healpy harmonic_ud_grade vs ud_grade comparison notebook (executed)

comparison_executed.ipynb

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    "# `harmonic_ud_grade` vs `ud_grade`: A Focused Comparison\n",
    "\n",
    "When working with HEALPix maps, it is often necessary to change the\n",
    "resolution \u2014 for example, to downgrade a high-resolution observation\n",
    "map to a lower Nside for comparison with a model, or to reduce\n",
    "computational cost for large-scale analyses.\n",
    "\n",
    "healpy provides two functions for this:\n",
    "\n",
    "- **`ud_grade`**: operates purely in **pixel space** by averaging\n",
    "  groups of sub-pixels. It is fast, but because it does not apply any\n",
    "  anti-aliasing filter, high-multipole power from the input grid\n",
    "  \"folds\" back into the output \u2014 a phenomenon known as\n",
    "  [aliasing](https://en.wikipedia.org/wiki/Aliasing).\n",
    "\n",
    "- **`harmonic_ud_grade`**: operates in **spherical-harmonic space**.\n",
    "  It decomposes the input map into $a_{\\ell m}$ coefficients, applies\n",
    "  the pixel-window correction following the prescription in\n",
    "  [Planck 2015 XVI Eq. 1](https://arxiv.org/abs/1502.01587), and\n",
    "  optionally scales the beam. Because the harmonic transform\n",
    "  naturally band-limits the output, aliasing is eliminated.\n",
    "\n",
    "## How `harmonic_ud_grade` works\n",
    "\n",
    "The function implements the following per-$\\ell$ transfer\n",
    "([Planck 2015 XVI Eq. 1](https://arxiv.org/abs/1502.01587)):\n",
    "\n",
    "$$\n",
    "a^{\\rm out}_{\\ell m}\n",
    "= \\frac{p^{\\rm out}_\\ell}{p^{\\rm in}_\\ell}\n",
    "  \\cdot \\frac{b^{\\rm out}_\\ell}{b^{\\rm in}_\\ell}\n",
    "  \\cdot a^{\\rm in}_{\\ell m}\n",
    "$$\n",
    "\n",
    "where $p_\\ell$ is the HEALPix pixel-window function and $b_\\ell$ is\n",
    "a Gaussian beam transfer function.  The algorithm proceeds in three\n",
    "steps:\n",
    "\n",
    "1. **Analysis:** The input map is decomposed into $a_{\\ell m}$ via\n",
    "   `map2alm` (using pixel weights by default for high accuracy).\n",
    "2. **Transfer:** Each $a_{\\ell m}$ is multiplied by the ratio of\n",
    "   output/input pixel windows and beams.  Modes above\n",
    "   $\\ell_{\\max}^{\\rm out}$ are discarded (band-limiting).\n",
    "3. **Synthesis:** The modified $a_{\\ell m}$ are synthesised into a\n",
    "   map at `nside_out` via `alm2map`.\n",
    "\n",
    "### Function signature and arguments\n",
    "\n",
    "```python\n",
    "hp.harmonic_ud_grade(\n",
    "    map_in,                  # Input map(s), RING ordering\n",
    "    nside_out,               # Target Nside\n",
    "    lmax=None,               # Max multipole (default: min(3*nside_out - 1, 3*nside_in - 1))\n",
    "    mmax=None,               # Max m (default: lmax)\n",
    "    iter=None,               # map2alm iterations (see below)\n",
    "    pol=True,                # Treat 3-component input as TQU/TEB\n",
    "    pixwin=True,             # Deconvolve/apply pixel windows\n",
    "    fwhm_in=0,               # Input beam FWHM [radians]\n",
    "    fwhm_out=None,           # Output beam FWHM [radians] (see below)\n",
    "    use_weights=False,       # Use ring weights in map2alm\n",
    "    datapath=None,           # Path to pixel-weight files\n",
    "    use_pixel_weights=True,  # Use full pixel weights (recommended)\n",
    "    dtype=None,              # Cast output to this dtype\n",
    ")\n",
    "```\n",
    "\n",
    "**Key defaults and their rationale:**\n",
    "\n",
    "| Argument | Default | Rationale |\n",
    "|----------|---------|-----------|\n",
    "| `lmax` | `min(...)` | Standard HEALPix bandlimit, capped by the input resolution. See explanation below. |\n",
    "| `iter` | `None` (auto) | Uses 0 iterations when pixel weights are active and `lmax <= 1.5*nside_in` (the regime where pixel weights alone are sufficient); otherwise 3 iterations. |\n",
    "| `pixwin` | `True` | Deconvolves the input pixel window $p^{\\rm in}_\\ell$ and applies the output pixel window $p^{\\rm out}_\\ell$, ensuring the output map has the correct effective resolution. |\n",
    "| `fwhm_in` | `0` | No input beam deconvolution. **Must be set** to the actual beam FWHM when working with beam-convolved data. |\n",
    "| `fwhm_out` | `None` (auto) | Auto-computed as ``PLANCK_K * nside2resol(nside_out)`` where ``PLANCK_K = 160.0 / (degrees(nside2resol(64)) * 60)`` (\u2248 2.91). This matches the exact FWHM-to-pixel ratio used consistently across all Planck resolutions. Pass `0` to disable output smoothing. |\n",
    "| `use_pixel_weights` | `True` | Uses full per-pixel weights for high-accuracy spherical harmonic transforms. If the weight files are not available, an error is raised (pass `False` to fall back to unweighted transforms). |\n",
    "\n",
    "*`lmax` default:* `3\u00b7nside_out \u2013 1` is the standard HEALPix bandlimit, but\n",
    "it is capped to `3\u00b7nside_in \u2013 1` when *upgrading* resolution.  This prevents\n",
    "the transform from requesting multipoles the input map cannot meaningfully\n",
    "provide.\n",
    "\n",
    "## Notebook overview\n",
    "\n",
    "This notebook compares the two methods through four tests:\n",
    "1. **Aliasing stress test** \u2014 a single high-$\\ell$ mode that should\n",
    "   vanish after downgrading.\n",
    "2. **Power-spectrum recovery** \u2014 a broadband synthetic signal where\n",
    "   spectral fidelity matters.\n",
    "3. **Noise aliasing** \u2014 a realistic scenario with \"blue\" noise that\n",
    "   grows at high $\\ell$, mimicking beam-deconvolved instrumental\n",
    "   noise.\n",
    "4. **Point sources** \u2014 a case where `ud_grade` is the better choice\n",
    "   due to Gibbs ringing in the harmonic approach."
