Parallel Beam Filtered Backprojection (FBP)

This example demonstrates 2D parallel-beam filtered backprojection on a Shepp-Logan phantom using the public call chain from diffct:

ParallelProjectorFunction.apply  -- differentiable parallel forward
ramp_filter_1d                   -- ramp filter along the detector axis
angular_integration_weights      -- per-view integration weights
parallel_weighted_backproject    -- voxel-driven FBP backprojection gather

The accompanying source file is examples/fbp_parallel.py.

Overview

Parallel-beam FBP is the classical analytical reconstruction method for 2D parallel CT. Because there is no source (rays are collimated), the pipeline is the simplest of the three analytical paths in diffct: no cosine pre-weight, no distance weight, no magnification correction. What remains is ramp filtering, a per-view angular integration weight, and a 1/(2*pi) analytical constant that the backprojection helper applies internally.

This example shows how to:

Sample a circular full-scan trajectory and a flat detector.
Generate parallel projections of a 2D Shepp-Logan phantom.
Apply ramp filtering with zero-padding and a Hann window.
Run the dedicated voxel-driven FBP gather kernel through parallel_weighted_backproject to reconstruct the image, already amplitude-calibrated.

Mathematical Background

Parallel-beam geometry

Rays at angle \(\theta\) are collimated in the direction \((\cos\theta, \sin\theta)\). A detector cell at position \(u\) on the detector axis \((-\sin\theta, \cos\theta)\) samples a ray passing through the point \((-u \sin\theta,\; u \cos\theta)\) in image-space coordinates centered on the isocenter. The corresponding line integral is the classical Radon transform

\[p(\theta, u) = \int_{-\infty}^{\infty} f\!\left(-u \sin\theta + s \cos\theta,\; u \cos\theta + s \sin\theta\right) ds.\]

The pixel \((x, y)\) projects to the detector at \(u_{xy} = -x \sin\theta + y \cos\theta\).

FBP algorithm

Ramp filtering along the detector axis:

\[p_f(\theta, u) = \mathcal{F}_u^{-1}\bigl\{\,|\omega_u|\, \mathcal{F}_u\{p(\theta, u)\}\bigr\}.\]
Voxel-driven backprojection without any distance weighting:

\[f(x, y) = \frac{1}{4\pi} \int_0^{2\pi} p_f\!\left(\theta, u_{xy}(\theta)\right) d\theta.\]

The 1/(4\pi) \cdot \int_0^{2\pi} factor is equivalent to \(\frac{1}{2\pi} \cdot \int_0^{\pi}\); diffct uses the full \([0, 2\pi)\) form with redundant_full_scan=True in angular_integration_weights to absorb the factor of 1/2.

parallel_weighted_backproject dispatches to a dedicated voxel-driven gather kernel _parallel_2d_fbp_backproject_kernel for this step. For each pixel and each view it computes \(u_{xy}\), linearly samples the filtered sinogram, and sums - no distance weighting, no magnification. The Siddon-based autograd parallel backprojector is reserved for the differentiable path used by iterative reconstruction and is not touched by this analytical pipeline.

Ramp Filter Options

ramp_filter_1d accepts the same high-precision optional kwargs as in the fan and cone examples: sample_spacing (set to detector_spacing), pad_factor (2 recommended), window (None / "ram-lak", "hann", "hamming", "cosine", "shepp-logan"), and use_rfft. The call is backward compatible so passing only (sino, dim=1) still works.

Analytical FBP scale

parallel_weighted_backproject multiplies its output by 1 / (2 * pi). This is the Fourier-convention constant that matches the |omega| ramp used by ramp_filter_1d. Unlike the fan and cone helpers there is no sdd/sid magnification factor because parallel beam has no source.

Implementation Steps

The main() function in examples/fbp_parallel.py follows the unified 8-step structure: volume geometry, source trajectory, detector geometry, CUDA transfer, forward projection, FBP pipeline (ramp filter / angular weights / voxel-gather backprojection / optional ReLU), quantitative summary (including a gradient sanity check), visualization.

Source

2D Parallel Beam FBP Example

"""Parallel-beam FBP reconstruction example.

