Added new tutorial on MRI imaging

Author: mrava87

testdata/brainphantom.mat

@@ -0,0 +1,143 @@
+r"""
+MRI Imaging and Segmentation of Brain
+=====================================
+This tutorial considers the well-known problem of MRI imaging, where given the availability of a sparsely sampled
+KK-spectrum, one is tasked to reconstruct the underline spatial luminosity of an object under observation. In
+this specific case, we will be using an example from `Corona et al., 2019, Enhancing joint reconstruction and
+segmentation with non-convex Bregman iteration`.
+
+We first consider the imaging problem defined by the following cost functuon
+
+.. math::
+    \argmin_\mathbf{x} \|\mathbf{y}-\mathbf{Ax}\|_2^2 + \alpha TV(\mathbf{x})
+
+where the operator :math:`\mathbf{A}` performs a 2D-Fourier transform followed by sampling of the KK plane, :math:`\mathbf{x}`
+is the object of interest and :math:`\mathbf{y}` the set of available Fourier coefficients.
+
+Once the model is reconstructed, we solve a second inverse problem with the aim of segmenting the retrieved object into
+:math:`N` classes of different luminosity.
+"""
+import numpy as np
+import matplotlib.pyplot as plt
+import pylops
+from scipy.io import loadmat
+
+import pyproximal
+
+plt.close('all')
+np.random.seed(10)
+
+###############################################################################
+# Let's start by loading the data and the sampling mask
+mat = loadmat('../testdata/brainphantom.mat')
+mat1 = loadmat('../testdata/spiralsampling.mat')
+gt = mat['gt']
+seggt = mat['gt_seg']
+sampling = mat1['samp']
+sampling1 = np.fft.ifftshift(sampling)
+
+fig, axs = plt.subplots(1, 3, figsize=(15, 6))
+axs[0].imshow(gt, cmap='gray')
+axs[0].axis('tight')
+axs[0].set_title("Object")
+axs[1].imshow(seggt, cmap='Accent')
+axs[1].axis('tight')
+axs[1].set_title("Segmentation")
+axs[2].imshow(sampling, cmap='gray')
+axs[2].axis('tight')
+axs[2].set_title("Sampling mask")
+plt.tight_layout()
+
+###############################################################################
+# We can now create the MRI operator
+Fop = pylops.signalprocessing.FFT2D(dims=gt.shape)
+Rop = pylops.Restriction(gt.size, np.where(sampling1.ravel() == 1)[0],
+                         dtype=np.complex128)
+Dop = Rop * Fop
+
+# KK spectrum
+GT = Fop * gt.ravel()
+GT = GT.reshape(gt.shape)
+
+# Data (Masked KK spectrum)
+d = Dop * gt.ravel()
+
+fig, axs = plt.subplots(1, 2, figsize=(12, 6))
+axs[0].imshow(np.fft.fftshift(np.abs(GT)), vmin=0, vmax=1, cmap='gray')
+axs[0].axis('tight')
+axs[0].set_title("Spectrum")
+axs[1].plot(np.fft.fftshift(np.abs(d)), 'k', lw=2)
+axs[1].axis('tight')
+axs[1].set_title("Masked Spectrum")
+plt.tight_layout()
+
+###############################################################################
+# Let's try now to reconstruct the object from its measurement. The simplest
+# approach entails simply filling the missing values in the KK spectrum with
+# zeros and applying inverse FFT.
+
+GTzero = sampling1 * GT
+gtzero = (Fop.H * GTzero).real
+
+fig, axs = plt.subplots(1, 2, figsize=(12, 6))
+axs[0].imshow(gt, cmap='gray')
+axs[0].axis('tight')
+axs[0].set_title("True Object")
+axs[1].imshow(gtzero, cmap='gray')
+axs[1].axis('tight')
+axs[1].set_title("Zero-filling Object")
+plt.tight_layout()
+
+###############################################################################
+# We can now do better if we introduce some prior information in the form of
+# TV on the solution
+
+with pylops.disabled_ndarray_multiplication():
+    sigma = 0.04
+    l1 = pyproximal.proximal.L21(ndim=2)
+    l2 = pyproximal.proximal.L2(Op=Dop, b=d.ravel(), niter=50, warm=True)
+    Gop = sigma * pylops.Gradient(dims=gt.shape, edge=True, kind='forward', dtype=np.complex)
+
+    L = sigma ** 2 * 8
+    tau = .99 / np.sqrt(L)
+    mu = .99 / np.sqrt(L)
+
+    gtpd = pyproximal.optimization.primaldual.PrimalDual(l2, l1, Gop, x0=np.zeros(gt.size),
+                                                         tau=tau, mu=mu, theta=1.,
+                                                         niter=100, show=True)
+    gtpd = np.real(gtpd.reshape(gt.shape))
+
+fig, axs = plt.subplots(1, 2, figsize=(12, 6))
+axs[0].imshow(gt, cmap='gray')
+axs[0].axis('tight')
+axs[0].set_title("True Object")
+axs[1].imshow(gtpd, cmap='gray')
+axs[1].axis('tight')
+axs[1].set_title("TV-reg Object")
+plt.tight_layout()
+
+###############################################################################
+# Finally we segment our reconstructed model into 4 classes.
+
+cl = np.array([0.01, 0.43, 0.65, 0.8])
+ncl = len(cl)
+segpd_prob, segpd = \
+    pyproximal.optimization.segmentation.Segment(gtpd, cl, 1., 0.001,
+                                                 niter=10, show=True,
+                                                 kwargs_simplex=dict(engine='numba'))
+
+fig, axs = plt.subplots(1, 2, figsize=(12, 6))
+axs[0].imshow(seggt, cmap='Accent')
+axs[0].axis('tight')
+axs[0].set_title("True Classes")
+axs[1].imshow(segpd, cmap='Accent')
+axs[1].axis('tight')
+axs[1].set_title("Estimated Classes")
+plt.tight_layout()
+
+fig, axs = plt.subplots(1, 4, figsize=(15, 6))
+for i, ax in enumerate(axs):
+    ax.imshow(segpd_prob[:, i].reshape(gt.shape), cmap='Reds')
+    axs[i].axis('tight')
+    axs[i].set_title(f"Class {i}")
+plt.tight_layout()