Fix fractional centers in CircularConvolve (#471)

* Add failing test

* Fix phase ramp

* Minor comment improvement

* Improve RNG key use

---------

Co-authored-by: Brendt Wohlberg <brendt@ieee.org>

scico/linop/_circconv.py

@@ -120,7 +120,6 @@ def __init__(
             output_dtype = snp.dtype(input_dtype)  # cannot infer from h_dft because it is complex
         else:
             fft_shape = input_shape[-self.ndims :]
-            pad = ()
             fft_axes = list(range(h.ndim - self.ndims, h.ndim))
             self.h_dft = snp.fft.fftn(h, s=fft_shape, axes=fft_axes)
             output_dtype = result_type(h.dtype, input_dtype)
@@ -140,7 +139,15 @@ def __init__(
                         offset = -snp.array(self.h_center)
                 shifts: Tuple[np.ndarray, ...] = np.ix_(
                     *tuple(
-                        np.exp(-1j * k * 2 * np.pi * np.fft.fftfreq(s))  # type: ignore
+                        np.select(
+                            # see doi:10.1109/78.700979 and doi:10.1109/LSP.2012.2191280
+                            [np.arange(s) < s / 2, np.arange(s) == s / 2, np.arange(s) > s / 2],
+                            [
+                                np.exp(-1j * k * 2 * np.pi * np.arange(s) / s),
+                                np.cos(k * np.pi),
+                                np.exp(1j * k * 2 * np.pi * (s - np.arange(s)) / s),
+                            ],  # type: ignore
+                        )
                         for k, s in zip(offset, input_shape[-self.ndims :])
                     )
                 )

scico/test/linop/test_circconv.py

@@ -33,7 +33,6 @@ def setup_method(self, method):
     @pytest.mark.parametrize("input_dtype", [np.float32, np.complex64])
     @pytest.mark.parametrize("axes_shape_spec", SHAPE_SPECS)
     def test_eval(self, axes_shape_spec, input_dtype, jit):
-
         x_shape, ndims, h_shape = axes_shape_spec
 
         h, key = randn(tuple(h_shape), dtype=input_dtype, key=self.key)
@@ -62,7 +61,6 @@ def test_eval(self, axes_shape_spec, input_dtype, jit):
     @pytest.mark.parametrize("input_dtype", [np.float32, np.complex64])
     @pytest.mark.parametrize("axes_shape_spec", SHAPE_SPECS)
     def test_adjoint(self, axes_shape_spec, input_dtype, jit):
-
         x_shape, ndims, h_shape = axes_shape_spec
 
         h, key = randn(tuple(h_shape), dtype=input_dtype, key=self.key)
@@ -160,6 +158,21 @@ def test_center(self, center):
             shift = -center[0]
         np.testing.assert_allclose(A @ x, snp.roll(B @ x, shift), atol=1e-5)
 
+    def test_fractional_center(self):
+        """A fractional center should keep outputs real."""
+        x, key = uniform(minval=-1, maxval=1, shape=(4, 5), key=self.key)
+        h, _ = uniform(minval=-1, maxval=1, shape=(2, 2), key=key)
+        A = CircularConvolve(h=h, input_shape=x.shape, h_center=[0.1, 2.7])
+
+        # taken from CircularConvolve._eval
+        x_dft = snp.fft.fftn(x, axes=A.x_fft_axes)
+        hx = snp.fft.ifftn(
+            A.h_dft * x_dft,
+            axes=A.ifft_axes,
+        )
+
+        np.testing.assert_allclose(hx, snp.real(hx))
+
     @pytest.mark.parametrize("axes_shape_spec", SHAPE_SPECS)
     @pytest.mark.parametrize("input_dtype", [np.float32, np.complex64])
     @pytest.mark.parametrize("jit_old_op", [True, False])