   ]
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   "source": [
    "import time\n",
    "import numpy as np\n",
    "import healpy as hp\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "plt.rcParams.update({\"figure.dpi\": 120, \"font.size\": 11})\n",
    "\n",
    "# Planck 2013 XXIII Table 1: exact FWHM-to-pixel ratio\n",
    "PLANCK_K = 160.0 / (np.degrees(hp.nside2resol(64)) * 60)\n",
    "print(f\"healpy {hp.__version__}\")\n",
    "print(f\"Planck FWHM/pixel ratio: {PLANCK_K:.4f}\")"
   ]
  },
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   "cell_type": "markdown",
   "id": "5188578f",
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    "## 1. Aliasing Stress Test: A Single High-$\\ell$ Mode\n",
    "\n",
    "The simplest way to reveal aliasing is to construct a pathological\n",
    "input: a map at `nside_in = 128` that contains power at **exactly\n",
    "one** spherical-harmonic multipole, $\\ell = 120$.\n",
    "\n",
    "The target resolution is `nside_out = 32`, whose maximum resolvable\n",
    "multipole is $\\ell_{\\max} = 3 \\times 32 - 1 = 95$.  Since\n",
    "$\\ell = 120 > \\ell_{\\max}$, a correct downgrade must produce an\n",
    "output whose angular power spectrum is identically zero.\n",
    "\n",
    "**Key parameter choices:**\n",
    "- `pixwin=True` is passed both when creating the input map\n",
    "  (`alm2map`) and when calling `harmonic_ud_grade`. This simulates\n",
    "  a realistic pixelised observation and allows `harmonic_ud_grade`\n",
    "  to properly deconvolve the input pixel window and apply the output\n",
    "  pixel window.\n",
    "- `fwhm_in=0` because the synthetic input has no instrumental beam."
   ]
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   "id": "b7632f2f",
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    "nside_in = 128\n",
    "nside_out = 32\n",
    "lmax_out = 3 * nside_out - 1\n",
    "\n",
    "# Single mode at ell=120\n",
    "alm_single = np.zeros(hp.Alm.getsize(120), dtype=np.complex128)\n",
    "alm_single[hp.Alm.getidx(120, 120, 0)] = 1.0\n",
    "# We generate the synthetic map and run downgrade methods with pixwin=True.\n",
    "# This simulates a realistic pixelized map and allows harmonic_ud_grade\n",
    "# to properly apply its pixel window correction (Eq 1).\n",
    "m_in = hp.alm2map(alm_single, nside=nside_in, pixwin=True)\n",
    "\n",
    "# Downgrade methods\n",
    "m_ud = hp.ud_grade(m_in, nside_out=nside_out)\n",
    "\n",
    "# fwhm_in is required\n",
    "# For this synthetic test there is no input beam \u2192 pass 0\n",
    "m_harm = hp.harmonic_ud_grade(\n",
    "    m_in,\n",
    "    nside_out=nside_out,\n",
    "    fwhm_in=0,\n",
    "    use_pixel_weights=False,\n",
    "    pixwin=True,\n",
    "    fwhm_out=0,\n",
    ")\n",
    "\n",
    "cl_in = hp.anafast(m_in)\n",
    "cl_ud = hp.anafast(m_ud)\n",
    "cl_harm = hp.anafast(m_harm)\n",
    "\n",
    "plt.figure(figsize=(10, 5))\n",
    "plt.semilogy(np.maximum(cl_in, 1e-8), label=f\"Input (Nside={nside_in})\", color=\"black\", linestyle=\"--\")\n",
    "plt.semilogy(np.maximum(cl_ud, 1e-8), label=f\"ud_grade (Nside={nside_out})\", alpha=0.8)\n",
    "plt.semilogy(np.maximum(cl_harm, 1e-8), label=f\"harmonic_ud_grade (Nside={nside_out})\", alpha=0.8)\n",
    "plt.axvline(lmax_out, color=\"red\", ls=\":\", label=r\"Output $\\ell_{\\max}$\")\n",
    "plt.xlabel(r\"$\\ell$\")\n",
    "plt.ylabel(r\"$C_\\ell$\")\n",
    "plt.xlim(0, 150)\n",
    "plt.ylim(1e-8, cl_in.max() * 5)\n",
    "plt.title(\"Aliasing of a single high-frequency mode ($\\\\ell=120$)\")\n",
    "plt.legend()\n",
    "plt.grid(alpha=0.3)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
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    "**Interpretation of the plot above:**\n",
    "\n",
    "- **Black dashed line (Input):** The input power spectrum has a sharp\n",
    "  peak at $\\ell = 120$ and is essentially zero everywhere else. The\n",
    "  satellite peaks visible around $\\ell = 120$ are due to the pixel\n",
    "  window of the `nside_in = 128` grid.\n",
    "- **Orange line (`harmonic_ud_grade`):** The output spectrum is\n",
    "  numerically zero across all multipoles \u2014 exactly the correct\n",
