));
 ylabel('Gradient');
 
-% Integrate ODE
+%% Integrate ODE
 T1 = 1500;
 T2 = 50;
 t0 = 0;
@@ -27,20 +27,20 @@ dt = 0.005;
 M0 = [0; 0; 1];
 [t, M] = ode45(@(t,M)bloch_ode(t, M, T1, T2), linspace(t0, t1, (t1-t0)/dt), M0);
 
-% Plot Results
+%% Plot Results
 % create figure
 figure();hold on;
 
 % plot x, y and z components of Magnetisation
-plot(t, M(:,1));
-plot(t, M(:,2));
-plot(t, M(:,3));
+plot(t, M(:,1), 'linewidth', 2);
+plot(t, M(:,2), 'linewidth', 2);
+plot(t, M(:,3), 'linewidth', 2);
 
 % add legend and grid
 legend({'Mx','My','Mz'});
 grid on;
 
-% define the bloch equation
+%% define the bloch equation
 function dM = bloch_ode(t, M, T1, T2)
     % get effective B-field at time t
     B   =   B_eff(t);                               
@@ -51,7 +51,7 @@ function dM = bloch_ode(t, M, T1, T2)
             M(1)*B(2) - M(2)*B(1) - (M(3)-1)/T1];   
 end
 
-% define effective B-field
+%% define effective B-field
 function b = B_eff(t)
     % Do nothing for 0.25 ms
     if t < 0.25             

talks/matlab_vs_python/partial_fourier/partial_fourier.ipynb

@@ -64,7 +64,8 @@
     "In this section:\n",
     "- np.random.randn\n",
     "- np.fft\n",
-    "- 0-based indexing"
+    "- 0-based indexing\n",
+    "- image plotting"
    ]
   },
   {
@@ -80,7 +81,11 @@
     "y = np.fft.fftshift(np.fft.fft2(img), axes=0) + n\n",
     "\n",
     "# set bottom 24/96 lines to 0\n",
-    "y[72:,:] = 0"
+    "y[72:,:] = 0\n",
+    "\n",
+    "# show sampling\n",
+    "_, ax = plt.subplots()\n",
+    "ax.imshow(np.log(np.abs(np.fft.fftshift(y, axes=1))))"
    ]
   },
   {
@@ -96,7 +101,8 @@
     "In this section:\n",
     "- np.pad\n",
     "- np.hanning\n",
-    "- reshaping 1D array to 2D array using np.newaxis (or None)"
+    "- reshaping 1D array to 2D array using np.newaxis (or None)\n",
+    "- subplots with titles"
    ]
   },
   {
@@ -117,7 +123,14 @@
     "low = np.fft.ifft2(np.fft.ifftshift(y*filt, axes=0))\n",
     "\n",
     "# get phase image\n",
-    "phs = np.exp(1j*np.angle(low))"
+    "phs = np.exp(1j*np.angle(low))\n",
+    "\n",
+    "# show phase estimate alongside true phase\n",
+    "_, ax = plt.subplots(1,2)\n",
+    "ax[0].imshow(np.angle(img))\n",
+    "ax[0].set_title('True image phase')\n",
+    "ax[1].imshow(np.angle(phs))\n",
+    "ax[1].set_title('Estimated phase')"
    ]
   },
   {
@@ -190,7 +203,7 @@
     "\n",
     "In this section:\n",
     "- print formatted strings to standard output\n",
-    "- plotting, with min/max scales\n",
+    "- 2D subplots with min/max scales, figure size\n",
     "- np.sum sums over all elements by default"
    ]
   },
@@ -212,15 +225,17 @@
     "print(f'RMSE for POCS recon: {err_pocs}')\n",
     "\n",
     "# plot both recons side-by-side\n",
-    "_, ax = plt.subplots(1,2,figsize=(16,16))\n",
+    "_, ax = plt.subplots(2,2,figsize=(16,16))\n",
     "\n",
     "# plot zero-filled\n",
-    "ax[0].imshow(np.abs(zf), vmin=0, vmax=1)\n",
-    "ax[0].set_title('Zero-filled')\n",
+    "ax[0,0].imshow(np.abs(zf), vmin=0, vmax=1)\n",
+    "ax[0,0].set_title('Zero-filled')\n",
+    "ax[1,0].plot(np.abs(zf[:,47]))\n",
     "\n",
     "# plot POCS\n",
-    "ax[1].imshow(est, vmin=0, vmax=1)\n",
-    "ax[1].set_title('POCS recon')"
+    "ax[0,1].imshow(est, vmin=0, vmax=1)\n",
+    "ax[0,1].set_title('POCS recon')\n",
+    "ax[1,1].plot(np.abs(est[:,47]))"
    ]
   }
  ],
 %% Cell type:markdown id: tags:
 
