Signs and Conventions

Signs and Conventions#

This page centralizes the sign and axis conventions used in Pyralysis for image-space and Fourier-space operations. It is intended as a practical reference for users implementing pipelines, custom estimators, or validating equations.

Image Pixel Size Convention (cellsize)#

Internally, Pyralysis enforces a signed 2-element cellsize in radians:

\[[\Delta l, \Delta m] = [-|\Delta l|,\; +|\Delta m|]\]

This is implemented by pyralysis.utils.cellsize and used by pyralysis.grids.base.Grid.

  • cellsize[0] maps to the horizontal image axis (\(l\)) and is stored negative.

  • cellsize[1] maps to the vertical image axis (\(m\)) and is stored positive.

Even when a scalar value is provided, Pyralysis expands it to this signed pair.

Direction Cosines Convention (l, m, n)#

Pyralysis uses the standard radio-interferometry direction-cosine convention relative to a phase center (pyralysis.utils.coordinate_transforms):

\[l = \cos(\delta)\sin(\alpha-\alpha_0)\]
\[m = \sin(\delta)\cos(\delta_0)-\cos(\delta)\sin(\delta_0)\cos(\alpha-\alpha_0)\]
\[n = \sqrt{1-l^2-m^2}\]

where \((\alpha,\delta)\) is sky position and \((\alpha_0,\delta_0)\) is the phase center.

UV Cell Size and Axis Mapping#

For padded image size \((N_y, N_x)\) and image-space cell sizes \((\Delta l, \Delta m)\), Pyralysis uses:

\[\Delta u = \frac{1}{N_x \,\Delta l}, \qquad \Delta v = \frac{1}{N_y \,\Delta m}\]

where:

  • \(N_x\) is the horizontal (width) pixel count.

  • \(N_y\) is the vertical (height) pixel count.

  • \(\Delta l < 0\) and \(\Delta m > 0\) by convention.

Therefore, uvcellsize[0] (du) is usually negative and uvcellsize[1] (dv) positive.

Nyquist Limits (Useful Rules of Thumb)#

Given image-space sampling \(\Delta l, \Delta m\), the approximate Nyquist support in UV is:

\[|u|_{\max} \approx \frac{1}{2|\Delta l|}, \qquad |v|_{\max} \approx \frac{1}{2|\Delta m|}\]

And Fourier pixel spacing is:

\[\delta u = \frac{1}{N_x |\Delta l|}, \qquad \delta v = \frac{1}{N_y |\Delta m|}\]

In practice, padding changes \(N_x, N_y\), so it changes Fourier sampling density (pixel spacing) but not the physical baseline coverage of the observed data.

FFT/IDFT Sign Convention#

Pyralysis exposes both sign conventions via sign_convention:

  • "negative" (default): forward transform uses \(e^{-2\pi i(\cdot)}\).

  • "positive": forward transform uses \(e^{+2\pi i(\cdot)}\).

In pyralysis.fft._fft2.fft2() / pyralysis.fft._fft2.ifft2(), the inverse uses the corresponding opposite-sign primitive mapping to keep the pair consistent for the selected normalization mode.

Phase-Shift Convention (Grid and Visibilities)#

For shifting Fourier grids or visibilities to a phase-center offset, Pyralysis uses:

\[\exp\{-2\pi i(\cdot)\}\]

specifically in pyralysis.utils.fourier.phase_shift:

\[F'(u,v)=F(u,v)\exp\{-2\pi i(ux_0+vy_0)\}\]
\[V'(u,v,w)=V(u,v,w)\exp\{-2\pi i(ul_0+vm_0+w(n_0-1))\}\]

This is consistent with the usual Fourier shift theorem for analysis/registration.

IDFT for Visibility to Image#

For direct inversion (pyralysis.dft.idft2()), Pyralysis evaluates:

\[I(l,m) = \sum_k \omega_k V_k \exp\left\{2\pi i\left(u_k l + v_k m + w_k\left(\sqrt{1-l^2-m^2}-1\right)\right)\right\}\]

with sign flipped if sign_convention="positive" is selected (i.e., IDFT uses the opposite sign of the forward DFT convention).

Hermitian Symmetry Convention#

When hermitian_symmetry=True, Pyralysis stores only the non-redundant half-plane in u (via rfft2/irfft2 path):

  • u is represented as non-negative indices on the stored grid.

  • v remains centered.

This is a storage/computation optimization for real image-domain inputs and should be interpreted with the same sign conventions above.

Sky-Model Source Placement Factor#

When evaluating analytic sky models into visibility space, Pyralysis uses a positive phase factor:

\[\phi_\text{src}(u,v,w)=\frac{\exp\{+2\pi i(ul_0+vm_0+w(n_0-1))\}}{n_0}\]

This appears in source modeling utilities (for example, source sky-position factors) and is intentional:

  • phase-shift operators that move existing data use \(\exp\{-2\pi i(\cdot)\}\)

  • source forward modeling that synthesizes visibilities from sky components uses \(\exp\{+2\pi i(\cdot)\}\) in this implementation

Keep this distinction in mind when mixing custom source terms and post-FFT/visibility shift operations.