    "  answer. The harmonic transform naturally band-limits the result to\n",
    "  $\\ell \\le \\ell_{\\max}$, so the $\\ell = 120$ mode is cleanly\n",
    "  removed.\n",
    "- **Blue line (`ud_grade`):** Significant spurious power appears\n",
    "  across the *entire* output multipole range.  This is **aliasing**:\n",
    "  because pixel-space averaging has no frequency cutoff, the\n",
    "  high-$\\ell$ mode is \"folded\" back into the lower multipoles,\n",
    "  corrupting the output with an oscillating pattern.\n",
    "- **Red dotted line:** The output bandlimit $\\ell_{\\max} = 95$.\n",
    "\n",
    "The RMS values below quantify this: `ud_grade` retains about 43 % of\n",
    "the original map RMS as pure aliasing artefact, while\n",
    "`harmonic_ud_grade` suppresses it to the numerical noise floor."
   ]
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    "print(f\"Input map RMS:              {np.std(m_in):.6e}\")\n",
    "print(f\"ud_grade output RMS:        {np.std(m_ud):.6e}  <-- aliasing\")\n",
    "print(f\"harmonic_ud_grade RMS:      {np.std(m_harm):.6e}  <-- suppressed\")"
   ]
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   "id": "edb355d8",
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    "## 2. Power-Spectrum Recovery \u2014 Why It Matters\n",
    "\n",
    "The single-mode test above is deliberately extreme. Real sky maps\n",
    "contain power at **all** multipoles, so aliasing from `ud_grade`\n",
    "is spread across the full spectrum, making it harder to spot by\n",
    "eye but no less damaging to science.\n",
    "\n",
    "In this section we synthesise a realistic broadband signal with\n",
    "$C_\\ell \\propto \\ell^{-2}$ (a spectrum often used for test\n",
    "purposes) at `nside_in = 256`, downgrade to `nside_out = 64`,\n",
    "and compare the recovered power spectrum against a **ground-truth\n",
    "reference**.\n",
    "\n",
    "The reference is built by truncating the original $a_{\\ell m}$ to\n",
    "$\\ell_{\\max}^{\\rm out}$ and synthesising directly at the output\n",
    "resolution, so both the reference and the downgraded maps contain\n",
    "the same pixel-window effects. Any difference between them is\n",
    "therefore purely due to the downgrade method."
   ]
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    "nside_in = 256\n",
    "nside_out = 64\n",
    "lmax_in = 3 * nside_in - 1\n",
    "lmax_out = 3 * nside_out - 1\n",
    "\n",
    "np.random.seed(42)\n",
    "# healpy synfast/synalm uses the global RNG\n",
    "ell = np.arange(lmax_in + 1, dtype=float)\n",
    "cl_in = np.zeros(lmax_in + 1)\n",
    "cl_in[2:] = ell[2:] ** (-2.0)\n",
    "elm = hp.synalm(cl_in, lmax=lmax_in)\n",
    "m_in = hp.alm2map(elm, nside=nside_in, lmax=lmax_in, pixwin=True)\n",
    "\n",
    "# Ground-truth reference: directly synthesise at output resolution\n",
    "elm_ref = hp.resize_alm(elm, lmax_in, lmax_in, lmax_out, lmax_out)\n",
    "m_ref = hp.alm2map(elm_ref, nside=nside_out, lmax=lmax_out, pixwin=True)\n",
    "\n",
    "m_ud = hp.ud_grade(m_in, nside_out=nside_out)\n",
    "\n",
    "# fwhm_in=0 for this synthetic simulation with no beam\n",
    "m_harm = hp.harmonic_ud_grade(\n",
    "    m_in,\n",
    "    nside_out=nside_out,\n",
    "    fwhm_in=0,\n",
    "    use_pixel_weights=False,\n",
    "    pixwin=True,\n",
    "    fwhm_out=0,\n",
    ")\n",
    "\n",
    "# Measure spectra\n",
    "cl_ref = hp.anafast(m_ref, lmax=lmax_out)\n",
    "cl_ud = hp.anafast(m_ud, lmax=lmax_out)\n",
    "cl_harm = hp.anafast(m_harm, lmax=lmax_out)\n",
    "ell_out = np.arange(lmax_out + 1)"
   ]
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    "fig, axes = plt.subplots(1, 2, figsize=(12, 4))\n",
    "\n",
    "axes[0].loglog(ell_out[2:], cl_ref[2:], \"k--\", lw=2, label=\"Reference $C_\\ell$ (truth)\")\n",
    "axes[0].loglog(ell_out[2:], cl_ud[2:], alpha=0.8, label=\"ud_grade\")\n",
    "axes[0].loglog(ell_out[2:], cl_harm[2:], alpha=0.8, label=\"harmonic_ud_grade\")\n",
    "axes[0].axvline(lmax_out, color=\"red\", ls=\":\", label=r\"$\\ell_{\\max}$\")\n",
    "axes[0].set_xlabel(r\"$\\ell$\")\n",
    "axes[0].set_ylabel(r\"$C_\\ell$\")\n",
    "axes[0].set_title(\"Power spectra after downgrade\")\n",
    "axes[0].legend()\n",
