Rebuild docs

.buildinfo

@@ -0,0 +1,4 @@
+# Sphinx build info version 1
+# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
+config: 8f9da24b5a9f7f395c81a6ce297fc0c7
+tags: 645f666f9bcd5a90fca523b33c5a78b7

_downloads/2db4d350e416e3adf54e890f00153125/convolution_2d.py

@@ -0,0 +1,235 @@
+"""
+Convolution in 2D
+=================
+"""
+
+
+#######################################################################################
+# Import packages
+# ---------------
+#
+# First, we import the packages we need for this example.
+
+import matplotlib.pyplot as plt
+import numpy as np
+import torch
+
+import pytorch_finufft
+
+#######################################################################################
+# Let's create a Gaussian convolutional filter as a function of x,y
+
+
+def gaussian_function(x, y, sigma=1):
+    return np.exp(-(x**2 + y**2) / (2 * sigma**2))
+
+
+#######################################################################################
+# Let's visualize this filter kernel. We will be using it to convolve with points
+# living on the $[0, 2*\pi] \times [0, 2*\pi]$ torus. So let's dimension it accordingly.
+
+shape = (128, 128)
+sigma = 0.5
+x = np.linspace(-np.pi, np.pi, shape[0], endpoint=False)
+y = np.linspace(-np.pi, np.pi, shape[1], endpoint=False)
+
+gaussian_kernel = gaussian_function(x[:, np.newaxis], y, sigma=sigma)
+
+fig, ax = plt.subplots()
+_ = ax.imshow(gaussian_kernel)
+
+#######################################################################################
+# In order for the kernel to not shift the signal, we need to place its mass at 0.
+# To do this, we ifftshift the kernel
+
+shifted_gaussian_kernel = np.fft.ifftshift(gaussian_kernel)
+
+fig, ax = plt.subplots()
+_ = ax.imshow(shifted_gaussian_kernel)
+
+
+#######################################################################################
+# Now let's create a point cloud on the torus that we can convolve with our filter
+
+N = 20
+points = np.random.rand(2, N) * 2 * np.pi
+
+fig, ax = plt.subplots()
+ax.set_xlim(0, 2 * np.pi)
+ax.set_ylim(0, 2 * np.pi)
+ax.set_aspect("equal")
+_ = ax.scatter(points[0], points[1], s=1)
+
+
+#######################################################################################
+# Now we can convolve the point cloud with the filter kernel.
+# To do this, we Fourier-transform both the point cloud and the filter kernel,
+# multiply them together, and then inverse Fourier-transform the result.
+# First we need to convert all data to torch tensors
+
+fourier_shifted_gaussian_kernel = torch.fft.fft2(
+    torch.from_numpy(shifted_gaussian_kernel)
+)
+fourier_points = pytorch_finufft.functional.finufft_type1(
+    torch.from_numpy(points), torch.ones(points.shape[1], dtype=torch.complex128), shape
+)
+
+fig, axs = plt.subplots(1, 3)
+axs[0].imshow(fourier_shifted_gaussian_kernel.real)
+axs[1].imshow(fourier_points.real, vmin=-10, vmax=10)
+_ = axs[2].imshow(
+    (
+        fourier_points
+        * fourier_shifted_gaussian_kernel
+        / fourier_shifted_gaussian_kernel[0, 0]
+    ).real,
+    vmin=-10,
+    vmax=10,
+)
+
+#######################################################################################
+# We now have two possibilities: Invert the Fourier transform on a grid, or on a point
+# cloud. We'll first invert the Fourier transform on a grid in order to be able to
+# visualize the effect of the convolution.
+
+convolved_points = torch.fft.ifft2(fourier_points * fourier_shifted_gaussian_kernel)
+
+fig, ax = plt.subplots()
+ax.imshow(convolved_points.real)
+_ = ax.scatter(
+    points[1] / 2 / np.pi * shape[0], points[0] / 2 / np.pi * shape[1], s=2, c="r"
+)
+
+#######################################################################################
+# We see that the convolution has smeared out the point cloud.
+# After a small coordinate change, we can also plot the original points
+# on the same plot as the convolved points.
+
+
+#######################################################################################
+# Next, we invert the Fourier transform on the same points as
+# our original point cloud. We will then compare this to direct evaluation
+# of the kernel on all pairwise difference vectors between the points.
+