 Imports
 
 %% Cell type:code id: tags:
 
 ``` python
 import h5py
 import matplotlib.pyplot as plt
 import numpy as np
 ```
 
 %% Cell type:markdown id: tags:
 
 # Load data
 
 Load complex image data from MATLAB mat-file (v7.3 or later), which is actually an HDF5 format
 
 Complex data is loaded as a (real, imag) tuple, so it neds to be explicitly converted to complex double
 
 In this section:
 - using h5py module
 - np.transpose
 - 1j as imaginary constant
 
 %% Cell type:code id: tags:
 
 ``` python
 # get hdf5 object for the mat-file
 h = h5py.File('data.mat','r')
 
 # get img variable from the mat-file
 dat = h.get('img')
 
 # turn array of (real, imag) tuples into an array of complex doubles
 # transpose to keep data in same orientation as MATLAB
 img = np.transpose(dat['real'] + 1j*dat['imag'])
 ```
 
 %% Cell type:markdown id: tags:
 
 # 6/8 Partial Fourier sampling
 
 Fourier transform the image to get k-space data, and add complex Gaussian noise
 
 To simulate 6/8 Partial Fourier sampling, zero out the bottom 1/4 of k-space
 
 In this section:
 - np.random.randn
 - np.fft
 - 0-based indexing
+- image plotting
 
 %% Cell type:code id: tags:
 
 ``` python
 # generate normally-distributed complex noise
 n = np.random.randn(96,96) + 1j*np.random.randn(96,96)
 
 # Fourier transform the image and add noise
 y = np.fft.fftshift(np.fft.fft2(img), axes=0) + n
 
 # set bottom 24/96 lines to 0
 y[72:,:] = 0
+
+# show sampling
+_, ax = plt.subplots()
+ax.imshow(np.log(np.abs(np.fft.fftshift(y, axes=1))))
 ```
 
 %% Cell type:markdown id: tags:
 
 # Estimate phase
 
 Filter the k-space data and extract a low-resolution phase estimate
 
 Filtering can help reduce ringing in the phase image
 
 In this section:
 - np.pad
 - np.hanning
 - reshaping 1D array to 2D array using np.newaxis (or None)
+- subplots with titles
 
 %% Cell type:code id: tags:
 
 ``` python
 # create zero-padded hanning filter for ky-filtering
 filt = np.pad(np.hanning(48),24,'constant')
 
 # reshape 1D array into 2D array
 filt = filt[:,np.newaxis]
 # or
 # filt = filt[:,None]
 
 # generate low-res image with inverse Fourier transform
 low = np.fft.ifft2(np.fft.ifftshift(y*filt, axes=0))
 
 # get phase image
 phs = np.exp(1j*np.angle(low))
+
+# show phase estimate alongside true phase
+_, ax = plt.subplots(1,2)
+ax[0].imshow(np.angle(img))
+ax[0].set_title('True image phase')
+ax[1].imshow(np.angle(phs))
+ax[1].set_title('Estimated phase')
 ```
 
 %% Cell type:markdown id: tags:
 
 # POCS reconstruction
 
 Perform the projection-onto-convex-sets (POCS) partial Fourier reconstruction method.
 
 POCS is an iterative scheme estimates the reconstructed image as any element in the intersection of the following two (convex) sets:
 1. Set of images consistent with the measured data
 2. Set of images that are non-negative real
 
 This requires prior knowledge of the image phase (hence the estimate above), and it works because although we have less than a full k-space of measurements, we're now only estimating half the number of free parameters (real values only, instead of real + imag), and we're no longer under-determined. Equivalently, consider the fact that real-valued images have conjugate symmetric k-spaces, so we only require half of k-space to reconstruct our image.
 
