Estimate Shifts w/ Symmetry

src/aspire/abinitio/commonline_base.py

@@ -212,7 +212,7 @@ def estimate_shifts(self):
         """
 
         # Generate approximated shift equations from estimated rotations
-        shift_equations, shift_b = self._get_shift_equations_approx()
+        shift_equations, shift_b = self._get_shift_equations()
 
         # Solve the linear equation, optionally printing numerical debug details.
         show = False
@@ -241,6 +241,22 @@ def estimate(self, **kwargs):
 
         return self.rotations, self.shifts
 
+    def _get_shift_equations(self):
+        """
+        Generate shift equations from the estimated rotations.
+
+        Dispatches to the legacy asymmetric shift-equation construction for C1
+        sources, and to the symmetry-expanded construction for sources with
+        nontrivial symmetry. This keeps the C1 code path unchanged while allowing
+        symmetric molecules to contribute multiple common-line equations per image
+        pair without introducing additional shift unknowns.
+
+        :return: The sparse shift-equation matrix and right-hand side vector.
+        """
+        if str(self.src.symmetry_group) == "C1":
+            return self._get_shift_equations_approx()
+        return self._get_shift_equations_approx_symmetric()
+
     def _get_shift_equations_approx(self):
         """
         Generate approximated shift equations from estimated rotations
@@ -380,15 +396,160 @@ def _get_shift_equations_approx(self):
 