Pipeline (matches the fan-beam and cone-beam analytical examples):

    ParallelProjectorFunction.apply  # forward projection (sinogram)
    ramp_filter_1d                   # ramp filter along detector axis
    angular_integration_weights      # per-view integration weights
    parallel_weighted_backproject    # voxel-driven FBP gather

Parallel beam has no source, so there is no cosine pre-weighting and
no distance weighting during backprojection - the FBP pipeline only
needs ramp filtering, angular weights, and the ``1/(2*pi)`` analytical
constant that ``parallel_weighted_backproject`` applies internally.
"""

import math

import matplotlib.pyplot as plt
import numpy as np
import torch
import torch.nn.functional as F

from diffct.differentiable import (
    ParallelProjectorFunction,
    angular_integration_weights,
    parallel_weighted_backproject,
    ramp_filter_1d,
)


def shepp_logan_2d(Nx, Ny):
    """2D Shepp-Logan phantom clipped to ``[0, 1]``."""
    Nx = int(Nx)
    Ny = int(Ny)
    phantom = np.zeros((Ny, Nx), dtype=np.float32)
    # (x0, y0, a, b, angle_deg, amplitude)
    ellipses = [
        (0.0,    0.0,    0.69,   0.92,    0.0,  1.0),
        (0.0,   -0.0184, 0.6624, 0.8740,  0.0, -0.8),
        (0.22,   0.0,    0.11,   0.31,  -18.0, -0.8),
        (-0.22,  0.0,    0.16,   0.41,   18.0, -0.8),
        (0.0,    0.35,   0.21,   0.25,    0.0,  0.7),
    ]
    cx = (Nx - 1) * 0.5
    cy = (Ny - 1) * 0.5
    for ix in range(Nx):
        for iy in range(Ny):
            xnorm = (ix - cx) / (Nx / 2)
            ynorm = (iy - cy) / (Ny / 2)
            val = 0.0
            for (x0, y0, a, b, angdeg, ampl) in ellipses:
                th = np.deg2rad(angdeg)
                xprime = (xnorm - x0) * np.cos(th) + (ynorm - y0) * np.sin(th)
                yprime = -(xnorm - x0) * np.sin(th) + (ynorm - y0) * np.cos(th)
                if xprime * xprime / (a * a) + yprime * yprime / (b * b) <= 1.0:
                    val += ampl
            phantom[iy, ix] = val
    return np.clip(phantom, 0.0, 1.0)


def main():
    # ------------------------------------------------------------------
    # 1. Image geometry
    # ------------------------------------------------------------------
    # ``Nx`` / ``Ny`` are the reconstruction grid size in pixels. The
    # phantom tensor has shape ``(Ny, Nx)`` (rows, cols) which matches
    # the ``(H, W)`` layout every 2D routine in diffct expects.
    Nx, Ny = 256, 256
    phantom = shepp_logan_2d(Nx, Ny)

    # ``voxel_spacing`` is the physical pixel pitch in the same units as
    # ``detector_spacing``. Only ratios matter internally.
    voxel_spacing = 1.0

    # ------------------------------------------------------------------
    # 2. Source trajectory (parallel beam on a circular orbit)
    # ------------------------------------------------------------------
    # For parallel beam the Radon inversion is periodic with period pi:
    # views at beta and beta+pi carry the same information up to a
    # flip of t. Sampling a full 2*pi range double-counts each ray and
    # we correct for it with ``redundant_full_scan=True`` in the
    # angular weights. You can equivalently use
    # ``np.linspace(0, np.pi, num_angles, endpoint=False)`` together
    # with ``redundant_full_scan=False``.
    num_angles = 360
    angles_np = np.linspace(
        0.0, 2.0 * np.pi, num_angles, endpoint=False
    ).astype(np.float32)

    # ------------------------------------------------------------------
    # 3. Detector geometry
    # ------------------------------------------------------------------
    # ``num_detectors`` is the number of detector cells. ``detector_spacing``
    # is their physical pitch. Parallel beam needs enough detector cells
    # to cover the diagonal of the field of view (approximately
    # ``sqrt(Nx^2 + Ny^2) * voxel_spacing / detector_spacing``) so that
    # no ray through the FOV ever projects outside the detector.
    num_detectors = 512
    detector_spacing = 1.0
    detector_offset = 0.0

    # ------------------------------------------------------------------
    # 4. Move everything to CUDA
    # ------------------------------------------------------------------
    if not torch.cuda.is_available():
        raise RuntimeError("This example requires CUDA.")
    device = torch.device("cuda")
    image_torch = torch.tensor(
        phantom, device=device, dtype=torch.float32, requires_grad=True
    )
    angles_torch = torch.tensor(angles_np, device=device, dtype=torch.float32)