    "axes[0].grid(alpha=0.3)\n",
    "\n",
    "# Fractional error\n",
    "frac_ud = (cl_ud[2:] - cl_ref[2:]) / cl_ref[2:]\n",
    "frac_harm = (cl_harm[2:] - cl_ref[2:]) / cl_ref[2:]\n",
    "\n",
    "axes[1].plot(ell_out[2:], frac_ud * 100, alpha=0.8, label=\"ud_grade\")\n",
    "axes[1].plot(ell_out[2:], frac_harm * 100, alpha=0.8, label=\"harmonic_ud_grade\")\n",
    "axes[1].axhline(0, color=\"k\", lw=0.5)\n",
    "axes[1].axvline(lmax_out, color=\"red\", ls=\":\")\n",
    "axes[1].set_xlabel(r\"$\\ell$\")\n",
    "axes[1].set_ylabel(\"Fractional error [%]\")\n",
    "axes[1].set_title(\"Spectral error vs truth\")\n",
    "axes[1].legend()\n",
    "axes[1].grid(alpha=0.3)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
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    "**Interpretation of the plots above:**\n",
    "\n",
    "**Left panel \u2014 Power spectra after downgrade:**\n",
    "All three curves agree well at low $\\ell$, but diverge at high\n",
    "multipoles.  The reference (black dashed) and `harmonic_ud_grade`\n",
    "(orange) both roll off smoothly beyond $\\ell \\approx 100$: this is\n",
    "the expected damping from the pixel window of the `nside_out = 64`\n",
    "grid.  `ud_grade` (blue), by contrast, shows a visible *excess* of\n",
    "power in this regime because aliased high-frequency structures from\n",
    "above $\\ell_{\\max}$ have been folded back into the output.\n",
    "\n",
    "**Right panel \u2014 Fractional spectral error relative to truth:**\n",
    "This panel makes the aliasing bias quantitative.\n",
    "- `harmonic_ud_grade` (orange) sits flat on 0 % error \u2014 it matches\n",
    "  the truth map almost perfectly.\n",
    "- `ud_grade` (blue) is systematically **positive**, meaning it\n",
    "  **over-estimates** the power.  This is expected: aliasing can only\n",
    "  *add* power, never subtract it. The error grows rapidly toward\n",
    "  $\\ell_{\\max}$, reaching 50\u201370 % near the bandlimit.\n",
    "\n",
    "Because both the reference and the downgraded maps include the same\n",
    "pixel window, window effects cancel exactly in the ratio. The\n",
    "residual is pure aliasing.\n",
    "\n",
    "The RMS fractional error below summarises this over the \"safe\"\n",
    "multipole band $\\ell \\in [2,\\, 2 N_{\\rm side}^{\\rm out}]$:"
   ]
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   "cell_type": "code",
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   "id": "2188611e",
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    "safe = slice(2, 2 * nside_out + 1)\n",
    "print(\n",
    "    f\"RMS frac. error over safe band \u2113 \u2208 [2, {2*nside_out}]:\\n\"\n",
    "    f\"  ud_grade:          {np.sqrt(np.mean(frac_ud[safe]**2))*100:.2f}%\\n\"\n",
    "    f\"  harmonic_ud_grade: {np.sqrt(np.mean(frac_harm[safe]**2))*100:.2f}%\"\n",
    ")"
   ]
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   "source": [
    "## 3. Noise Aliasing \u2014 The Real-World Failure Mode\n",
    "\n",
    "The tests above used noiseless inputs, but real observational data\n",
    "always contain instrumental noise.  In CMB data processing a\n",
    "particularly important case is **blue noise** \u2014 noise whose power\n",
    "spectrum *grows* with $\\ell$ (e.g., $C_\\ell^{\\rm noise} \\propto\n",
    "\\ell^{2}$). This commonly arises when a map is beam-deconvolved:\n",
    "dividing out the beam's Gaussian roll-off amplifies the\n",
    "high-frequency noise enormously.\n",
    "\n",
    "Because `ud_grade` operates in pixel space with no frequency cutoff,\n",
    "**all** of that amplified high-$\\ell$ noise above\n",
    "$\\ell_{\\max}^{\\rm out}$ is aliased back into the signal band,\n",
    "dramatically raising the noise floor.  `harmonic_ud_grade` avoids\n",
    "this by band-limiting the map before downgrading.\n",
    "\n",
    "Below we construct a signal + blue-noise map at `nside_in = 512`,\n",
    "downgrade to `nside_out = 64` (a factor-8 step), and isolate the\n",
    "noise contribution in each output.  The larger resolution ratio\n",
    "means there is a vast reservoir of high-$\\ell$ noise power above\n",
    "$\\ell_{\\max}^{\\rm out} = 191$ that can potentially alias back."