         return shift_equations, shift_b
 
-    def _estimate_num_shift_equations(self, n_img):
+    def _get_shift_equations_approx_symmetric(self):
+        """
+        Generate symmetry-expanded approximate shift equations from estimated rotations.
+
+        For each sampled image pair, this method computes the common lines induced by
+        the first image rotation and every symmetry-transformed copy of the second
+        image rotation. Each symmetry copy contributes one shift equation involving
+        the same two 2D image-shift unknowns, adding constraints without duplicating
+        images or introducing independent shift variables for symmetry copies.
+
+        :return: The sparse shift-equation matrix and right-hand side vector.
+        """
+
+        n_theta_half = self.n_theta // 2
+        n_img = self.n_img
+        pf = self.pf.copy()
+
+        # `estimate_shifts()` requires that rotations have already been estimated.
+        rotations = self.rotations
+
+        # Apply symmetry group to rotations.
+        # Symmetry copies add more equations for the same per-image shift unknowns.
+        sym_rots = self.src.symmetry_group.matrices.astype(self.dtype, copy=False)
+        n_sym = len(sym_rots)
+
+        # Estimate base image-pair equations, then expand each pair by symmetry.
+        n_pair_equations = self._estimate_num_shift_equations(n_img, n_sym=n_sym)
+        n_equations = n_pair_equations * n_sym
+
+        # Allocate local variables for estimating 2D shifts based on the estimated number
+        # of equations. The shift equations are represented using a sparse matrix,
+        # since each row in the system contains four non-zeros (as it involves
+        # exactly four unknowns). The variables below are used to construct
+        # this sparse system. The k'th non-zero element of the equations matrix
+        # is stored at index (shift_i(k),shift_j(k)).
+        shift_i = np.zeros((n_equations, 4), dtype=self.dtype)
+        shift_j = np.zeros((n_equations, 4), dtype=self.dtype)
+        shift_eq = np.zeros((n_equations, 4), dtype=self.dtype)
+        shift_b = np.zeros(n_equations, dtype=self.dtype)
+
+        # Prepare the shift phases to try and generate filter for common-line detection
+        # The shift phases are pre-defined in a range of max_shift that can be
+        # applied to maximize the common line calculation. The common-line filter
+        # is also applied to the radial direction for easier detection.
+        r_max = pf.shape[2]
+        _, shift_phases, h = _generate_shift_phase_and_filter(
+            r_max, self.offsets_max_shift, self.offsets_shift_step, self.dtype
+        )
+
+        d_theta = np.pi / n_theta_half
+
+        # Generate base [i, j] image pairs before symmetry expansion.
+        idx_i, idx_j = self._generate_index_pairs(n_pair_equations)
+
+        # Filter, normalize, and conjugate all rays once instead of once per equation.
+        # Conjugation uses ray from opposite side of origin.
+        # Correpsonds to `freqs` convention in PFT,
+        #   where the legacy code used a negated frequency grid.
+        pf = np.conj(self._apply_filter_and_norm("ijk, k -> ijk", pf, r_max, h))
+
+        # Iterate over image pairs; each iteration fills one block of n_sym
+        # symmetry-induced common-line shift equations.
+        for pair_eq_idx in range(n_pair_equations):
+            i = idx_i[pair_eq_idx]
+            j = idx_j[pair_eq_idx]
+            rows = pair_eq_idx + np.arange(n_sym) * n_pair_equations
+
+            # Common lines for Ri against all symmetry copies g @ Rj.
+            Rjs = sym_rots @ rotations[j]
+            c_ij, c_ji = self._get_cl_indices_from_rot_pairs(
+                rotations[i], Rjs, n_theta_half
+            )
+
+            # Extract the Fourier rays that correspond to the common lines
+            pf_i = pf[i, c_ij]  # shape (n_sym, n_rad)
+
+            # Track which symmetry-induced rays in image j use the opposite ray direction.
+            is_pf_j_flipped = c_ji >= n_theta_half
+            pf_j = pf[j, c_ji % n_theta_half]
+
+            # Apply candidate 1D shifts to all symmetry-induced rays for this image pair.
+            pf_i_stack = pf_i[:, :, None] * shift_phases.T[None, :, :]
+            pf_i_flipped_stack = np.conj(pf_i)[:, :, None] * shift_phases.T[None, :, :]
+
+            c1 = 2 * np.sum(pf_i_stack.conj() * pf_j[:, :, None], axis=1).real
+            c2 = 2 * np.sum(pf_i_flipped_stack.conj() * pf_j[:, :, None], axis=1).real
+
+            # Pick the best candidate shift for each symmetry-induced ray pair.
+            sidx1 = np.argmax(c1, axis=1)
+            sidx2 = np.argmax(c2, axis=1)
+
+            score1 = c1[np.arange(n_sym), sidx1]
+            score2 = c2[np.arange(n_sym), sidx2]
+            sidx = np.where(score1 > score2, sidx1, sidx2)
+            dx = -self.offsets_max_shift + sidx * self.offsets_shift_step
+
+            # Angle(s) of common ray(s) in image i
+            shift_alpha = c_ij * d_theta
+            # Angle(s) of common ray(s) in image j
+            shift_beta = c_ji * d_theta
+            # Row indices to construct the sparse equations
+            shift_i[rows] = rows[:, None]
+            # All symmetry rows for this pair use the same set of image shift unknowns.
+            shift_j[rows] = [2 * i, 2 * i + 1, 2 * j, 2 * j + 1]
+            # Right hand side of the current equation(s)
+            shift_b[rows] = dx
+
+            # Initialize shift equation block.
+            # One four-coefficient equation row per symmetry-induced common line.
+            eq = np.empty((n_sym, 4), dtype=self.dtype)
+
+            # Compute the coefficients of the current block of equations.
+            not_flipped = ~is_pf_j_flipped
+            eq[not_flipped] = np.column_stack(
+                (
+                    np.sin(shift_alpha[not_flipped]),
+                    np.cos(shift_alpha[not_flipped]),
+                    -np.sin(shift_beta[not_flipped]),
+                    -np.cos(shift_beta[not_flipped]),
+                )
+            )
+
+            beta_flipped = shift_beta[is_pf_j_flipped] - np.pi
+            eq[is_pf_j_flipped] = np.column_stack(
+                (
+                    -np.sin(shift_alpha[is_pf_j_flipped]),
+                    -np.cos(shift_alpha[is_pf_j_flipped]),
+                    -np.sin(beta_flipped),
+                    -np.cos(beta_flipped),
+                )
+            )
+
+            shift_eq[rows] = eq
+
+        # create sparse matrix object only containing non-zero elements
+        shift_equations = sparse.csr_matrix(
+            (shift_eq.flatten(), (shift_i.flatten(), shift_j.flatten())),
+            shape=(n_equations, 2 * n_img),
+            dtype=self.dtype,
+        )
+
+        return shift_equations, shift_b
+
+    def _estimate_num_shift_equations(self, n_img, n_sym=1):
         """
         Estimate total number of shift equations in images
 
         The function computes total number of shift equations based on
         number of images and preselected memory factor.
 