    # ------------------------------------------------------------------
    # 5. Forward projection: image -> parallel sinogram
    # ------------------------------------------------------------------
    # ``ParallelProjectorFunction`` is the differentiable Siddon-based
    # parallel-beam forward projector. It returns a
    # (num_angles, num_detectors) sinogram. The call is autograd-aware
    # so the same function can be used inside an iterative reconstruction
    # loop (see ``iterative_reco_parallel.py``).
    sinogram = ParallelProjectorFunction.apply(
        image_torch,
        angles_torch,
        num_detectors,
        detector_spacing,
        voxel_spacing,
    )

    # ==================================================================
    # 6. FBP analytical reconstruction
    # ==================================================================

    # --- 6.1  1D ramp filter along the detector axis -----------------
    # Parallel beam does not need a cosine pre-weight (there is no fan
    # angle). The only filtering step is the 1D ramp along the
    # detector-u axis. See the fdk_cone.py example for the full list
    # of ``ramp_filter_1d`` options.
    sinogram_filt = ramp_filter_1d(
        sinogram,
        dim=1,
        sample_spacing=detector_spacing,
        pad_factor=2,
        window="hann",
    ).contiguous()

    # --- 6.2  Per-view angular integration weights -------------------
    # For a full 2*pi scan uniformly sampled, each view contributes
    # ``pi / num_angles`` to the FBP integral (the ``1/2`` redundancy
    # factor is absorbed by ``redundant_full_scan=True``).
    d_beta = angular_integration_weights(
        angles_torch, redundant_full_scan=True
    ).view(-1, 1)
    sinogram_filt = sinogram_filt * d_beta

    # --- 6.3  Voxel-driven FBP backprojection ------------------------
    # ``parallel_weighted_backproject`` dispatches to the dedicated
    # parallel-beam FBP gather kernel. For each pixel and each view it
    # computes the detector-u coordinate the pixel projects to,
    # linearly samples the filtered sinogram and accumulates. The
    # ``1/(2*pi)`` Fourier-convention constant is applied inside the
    # wrapper so the returned image is already amplitude-calibrated.
    reconstruction_raw = parallel_weighted_backproject(
        sinogram_filt,
        angles_torch,
        detector_spacing,
        Ny,
        Nx,
        voxel_spacing=voxel_spacing,
        detector_offset=detector_offset,
    )
    reconstruction = F.relu(reconstruction_raw)

    # ------------------------------------------------------------------
    # 7. Quantitative summary + a gradient sanity check
    # ------------------------------------------------------------------
    raw_loss = torch.mean((reconstruction_raw - image_torch) ** 2)
    clamped_loss = torch.mean((reconstruction - image_torch) ** 2)
    # Trigger autograd to make sure the differentiable forward still
    # flows a gradient back to ``image_torch`` after the pipeline.
    clamped_loss.backward()

    print("Parallel Beam FBP example (circular full scan):")
    print(f"  Raw MSE:              {raw_loss.item():.6f}")
    print(f"  Clamped MSE:          {clamped_loss.item():.6f}")
    print(f"  Reconstruction shape: {tuple(reconstruction.shape)}")
    print(
        "  Raw reco data range:  "
        f"[{reconstruction_raw.min().item():.4f}, {reconstruction_raw.max().item():.4f}]"
    )
    print(
        "  Clamped reco range:   "
        f"[{reconstruction.min().item():.4f}, {reconstruction.max().item():.4f}]"
    )
    print(
        "  Phantom data range:   "
        f"[{float(phantom.min()):.4f}, {float(phantom.max()):.4f}]"
    )
    print(
        "  |phantom.grad|.mean:  "
        f"{image_torch.grad.abs().mean().item():.6e}"
    )

    # ------------------------------------------------------------------
    # 8. Visualization
    # ------------------------------------------------------------------
    sinogram_cpu = sinogram.detach().cpu().numpy()
    reco_cpu = reconstruction.detach().cpu().numpy()

    plt.figure(figsize=(12, 4))
    plt.subplot(1, 3, 1)
    plt.imshow(phantom, cmap="gray")
    plt.title("Phantom")
    plt.axis("off")
    plt.subplot(1, 3, 2)
    plt.imshow(sinogram_cpu, aspect="auto", cmap="gray")
    plt.title("Parallel Sinogram")
    plt.axis("off")
    plt.subplot(1, 3, 3)
    plt.imshow(reco_cpu, cmap="gray")
    plt.title("Parallel Reconstruction")
    plt.axis("off")
    plt.tight_layout()
    plt.show()


if __name__ == "__main__":
    main()