Image data example

demo/demo_image/README.md

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+The file `leopard.jpg` employed in the demo `demo_image.py`.
+
+![Leopard Image](leopard.jpg)
+
+"Leopard" by Mark Kent (flamesworddragon) is licensed under CC BY-SA 2.0.
+
+* [Flickr](https://www.flickr.com/photos/31343702@N06/5710239437)
+* [openverse](https://openverse.org/image/94b35018-0cb0-4639-9dda-83e9b052c403?q=leopard)

demo/demo_image/demo_image.py

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+# Copyright (c) 2023 Nathan Sime
+# This file is part of LEoPart-X, a particle-in-cell package for DOLFIN-X
+# License: GNU Lesser GPL version 3 or any later version
+# SPDX-License-Identifier:    LGPL-3.0-or-later
+
+from mpi4py import MPI
+from petsc4py import PETSc
+
+import imageio
+import matplotlib.pyplot as plt
+import numpy as np
+
+import dolfinx
+import dolfinx.nls.petsc
+import leopart.cpp as pyleopart
+import ufl
+
+# Initial mesh matches the resolution of the input image
+mesh = dolfinx.mesh.create_unit_square(
+    MPI.COMM_WORLD, 150, 150, cell_type=dolfinx.mesh.CellType.quadrilateral,
+    ghost_mode=dolfinx.mesh.GhostMode.none)
+
+# Load the data on rank 0
+if mesh.comm.rank == 0:
+    # "Leopard" by Mark Kent (flamesworddragon) is licensed under CC BY-SA 2.0.
+    data = imageio.v2.imread("leopard.jpg")
+    img_size = (data.shape[0], data.shape[1])
+    L, H = data.shape[:2]
+    x, y = np.meshgrid(np.linspace(0.0, 1.0, L), np.linspace(0.0, 1.0, H))
+    xp = np.c_[x.ravel(), y.ravel(), np.zeros_like(x.ravel())]
+else:
+    img_size = None
+    xp = np.zeros(0, dtype=np.double)
+    data = np.zeros((0, 0, 3), dtype=np.double)
+
+p2c = [0] * xp.shape[0]
+ptcls = pyleopart.Particles(xp, p2c)
+ptcls.add_field("data", [3])
+ptcls.add_field("r", [1])
+ptcls.add_field("c", [1])
+if mesh.comm.rank == 0:
+    ptcls.field("data").data()[:] = data.reshape((-1, 3))
+    indices = np.indices(img_size)
+    ptcls.field("r").data().T[:] = indices[0].ravel()
+    ptcls.field("c").data().T[:] = indices[1].ravel()
+
+# Distribute data across processes
+ptcls.relocate_bbox(mesh._cpp_object, np.arange(len(p2c)))
+
+# Compute grayscale of data and add artificial noise
+gray_label = "gray"
+noise_label = "noise"
+ptcls.add_field(gray_label, [1])
+ptcls.add_field(noise_label, [1])
+
+ptcl_data = ptcls.field("data").data()
+grayscale = np.dot(ptcl_data, np.array([0.299, 0.587, 0.114])) / 255.0
+noise = np.random.default_rng(1).normal(0.0, 0.05, grayscale.shape)
+ptcls.field(gray_label).data().T[:] = grayscale
+ptcls.field(noise_label).data().T[:] = np.clip(grayscale + noise, 0.0, 1.0)
+
+# DG0 continuum for image data
+DG0 = dolfinx.fem.FunctionSpace(mesh, ("DG", 0))
+u_img = dolfinx.fem.Function(DG0)
+
+# FE space and problem for data diffusion
+V = dolfinx.fem.FunctionSpace(mesh, ("CG", 1))
+uh = dolfinx.fem.Function(V)
+um = dolfinx.fem.Function(V)
+uth = 0.5 * (uh + um)
+v = ufl.TestFunction(V)
+
+# FE formulation of Peronal Malik model
+sigma_d0 = dolfinx.fem.Constant(mesh, 0.0)
+pm_d = ufl.exp(
+    - ufl.inner(ufl.grad(uth), ufl.grad(uth)) / (2 * sigma_d0**2))
+pmd_expr = dolfinx.fem.Expression(pm_d, DG0.element.interpolation_points())
+pmd_dg0 = dolfinx.fem.Function(DG0)
+
+dt = dolfinx.fem.Constant(mesh, 0.0)
+F = ufl.inner((uh - um) / dt, v) * ufl.dx
+F += ufl.inner(pm_d * ufl.grad(uth), ufl.grad(v)) * ufl.dx
+
+# Nonlinear solver and parameters
+problem = dolfinx.fem.petsc.NonlinearProblem(F, uh)
+solver = dolfinx.nls.petsc.NewtonSolver(MPI.COMM_WORLD, problem)
+solver.max_it = 10
+solver.rtol = 1e-5
+
+ksp = solver.krylov_solver
+opts = PETSc.Options()