         :param n_img:  The total number of input images
-        :return: Estimated number of shift equations
+        :param n_sym: Number of symmetry-expanded rows generated per sampled image pair.
+            Defaults to 1 for the legacy asymmetric path.
+        :return: Number of base image-pair equations to sample before any symmetry expansion.
         """
         # Number of equations that will be used to estimation the shifts
         n_equations_total = int(np.ceil(n_img * (self.n_check - 1) / 2))
@@ -397,7 +558,7 @@ def _estimate_num_shift_equations(self, n_img):
         # This ignores the sparsity of the system, since backslash seems to
         # ignore it.
         memory_total = self.offsets_equations_factor * (
-            n_equations_total * 2 * n_img * self.dtype.itemsize
+            n_equations_total * n_sym * 2 * n_img * self.dtype.itemsize
         )
 
         if memory_total < (self.offsets_max_memory * 10**6):
@@ -458,6 +619,29 @@ def _get_cl_indices(self, rotations, i, j, n_theta):
 
         return c_ij, c_ji
 
+    def _get_cl_indices_from_rot_pairs(self, Ri, Rjs, n_theta):
+        """
+        Get common-line indices for one rotation Ri and multiple rotations Rjs.
+        """
+        ell = 2 * n_theta
+
+        # Match _get_cl_indices, which calls
+        # Rotation(np.stack((Ri, Rj))).invert().common_lines(i, j, ...).
+        ut = np.swapaxes(Rjs, -1, -2) @ Ri
+
+        alpha_ij = np.arctan2(ut[:, 2, 0], -ut[:, 2, 1]) + np.pi
+        alpha_ji = np.arctan2(-ut[:, 0, 2], ut[:, 1, 2]) + np.pi
+
+        c_ij = np.mod(np.round(alpha_ij * ell / (2 * np.pi)), ell).astype(int)
+        c_ji = np.mod(np.round(alpha_ji * ell / (2 * np.pi)), ell).astype(int)
+
+        mask = c_ij >= n_theta
+        c_ij[mask] -= n_theta
+        c_ji[mask] -= n_theta
+        c_ji[c_ji < 0] += ell
+
+        return c_ij, c_ji
+
     def _apply_filter_and_norm(self, subscripts, pf, r_max, h):
         """
         Apply common line filter and normalize each ray