   ]
  },
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    "nside_in = 512\n",
    "nside_out = 64\n",
    "lmax_out = 3 * nside_out - 1\n",
    "\n",
    "np.random.seed(42)\n",
    "# healpy synfast/synalm uses the global RNG\n",
    "ell_in = np.arange(3 * nside_in + 1, dtype=float)\n",
    "cl_signal = np.zeros(3 * nside_in + 1)\n",
    "cl_signal[2:] = ell_in[2:] ** (-2.0)\n",
    "map_signal = hp.synfast(cl_signal, nside_in, lmax=3 * nside_in, new=True, pixwin=True)\n",
    "\n",
    "# Blue noise: noise dominates at high \u2113 (e.g. beam deconvolved noise)\n",
    "ell_knee = 50\n",
    "cl_noise = np.zeros(3 * nside_in + 1)\n",
    "cl_noise[2:] = cl_signal[ell_knee] * (ell_in[2:] / ell_knee) ** 2\n",
    "# pixwin=False: noise is a pixel-level quantity, not a smooth sky signal.\n",
    "# Applying the pixel window to noise would incorrectly suppress its\n",
    "# high-ell power, understating the aliasing problem.\n",
    "map_noise = hp.synfast(cl_noise, nside_in, lmax=3 * nside_in, new=True, pixwin=False)\n",
    "map_noisy = map_signal + map_noise\n",
    "\n",
    "# Reference: downgrade signal-only with harmonic\n",
    "m_ref_signal = hp.harmonic_ud_grade(\n",
    "    map_signal,\n",
    "    nside_out=nside_out,\n",
    "    fwhm_in=0,\n",
    "    use_pixel_weights=False,\n",
    "    pixwin=True,\n",
    "    fwhm_out=0,\n",
    ")\n",
    "\n",
    "# Downgrade methods\n",
    "m_ud = hp.ud_grade(map_noisy, nside_out=nside_out)\n",
    "m_harm = hp.harmonic_ud_grade(\n",
    "    map_noisy,\n",
    "    nside_out=nside_out,\n",
    "    fwhm_in=0,\n",
    "    use_pixel_weights=False,\n",
    "    pixwin=True,\n",
    "    fwhm_out=0,\n",
    ")\n",
    "\n",
    "# Isolate noise contribution\n",
    "cl_noise_ud = hp.anafast(m_ud - hp.ud_grade(map_signal, nside_out=nside_out), lmax=lmax_out)\n",
    "cl_noise_harm = hp.anafast(m_harm - m_ref_signal, lmax=lmax_out)\n",
    "cl_signal_out = hp.anafast(m_ref_signal, lmax=lmax_out)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e6cffb7d",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.figure(figsize=(8, 5))\n",
    "\n",
    "plt.loglog(ell_out[2:], cl_signal_out[2:], \"k--\", lw=2, label=\"Signal (truth)\")\n",
    "plt.loglog(ell_out[2:], cl_noise[:lmax_out + 1][2:], \"gray\", ls=\":\", lw=2, label=\"Input Noise\")\n",
    "plt.loglog(ell_out[2:], cl_noise_ud[2:], alpha=0.8, label=\"Noise in ud_grade\")\n",
    "plt.loglog(ell_out[2:], cl_noise_harm[2:], alpha=0.8, label=\"Noise in harmonic_ud_grade\")\n",
    "plt.xlabel(r\"$\\ell$\")\n",
    "plt.ylabel(r\"$C_\\ell$ (noise)\")\n",
    "plt.title(\"Noise contamination after downgrade\")\n",
    "plt.legend()\n",
    "plt.grid(alpha=0.3)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36e7c4c0",
   "metadata": {},
   "source": [
    "**Interpretation of the plot above:**\n",
    "\n",
    "This plot isolates the **noise-only** component after downgrading\n",
    "from `nside = 512` to `nside = 64` (a factor-8 resolution step).\n",
    "Each curve has a specific meaning:\n",
    "\n",
    "- **Black dashed (Signal truth):** The true signal power spectrum at\n",
    "  the output resolution.  It is shown for context so you can judge\n",
    "  where noise starts to dominate over signal.\n",
    "- **Grey dotted (Input Noise):** The raw analytic noise spectrum\n",
    "  $C_\\ell^{\\rm noise} \\propto \\ell^{2}$ as injected into the\n",
    "  input map.  It crosses the signal near $\\ell \\approx 50$\n",
    "  (the chosen knee multipole) and grows steeply beyond.\n",
    "- **Orange (`harmonic_ud_grade`):** The noise residual after\n",
    "  harmonic downgrading.  It tracks the input noise closely at low\n",
    "  $\\ell$ and rolls off above $\\ell \\approx 100$ due to the target\n",
    "  pixel window \u2014 exactly as expected.\n",
    "- **Blue (`ud_grade`):** The noise residual after pixel-space\n",
    "  downgrading.  Across the full multipole range, the `ud_grade`\n",
    "  noise is roughly **5\u201310\u00d7 higher** than the `harmonic_ud_grade`\n",
    "  noise.  This excess is entirely due to aliased high-$\\ell$ noise\n",
    "  being folded into the signal band.  The effect grows with the\n",
    "  resolution ratio: a larger step (here 8\u00d7) means more high-$\\ell$\n",
    "  modes available to alias, producing a larger noise floor uplift.\n",
    "\n",
    "**Notes on the noise model:**\n",
    "\n",