+option_prefix = ksp.getOptionsPrefix()
+opts[f"{option_prefix}ksp_type"] = "preonly"
+opts[f"{option_prefix}pc_type"] = "lu"
+opts[f"{option_prefix}pc_factor_mat_solver_type"] = "mumps"
+ksp.setFromOptions()
+
+# Utility functions for storing snapshot data
+def create_timestamp(t):
+    return f"t={t:.2e}"
+
+
+def record_snapshot(name):
+    uname = "u_" + name
+    ptcls.add_field(uname, [1])
+    pyleopart.transfer_to_particles(ptcls, ptcls.field(uname), uh._cpp_object)
+
+    pmd_name = "pmd_" + name
+    pmd_dg0.interpolate(pmd_expr)
+    ptcls.add_field(pmd_name, [1])
+    pyleopart.transfer_to_particles(
+        ptcls, ptcls.field(pmd_name), pmd_dg0._cpp_object)
+
+
+# Adaptive time stepping parameters
+dt_scl = 0.97  # Scale factor in (0.0, 1.0]
+du_max = 0.1   # Maximum permitted change between steps
+dt_min = 1e-6  # Minimum time step
+dt_max = 1.0   # Maximum time step
+euler_o = 1.0  # Euler method order
+
+# Initial parameters
+dt.value = 1e-5
+t = 0.0
+max_steps = 5
+sigma_d0.value = 1e1
+
+# Data snapshot label storage
+t_snapshots = []
+snapshot_interval = 2
+
+# Transfer discrete initial data to FE continuum
+pyleopart.transfer_to_function(
+    u_img._cpp_object, ptcls, ptcls.field(noise_label))
+uh.interpolate(u_img)
+uh.x.scatter_forward()
+
+# Main loop
+for n in range(max_steps):
+    if n > 0:
+        # Adapt time step
+        diff = uh.vector - um.vector
+        diff.abs()
+        du = diff.max()[1]
+        dt_new = dt_scl * dt.value * (du_max / (euler_o * du))
+        dt_new = min(max(dt_new, dt_min), dt_max)
+        dt.value = dt_new
+        PETSc.Sys.Print(
+            f"Adapting time step: step={n}, dt={dt.value:.3e}, t={t:.3e}")
+
+    # Solve system
+    t += dt.value
+    um.x.array[:] = uh.x.array
+    um.x.scatter_forward()
+    solver.solve(uh)
+    if n % snapshot_interval == 0 or n == max_steps - 1:
+        record_snapshot(create_timestamp(t))
+        t_snapshots += [t]
+
+# Gather data on process 0
+r = mesh.comm.gather(ptcls.field("r").data())
+c = mesh.comm.gather(ptcls.field("c").data())
+if mesh.comm.rank == 0:
+    r, c = (np.asarray(np.concatenate(x), dtype=int) for x in (r, c))
+
+
+def gather_img_rank0(field_name, dtype):
+    value_shape = ptcls.field(field_name).value_shape
+    data = mesh.comm.gather(ptcls.field(field_name).data())
+    img = None
+    if mesh.comm.rank == 0:
+        data = np.concatenate(data)
+        img = np.zeros((*img_size, value_shape[0]), dtype=dtype)
+        img[r.ravel(), c.ravel(), ...] = data
+    return img
+
+
+img_gray = gather_img_rank0(gray_label, dtype=np.double)
+img_gray_noise = gather_img_rank0(noise_label, dtype=np.double)
+img_tstep_gray = [gather_img_rank0("u_" + create_timestamp(t), dtype=np.double)
+                  for t in t_snapshots]
+img_tstep_pmd = [gather_img_rank0("pmd_" + create_timestamp(t), dtype=np.double)
+                  for t in t_snapshots]
+
+# Plot results
+if mesh.comm.rank == 0:
+    n_figs = len(t_snapshots) + 1
+    fig, axs = plt.subplots(2, n_figs, figsize=(16, 2 * 16 / n_figs))
+    axs[0, 0].imshow(img_gray, "gray")
+    axs[0, 0].set_title("Original grayscale")
+    axs[1, 0].imshow(img_gray_noise, "gray")
+    axs[1, 0].set_title("Noisy grayscale")
+
+    for i, t in enumerate(t_snapshots):
+        axs[0, i + 1].imshow(img_tstep_gray[i], "gray")
+        axs[0, i + 1].set_title(f"u({create_timestamp(t)})")
+
+        axs[1, i + 1].imshow(img_tstep_pmd[i], "gray")
+        axs[1, i + 1].set_title(rf"$A(\nabla u, {create_timestamp(t)})$")
+
+    for ax in axs.ravel():
+        ax.axis("off")
+
+    plt.tight_layout()
+    plt.show()