def find_homography_dlt(
points1: torch.Tensor, points2: torch.Tensor, weights: Optional[torch.Tensor] = None
) -> torch.Tensor:
r"""Computes the homography matrix using the DLT formulation.
The linear system is solved by using the Weighted Least Squares Solution for the 4 Points algorithm.
Args:
points1 (torch.Tensor): A set of points in the first image with a tensor shape :math:`(B, N, 2)`.
points2 (torch.Tensor): A set of points in the second image with a tensor shape :math:`(B, N, 2)`.
weights (torch.Tensor, optional): Tensor containing the weights per point correspondence with a shape of
:math:`(B, N)`. Defaults to all ones.
Returns:
torch.Tensor: the computed homography matrix with shape :math:`(B, 3, 3)`.
"""
assert points1.shape == points2.shape, points1.shape
assert len(points1.shape) >= 1 and points1.shape[-1] == 2, points1.shape
assert points1.shape[1] >= 4, points1.shape
device, dtype = _extract_device_dtype([points1, points2])
eps: float = 1e-8
points1_norm, transform1 = normalize_points(points1)
points2_norm, transform2 = normalize_points(points2)
x1, y1 = torch.chunk(points1_norm, dim=-1, chunks=2) # BxNx1
x2, y2 = torch.chunk(points2_norm, dim=-1, chunks=2) # BxNx1
ones, zeros = torch.ones_like(x1), torch.zeros_like(x1)
# DIAPO 11: https://www.uio.no/studier/emner/matnat/its/nedlagte-emner/UNIK4690/v16/forelesninger/lecture_4_3-estimating-homographies-from-feature-correspondences.pdf # noqa: E501
ax = torch.cat([zeros, zeros, zeros, -x1, -y1, -ones, y2 * x1, y2 * y1, y2], dim=-1)
ay = torch.cat([x1, y1, ones, zeros, zeros, zeros, -x2 * x1, -x2 * y1, -x2], dim=-1)
A = torch.cat((ax, ay), dim=-1).reshape(ax.shape[0], -1, ax.shape[-1])
if weights is None:
# All points are equally important
A = A.transpose(-2, -1) @ A
else:
# We should use provided weights
assert len(weights.shape) == 2 and weights.shape == points1.shape[:2], weights.shape
w_diag = torch.diag_embed(weights.unsqueeze(dim=-1).repeat(1, 1, 2).reshape(weights.shape[0], -1))
A = A.transpose(-2, -1) @ w_diag @ A
try:
U, S, V = torch.svd(A)
except:
warnings.warn('SVD did not converge', RuntimeWarning)
return torch.empty((points1_norm.size(0), 3, 3), device=device, dtype=dtype)
H = V[..., -1].view(-1, 3, 3)
H = transform2.inverse() @ (H @ transform1)
H_norm = H / (H[..., -1:, -1:] + eps)
return H_norm
def find_homography_dlt(points1: torch.Tensor, points2: torch.Tensor,
weights: torch.Tensor) -> torch.Tensor:
r"""Computes the homography matrix using the DLT formulation.
The linear system is solved by using the Weighted Least Squares Solution for the 4 Points algorithm.
Args:
points1 (torch.Tensor): A set of points in the first image with a tensor shape :math:`(B, N, 2)`.
points2 (torch.Tensor): A set of points in the second image with a tensor shape :math:`(B, N, 2)`.
weights (torch.Tensor): Tensor containing the weights per point correspondence with a shape of :math:`(B, N)`.
Returns:
torch.Tensor: the computed homography matrix with shape :math:`(B, 3, 3)`.
"""
assert points1.shape == points2.shape, points1.shape
assert len(points1.shape) >= 1 and points1.shape[-1] == 2, points1.shape
assert points1.shape[1] >= 4, points1.shape
eps: float = 1e-8
points1_norm, transform1 = normalize_points(points1)
points2_norm, transform2 = normalize_points(points2)
x1, y1 = torch.chunk(points1_norm, dim=-1, chunks=2) # BxNx1
x2, y2 = torch.chunk(points2_norm, dim=-1, chunks=2) # BxNx1
ones, zeros = torch.ones_like(x1), torch.zeros_like(x1)
# DIAPO 11: https://www.uio.no/studier/emner/matnat/its/nedlagte-emner/UNIK4690/v16/forelesninger/lecture_4_3-estimating-homographies-from-feature-correspondences.pdf # noqa: E501
ax = torch.cat(
[zeros, zeros, zeros, -x1, -y1, -ones, y2 * x1, y2 * y1, y2], dim=-1)
ay = torch.cat(
[x1, y1, ones, zeros, zeros, zeros, -x2 * x1, -x2 * y1, -x2], dim=-1)
w_list = []
axy_list = []
for i in range(points1.shape[1]):
axy_list.append(ax[:, i])
axy_list.append(ay[:, i])
w_list.append(weights[:, i])
w_list.append(weights[:, i])
A = torch.stack(axy_list, dim=1)
w = torch.stack(w_list, dim=1)
# apply weights
w_diag = torch.diag_embed(w)
A = A.transpose(-2, -1) @ w_diag @ A
try:
U, S, V = torch.svd(A)
except:
return torch.empty(points1_norm.size(0), 3, 3)
H = V[..., -1].view(-1, 3, 3)
H = transform2.inverse() @ (H @ transform1)
H_norm = H / (H[..., -1:, -1:] + eps)
return H_norm
def test_mean_std(self, device, dtype):
points = torch.tensor([[[0.0, 0.0], [0.0, 2.0], [1.0, 1.0], [1.0, 3.0]]], device=device, dtype=dtype)
points_norm, _ = epi.normalize_points(points)
points_std, points_mean = torch.std_mean(points_norm, dim=1)
assert_close(points_mean, torch.zeros_like(points_mean))
assert (points_std < 2.0).all()
def test_mean_std(self, device, dtype):
points = torch.tensor([[[0., 0.], [0., 2.], [1., 1.], [1., 3.]]],
device=device,
dtype=dtype)
points_norm, trans = epi.normalize_points(points)
points_std, points_mean = torch.std_mean(points_norm, dim=1)
assert_allclose(points_mean, torch.zeros_like(points_mean))
assert (points_std < 2.).all()
def test_shape(self, batch_size, num_points, device, dtype):
B, N = batch_size, num_points
points = torch.rand(B, N, 2, device=device, dtype=dtype)
output = epi.normalize_points(points)
assert output[0].shape == (B, N, 2)
assert output[1].shape == (B, 3, 3)
def test_smoke(self, device, dtype):
points = torch.rand(1, 1, 2, device=device, dtype=dtype)
output = epi.normalize_points(points)
assert len(output) == 2
assert output[0].shape == (1, 1, 2)
assert output[1].shape == (1, 3, 3)