    "- Noise is generated with `pixwin=False` because instrumental noise\n",
    "  is a pixel-level quantity \u2014 it does not have a smooth pixel window\n",
    "  like sky emission.  Using `pixwin=True` for noise would incorrectly\n",
    "  suppress its high-$\\ell$ power and understate the aliasing problem.\n",
    "- The $C_\\ell \\propto \\ell^{2}$ spectrum is a proxy for beam-deconvolved\n",
    "  noise, but is not exactly equivalent: a truly deconvolved noise map\n",
    "  would have its power amplified by $1/b_\\ell^{2}$, which diverges\n",
    "  at high $\\ell$.  For a precise simulation, generate white noise and\n",
    "  then explicitly deconvolve the beam.  The $\\ell^{2}$ proxy is\n",
    "  sufficient here to demonstrate the aliasing effect.\n",
    "- For white noise ($C_\\ell = \\mathrm{const}$), `ud_grade` works well\n",
    "  because the pixel-averaging naturally reduces the noise variance by\n",
    "  the ratio of pixel areas.  The aliasing advantage of\n",
    "  `harmonic_ud_grade` is specific to noise with rising high-$\\ell$\n",
    "  power spectra.\n",
    "\n",
    "**Why this matters in practice:** When analysing real CMB or\n",
    "astrophysical data, any aliased noise that leaks into the low-$\\ell$\n",
    "modes will bias power-spectrum estimates, cross-correlations, and\n",
    "component-separation results.  `harmonic_ud_grade` prevents this by\n",
    "applying a strict harmonic bandlimit before re-gridding to the\n",
    "output resolution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2bc9d1f4",
   "metadata": {},
   "source": [
    "## 4. When `ud_grade` Wins: Point Sources and Gibbs Ringing\n",
    "\n",
    "The previous sections show that `harmonic_ud_grade` is clearly\n",
    "superior for broadband or noisy signals.  But there is an important\n",
    "case where **`ud_grade` is the better choice**: maps dominated by\n",
    "**compact, localised features** such as point sources or binary\n",
    "masks.\n",
    "\n",
    "A point source is a delta function on the sky, which means it has\n",
    "power at *all* multipoles.  When `harmonic_ud_grade` band-limits\n",
    "the map to $\\ell_{\\max}^{\\rm out}$, it truncates the harmonic\n",
    "expansion abruptly, producing oscillating **Gibbs ringing** around\n",
    "each source.  `ud_grade`, operating purely in pixel space, simply\n",
    "averages the sub-pixels and preserves the compact, positive-definite\n",
    "nature of the source.\n",
    "\n",
    "To make this test realistic, we simulate point sources **as an\n",
    "instrument would observe them**: we paint a Gaussian beam profile\n",
    "directly in pixel space at each source position, using the\n",
    "Planck-suggested FWHM for the input resolution.  We then compare\n",
    "three downgrade strategies:\n",
    "1. `ud_grade` \u2014 pixel-space averaging.\n",
    "2. `harmonic_ud_grade` with `fwhm_out = 0` \u2014 band-limit only, no\n",
    "   additional smoothing.\n",
    "3. `harmonic_ud_grade` with default `fwhm_out` \u2014 Planck-scaled\n",
    "   output beam."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "be3700bc",
   "metadata": {},
   "outputs": [],
   "source": [
    "nside_in = 256\n",
    "nside_out = 64\n",
    "\n",
    "# Planck-suggested beam for this resolution\n",
    "fwhm_in_pts = PLANCK_K * hp.nside2resol(nside_in)\n",
    "sigma_in = fwhm_in_pts / (2 * np.sqrt(2 * np.log(2)))\n",
    "print(f\"Nside_in = {nside_in}, fwhm_in = {np.degrees(fwhm_in_pts)*60:.1f} arcmin \"\n",
    "      f\"(Planck ratio)\")\n",
    "\n",
    "# Simulate point sources as an instrument would see them:\n",
    "# paint a Gaussian beam profile directly in pixel space.\n",
    "m_pts = np.zeros(hp.nside2npix(nside_in))\n",
    "rng = np.random.default_rng(123)\n",
    "src_pixels = rng.choice(hp.nside2npix(nside_in), size=5, replace=False)\n",
    "src_vecs = np.array(hp.pix2vec(nside_in, src_pixels)).T  # (5, 3)\n",
    "all_vecs = np.array(hp.pix2vec(nside_in, np.arange(hp.nside2npix(nside_in)))).T\n",
    "\n",
    "for src_vec in src_vecs:\n",
    "    cos_dist = np.dot(all_vecs, src_vec)\n",
    "    cos_dist = np.clip(cos_dist, -1, 1)\n",
    "    ang_dist = np.arccos(cos_dist)\n",
    "    m_pts += 100.0 * np.exp(-0.5 * (ang_dist / sigma_in) ** 2)\n",
    "\n",
    "# Downgrade: three methods\n",
    "m_pts_ud = hp.ud_grade(m_pts, nside_out=nside_out)\n",
    "\n",