 In this section:
 - np.zeros
 - range() builtin
 - point-wise multiplication (*)
 - np.fft operations default to last axis, not first
 - np.maximum vs np.max
 
 %% Cell type:code id: tags:
 
 ``` python
 # initialise image estimate to be zeros
 est = np.zeros((96,96))
 
 # set the number of iterations
 iters = 10
 
 # each iteration cycles between projections
 for i in range(iters):
 # projection onto data-consistent set:
     # use a-priori phase to get complex image
     est = est*phs
 
     # Fourier transform to get k-space
     est = np.fft.fftshift(np.fft.fft2(est), axes=0)
 
     # replace data with measured lines
     est[:72,:] = y[:72,:]
 
     # inverse Fourier transform to get back to image space
     est = np.fft.ifft2(np.fft.ifftshift(est, axes=0))
 
 # projection onto non-negative reals:
     # remove a-priori phase
     est = est*np.conj(phs)
 
     # get real part
     est = np.real(est)
 
     # ensure output is non-negative
     est = np.maximum(est, 0)
 ```
 
 %% Cell type:markdown id: tags:
 
 # Display error and plot reconstruction
 
 The POCS reconstruction is compared to a zero-filled reconstruction (i.e., where the missing data is zeroed prior to inverse Fourier Transform)
 
 In this section:
 - print formatted strings to standard output
-- plotting, with min/max scales
+- 2D subplots with min/max scales, figure size
 - np.sum sums over all elements by default
 
 %% Cell type:code id: tags:
 
 ``` python
 # compute zero-filled recon
 zf = np.fft.ifft2(np.fft.ifftshift(y, axes=0))
 
 # compute rmse for zero-filled and POCS recon
 err_zf = np.sqrt(np.sum(np.abs(zf - img)**2))
 err_pocs = np.sqrt(np.sum(np.abs(est*phs - img)**2))
 
 # print errors
 print(f'RMSE for zero-filled recon: {err_zf}')
 print(f'RMSE for POCS recon: {err_pocs}')
 
 # plot both recons side-by-side
-_, ax = plt.subplots(1,2,figsize=(16,16))
+_, ax = plt.subplots(2,2,figsize=(16,16))
 
 # plot zero-filled
-ax[0].imshow(np.abs(zf), vmin=0, vmax=1)
-ax[0].set_title('Zero-filled')
+ax[0,0].imshow(np.abs(zf), vmin=0, vmax=1)
+ax[0,0].set_title('Zero-filled')
+ax[1,0].plot(np.abs(zf[:,47]))
 
 # plot POCS
-ax[1].imshow(est, vmin=0, vmax=1)
-ax[1].set_title('POCS recon')
+ax[0,1].imshow(est, vmin=0, vmax=1)
+ax[0,1].set_title('POCS recon')
+ax[1,1].plot(np.abs(est[:,47]))
 ```

talks/matlab_vs_python/partial_fourier/partial_fourier.m

@@ -15,6 +15,10 @@ y = fftshift(fft2(img),1) + n;
 % set bottom 24/96 lines to 0
 y(73:end,:) = 0;
 
+% show sampling
+figure();
+imshow(log(abs(fftshift(y,2))), [], 'colormap', jet)
+
 %% Estimate phase
 % create zero-padded hanning filter for ky-filtering
 filt = padarray(hann(48),24);
@@ -25,6 +29,16 @@ low = ifft2(ifftshift(y.*filt,1));
 % get phase image
 phs = exp(1j*angle(low));
 
+% show phase estimate alongside true phase
+figure();
+subplot(1,2,1);
+imshow(angle(img), [-pi,pi], 'colormap', hsv);
+title('True image phase');
+
+subplot(1,2,2);
+imshow(angle(phs), [-pi,pi], 'colormap', hsv)
+title('Estimated phase');
+
 %% POCS reconstruction
 % initialise image estimate to be zeros
 est = zeros(96);
@@ -74,11 +88,15 @@ fprintf(1, 'RMSE for POCS recon: %f\n', err_pocs);
 figure();
 
 % plot zero-filled
-subplot(1,2,1);
+subplot(2,2,1);
 imshow(abs(zf), [0 1]);
 title('Zero-Filled');
+subplot(2,2,3);
+plot(abs(zf(:,48)), 'linewidth', 2);
 
 % plot POCS
-subplot(1,2,2);
+subplot(2,2,2);
 imshow(est, [0 1]);
 title('POCS recon');
+subplot(2,2,4);
+plot(abs(est(:,48)), 'linewidth', 2);