tests/test_estimate_shifts.py

@@ -0,0 +1,162 @@
+import numpy as np
+import pytest
+from scipy import sparse
+
+from aspire.abinitio import Orient3D
+from aspire.source import Simulation
+from aspire.volume import (
+    AsymmetricVolume,
+    CnSymmetricVolume,
+    DnSymmetricVolume,
+    OSymmetricVolume,
+    TSymmetricVolume,
+)
+
+DTYPES = [np.float64]
+RES = [89]
+SYMMETRIES = [
+    None,
+    "C2",
+    "C3",
+    "C4",
+    "C5",
+    "C6",
+    "D2",
+    "D3",
+    "D4",
+    "D5",
+    "D6",
+    "D7",
+    "T",
+    "O",
+]
+N_IMGS = 100
+SEED = 1980
+
+
+@pytest.fixture(params=RES, ids=lambda x: f"resolution={x}", scope="module")
+def resolution(request):
+    return request.param
+
+
+@pytest.fixture(params=SYMMETRIES, ids=lambda x: f"symmetry={x}", scope="module")
+def symmetry(request):
+    return request.param
+
+
+@pytest.fixture(params=DTYPES, ids=lambda x: f"dtype={x}", scope="module")
+def dtype(request):
+    return request.param
+
+
+@pytest.fixture(scope="module")
+def volume(resolution, symmetry, dtype):
+    if symmetry is None:
+        return AsymmetricVolume(
+            L=resolution, C=1, K=25, dtype=dtype, seed=SEED
+        ).generate()
+
+    if symmetry.startswith("C"):
+        order = int(symmetry[1:])
+        return CnSymmetricVolume(
+            L=resolution, C=1, order=order, K=25, dtype=dtype, seed=SEED
+        ).generate()
+
+    if symmetry.startswith("D"):
+        order = int(symmetry[1:])
+        return DnSymmetricVolume(
+            L=resolution, C=1, order=order, K=25, dtype=dtype, seed=SEED
+        ).generate()
+
+    if symmetry == "T":
+        return TSymmetricVolume(
+            L=resolution, C=1, K=25, dtype=dtype, seed=SEED
+        ).generate()
+
+    if symmetry == "O":
+        return OSymmetricVolume(
+            L=resolution, C=1, K=25, dtype=dtype, seed=SEED
+        ).generate()
+
+
+@pytest.fixture(scope="module")
+def estimator(volume):
+    """
+    Build a simulated source and use ground-truth rotations so this test isolates
+    shift-equation construction and shift recovery from orientation estimation error.
+    """
+    offset_scale = 1.5  # standard deviation of shifts
+    offsets = np.random.normal(scale=offset_scale, size=(N_IMGS, 2))
+
+    src = Simulation(
+        n=N_IMGS,
+        vols=volume,
+        amplitudes=1,
+        offsets=offsets,
+        seed=SEED,
+    ).cache()
+
+    orient_est = Orient3D(src)
+    orient_est.rotations = src.rotations
+
+    return orient_est
+
+
+def test_estimate_shifts(estimator):
+    """
+    Compare estimated shifts to ground truth after removing the nullspace of
+    the shift equation matrix. See the following publication for more info on
+    measuring shift estimation error:
+
+    Y. Shkolnisky and A. Singer,
+    Center of Mass Operators for Cryo-EM - Theory and Implementation,
+    Modeling Nanoscale Imaging in Electron Microscopy,
+    T. Vogt, W. Dahmen, and P. Binev (Eds.)
+    Nanostructure Science and Technology Series,
+    Springer, 2012, pp. 147–177
+    """
+    # Build the sparse common-line shift system Ax = b and solve it directly,
+    # matching the solver used by estimate_shifts().
+    A, b = estimator._get_shift_equations()
+    lsqr_result = sparse.linalg.lsqr(A, b, atol=1e-8, btol=1e-8, iter_lim=100)
+    x_est = lsqr_result[0]
+
+    # Convert Simulation offsets to the internal LSQR convention:
+    # estimate_shifts returns -x_est.reshape(n, 2)[:, ::-1].
+    x_ref_internal = (-estimator.src.offsets[:, ::-1]).reshape(-1)
+
+    # Use the SVD to separate the constrained directions from the nullspace,
+    # which corresponds to global 3D translation ambiguity.
+    _, s, Vt = np.linalg.svd(A.toarray(), full_matrices=False)
+
+    # Estimate the effective rank of A and keep the constrained directions.
+    sv_tol = 1e-2
+    rank = int(np.sum(s > sv_tol * s[0]))
+    V_nonnull = Vt[:rank].T
+
+    # Compute relative error after projecting out the nullspace.
+    num = np.linalg.norm(V_nonnull.T @ (x_ref_internal - x_est))
+    den = np.linalg.norm(V_nonnull.T @ x_ref_internal)
+    projected_rel_err = num / den
+
+    # Check the shift error is within 15% of the reference shift norm.
+    np.testing.assert_array_less(projected_rel_err, 0.15)
+
+    # The projected relative error follows the legacy diagnostic, but it is not
+    # a pixel-scale quantity. Below we check the same solution after aligning away
+    # the nullspace component so the error is easier to interpret.
+    V_null = Vt[rank:].T
+
+    # Add the nullspace component to the estimate before comparing directly
+    # against the reference shifts.
+    x_err = x_ref_internal - x_est
+    x_est_aligned = x_est + V_null @ (V_null.T @ x_err)
+
+    # Convert back to ASPIRE shift convention and compute per-image Euclidean
+    # shift error in pixels.
+    est_shifts_aligned = -x_est_aligned.reshape(estimator.src.n, 2)[:, ::-1]
+    per_img_err = np.linalg.norm(estimator.src.offsets - est_shifts_aligned, axis=1)
+    mean_aligned_px = per_img_err.mean()
+
+    # Check that aligned estimate errors are within 0.25 pixels on average.
+    np.testing.assert_array_less(mean_aligned_px, 0.25)