    "# harmonic_ud_grade with NO additional smoothing (band-limit only)\n",
    "m_pts_harm_nosmooth = hp.harmonic_ud_grade(\n",
    "    m_pts,\n",
    "    nside_out=nside_out,\n",
    "    fwhm_in=fwhm_in_pts,\n",
    "    use_pixel_weights=False,\n",
    "    pixwin=True,\n",
    "    fwhm_out=0,\n",
    ")\n",
    "\n",
    "# harmonic_ud_grade with Planck default output beam\n",
    "m_pts_harm_smooth = hp.harmonic_ud_grade(\n",
    "    m_pts,\n",
    "    nside_out=nside_out,\n",
    "    fwhm_in=fwhm_in_pts,\n",
    "    use_pixel_weights=False,\n",
    "    pixwin=True,\n",
    ")\n",
    "\n",
    "print(f\"ud_grade:            min={m_pts_ud.min():.4f}, max={m_pts_ud.max():.2f}, \"\n",
    "      f\"negative pixels: {(m_pts_ud < 0).sum()}\")\n",
    "print(f\"harmonic (no smooth): min={m_pts_harm_nosmooth.min():.4f}, \"\n",
    "      f\"max={m_pts_harm_nosmooth.max():.2f}, \"\n",
    "      f\"negative pixels: {(m_pts_harm_nosmooth < 0).sum()}\")\n",
    "print(f\"harmonic (Planck):   min={m_pts_harm_smooth.min():.4f}, \"\n",
    "      f\"max={m_pts_harm_smooth.max():.2f}, \"\n",
    "      f\"negative pixels: {(m_pts_harm_smooth < 0).sum()}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2cbe02e5",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Zoom in on one source with gnomview\n",
    "theta, phi = hp.pix2ang(nside_in, src_pixels[0])\n",
    "rot = (np.degrees(phi), 90 - np.degrees(theta))\n",
    "\n",
    "fig = plt.figure(figsize=(16, 4))\n",
    "\n",
    "ax1 = fig.add_subplot(141)\n",
    "hp.gnomview(m_pts, rot=rot, reso=5, xsize=200,\n",
    "            title=f\"Input (Nside={nside_in})\", hold=True, notext=True)\n",
    "\n",
    "ax2 = fig.add_subplot(142)\n",
    "hp.gnomview(m_pts_ud, rot=rot, reso=5, xsize=200,\n",
    "            title=f\"ud_grade (Nside={nside_out})\", hold=True, notext=True)\n",
    "\n",
    "ax3 = fig.add_subplot(143)\n",
    "hp.gnomview(m_pts_harm_nosmooth, rot=rot, reso=5, xsize=200,\n",
    "            title=\"harmonic (no smooth)\", hold=True, notext=True)\n",
    "\n",
    "ax4 = fig.add_subplot(144)\n",
    "hp.gnomview(m_pts_harm_smooth, rot=rot, reso=5, xsize=200,\n",
    "            title=\"harmonic (Planck beam)\", hold=True, notext=True)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aea3ae0c",
   "metadata": {},
   "source": [
    "**Interpretation of the plots above:**\n",
    "\n",
    "The four panels zoom in on one point source that was simulated with\n",
    "a realistic Gaussian beam painted directly in pixel space (FWHM\n",
    "\u2248 40 arcmin for Nside = 256).\n",
    "\n",
    "- **Panel 1 (Input):** The beam-convolved point source at the input\n",
    "  resolution \u2014 a smooth Gaussian profile, as an instrument would\n",
    "  observe it.\n",
    "- **Panel 2 (`ud_grade`):** Pixel-space averaging preserves the\n",
    "  smooth, positive-definite profile of the beam.  No ringing, no\n",
    "  negative pixels.\n",
    "- **Panel 3 (`harmonic_ud_grade`, no smoothing):** Band-limiting to\n",
    "  $\\ell_{\\max} = 191$ without additional smoothing produces visible\n",
    "  **Gibbs ringing** \u2014 oscillations with negative values around the\n",
    "  source.  Even though the input is beam-convolved (not a true\n",
    "  delta), the beam is narrow enough that significant power remains\n",
    "  above $\\ell_{\\max}^{\\rm out}$, and truncating it causes ringing.\n",
    "- **Panel 4 (`harmonic_ud_grade`, Planck beam):** The default\n",
    "  Planck-scaled output beam adds extra smoothing that suppresses the\n",
    "  ringing significantly.  This is the recommended harmonic approach\n",
    "  for diffuse science \u2014 but for point-source work, `ud_grade` still\n",
    "  produces a cleaner, more compact result.\n",
    "\n",
    "**Takeaway:** `harmonic_ud_grade` is optimal for **band-limited**\n",
    "signals (CMB, diffuse emission).  `ud_grade` is preferable for\n",
    "**pixel-localised** features (point sources, binary masks,\n",
    "hit-count maps)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "390eb3b8",
   "metadata": {},
   "source": [
    "### 4.1 Polarization (Q/U) with Point Sources\n",
    "\n",
    "The Gibbs-ringing problem is not limited to temperature maps.  For\n",
    "polarisation maps (TQU) containing compact features, the same\n",
    "band-limiting that harms point sources in $T$ also produces ringing\n",