+convolved_at_points = pytorch_finufft.functional.finufft_type2(
+    torch.from_numpy(points),
+    fourier_points * fourier_shifted_gaussian_kernel,
+    isign=1,
+).real / np.prod(shape)
+
+fig, ax = plt.subplots()
+ax.imshow(convolved_points.real)
+_ = ax.scatter(
+    points[1] / 2 / np.pi * shape[0],
+    points[0] / 2 / np.pi * shape[1],
+    s=10 * convolved_at_points,
+    c="r",
+)
+
+#######################################################################################
+# To compute the convolution directly, we need to evaluate the kernel on all pairwise
+# difference vectors between the points. Note the points that will be off the diagonal.
+# These will be due to the periodic boundary conditions of the convolution.
+
+pairwise_diffs = points[:, np.newaxis] - points[:, :, np.newaxis]
+kernel_diff_evals = gaussian_function(*pairwise_diffs, sigma=sigma)
+convolved_by_hand = kernel_diff_evals.sum(1)
+
+fig, ax = plt.subplots()
+ax.plot(convolved_at_points.numpy(), convolved_by_hand, ".")
+ax.plot([1, 3], [1, 3])
+
+relative_difference = torch.norm(
+    convolved_at_points - convolved_by_hand
+) / np.linalg.norm(convolved_by_hand)
+print(
+    "Relative difference between fourier convolution and direct convolution "
+    f"{relative_difference}"
+)
+
+
+#######################################################################################
+# Now let's see if we can learn the convolution kernel from the input and output point
+# clouds. To this end, let's first make a pytorch object that can compute a kernel
+# convolution on a point cloud.
+
+
+class FourierPointConvolution(torch.nn.Module):
+    def __init__(self, fourier_kernel_shape):
+        super().__init__()
+        self.fourier_kernel_shape = fourier_kernel_shape
+
+        self.build()
+
+    def build(self):
+        self.register_parameter(
+            "fourier_kernel",
+            torch.nn.Parameter(
+                torch.randn(self.fourier_kernel_shape, dtype=torch.complex128)
+            ),
+        )
+        # ^ think about whether we need to scale this init in some better way
+
+    def forward(self, points, values):
+        fourier_transformed_input = pytorch_finufft.functional.finufft_type1(
+            points, values, self.fourier_kernel_shape
+        )
+        fourier_convolved = fourier_transformed_input * self.fourier_kernel
+        convolved = pytorch_finufft.functional.finufft_type2(
+            points,
+            fourier_convolved,
+            isign=1,
+        ).real / np.prod(self.fourier_kernel_shape)
+        return convolved
+
+
+#######################################################################################
+# Now we can use this object in a pytorch training loop to learn the kernel from the
+# input and output point clouds. We will use the mean squared error as a loss function.
+
+fourier_point_convolution = FourierPointConvolution(shape)
+optimizer = torch.optim.AdamW(
+    fourier_point_convolution.parameters(), lr=0.005, weight_decay=0.001
+)
+
+ones = torch.ones(points.shape[1], dtype=torch.complex128)
+
+losses = []
+for i in range(10000):
+    # Make new set of points and compute forward model
+    points = np.random.rand(2, N) * 2 * np.pi
+    torch_points = torch.from_numpy(points)
+    fourier_points = pytorch_finufft.functional.finufft_type1(
+        torch.from_numpy(points), ones, shape
+    )
+    convolved_at_points = pytorch_finufft.functional.finufft_type2(
+        torch.from_numpy(points),
+        fourier_points * fourier_shifted_gaussian_kernel,
+        isign=1,
+    ).real / np.prod(shape)
+
+    # Learning step
+    optimizer.zero_grad()
+    convolved = fourier_point_convolution(torch_points, ones)
+    loss = torch.nn.functional.mse_loss(convolved, convolved_at_points)
+    losses.append(loss.item())
+    loss.backward()
+    optimizer.step()
+
+    if i % 100 == 0:
+        print(f"Iteration {i:05d}, Loss: {loss.item():1.4f}")
+
+
+fig, ax = plt.subplots()
+ax.plot(losses)
+ax.set_ylabel("Loss")
+ax.set_xlabel("Iteration")
+ax.set_yscale("log")
+
+fig, ax = plt.subplots()
+im = ax.imshow(
+    torch.real(torch.fft.fftshift(fourier_point_convolution.fourier_kernel.data))[
+        48:80, 48:80
+    ]
+)
+_ = fig.colorbar(im, ax=ax)