    "in $Q$ and $U$.  This is particularly relevant for maps that\n",
    "contain both diffuse polarised emission (where `harmonic_ud_grade`\n",
    "excels) and localised polarised sources (where `ud_grade` is better).\n",
    "\n",
    "Below we create a TQU map with point sources in all three components\n",
    "and compare the two methods."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "986da899",
   "metadata": {},
   "outputs": [],
   "source": [
    "nside_in = 128\n",
    "nside_out = 32\n",
    "\n",
    "fwhm_in_pts = PLANCK_K * hp.nside2resol(nside_in)\n",
    "sigma_in = fwhm_in_pts / (2 * np.sqrt(2 * np.log(2)))\n",
    "\n",
    "# Create TQU maps with point sources\n",
    "rng = np.random.default_rng(456)\n",
    "m_pts_tqu = np.zeros((3, hp.nside2npix(nside_in)))\n",
    "src_pixels = rng.choice(hp.nside2npix(nside_in), size=3, replace=False)\n",
    "src_vecs = np.array(hp.pix2vec(nside_in, src_pixels)).T\n",
    "all_vecs = np.array(hp.pix2vec(nside_in, np.arange(hp.nside2npix(nside_in)))).T\n",
    "\n",
    "for src_vec in src_vecs:\n",
    "    cos_dist = np.dot(all_vecs, src_vec)\n",
    "    cos_dist = np.clip(cos_dist, -1, 1)\n",
    "    ang_dist = np.arccos(cos_dist)\n",
    "    gaussian = 100.0 * np.exp(-0.5 * (ang_dist / sigma_in) ** 2)\n",
    "    m_pts_tqu[0] += gaussian       # T\n",
    "    m_pts_tqu[1] += 0.5 * gaussian  # Q\n",
    "    m_pts_tqu[2] += 0.3 * gaussian  # U\n",
    "\n",
    "m_tqu_ud = hp.ud_grade(m_pts_tqu, nside_out=nside_out)\n",
    "m_tqu_harm = hp.harmonic_ud_grade(\n",
    "    m_pts_tqu, nside_out=nside_out, pol=True,\n",
    "    fwhm_in=fwhm_in_pts, use_pixel_weights=False,\n",
    "    pixwin=True, fwhm_out=0,\n",
    ")\n",
    "\n",
    "print(\"Point sources in TQU:\")\n",
    "print(f\"  ud_grade Q  \u2014 negative pixels: {(m_tqu_ud[1] < 0).sum()}\")\n",
    "print(f\"  harmonic Q  \u2014 negative pixels: {(m_tqu_harm[1] < 0).sum()}\")\n",
    "print(f\"  ud_grade U  \u2014 negative pixels: {(m_tqu_ud[2] < 0).sum()}\")\n",
    "print(f\"  harmonic U  \u2014 negative pixels: {(m_tqu_harm[2] < 0).sum()}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd639560",
   "metadata": {},
   "source": [
    "### 4.2 Choosing the Right Method for Mixed-Signal Maps\n",
    "\n",
    "Real maps often contain **both** diffuse emission (which benefits\n",
    "from `harmonic_ud_grade`) and compact features (which benefit from\n",
    "`ud_grade`).  Neither method is perfect in this regime:\n",
    "\n",
    "- `ud_grade` handles point sources well but **aliases** the diffuse\n",
    "  signal, corrupting the power spectrum and introducing artefacts\n",
    "  in the low-$\\ell$ modes.\n",
    "- `harmonic_ud_grade` preserves the diffuse signal but produces\n",
    "  **Gibbs ringing** around compact sources.  For polarisation maps,\n",
    "  this ringing appears in all three (T, Q, U) components.\n",
    "\n",
    "**Possible mitigations for mixed-signal maps:**\n",
    "\n",
    "1. **Separate and recombine:** Detect and subtract point sources\n",
    "   from the map, downgrade the diffuse component with\n",
    "   `harmonic_ud_grade` and the point-source component with\n",
    "   `ud_grade`, then add them back together.\n",
    "2. **Use `harmonic_ud_grade` with smoothing:** The default\n",
    "   `fwhm_out` applies a Planck-scaled beam that suppresses Gibbs\n",
    "   ringing, at the cost of slightly broadening compact sources.\n",
    "3. **For masks and hit-count maps:** always use `ud_grade`, since\n",
    "   these are pixel-localised by nature and harmonic methods would\n",
    "   produce unphysical ringing.\n",
    "\n",
    "The choice ultimately depends on the scientific application: if\n",
    "power-spectrum fidelity matters more than point-source compactness,\n",
    "use `harmonic_ud_grade`; if point-source photometry or masking is\n",
    "the priority, use `ud_grade`."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff7aaa87",
   "metadata": {},
   "source": [
    "## 5. Required `fwhm_in`\n",
    "\n",
    "`harmonic_ud_grade` requires `fwhm_in` so the resolution scaling\n",
    "is always explicit.  The default `fwhm_out` preserves the Planck\n",
    "FWHM-to-pixel ratio across resolutions.\n",
    "\n",
    "### 5.1 Default beam scaling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8f658f9b",
   "metadata": {},
   "outputs": [],
   "source": [
    "nside_in = 2048\n",
    "nside_out = 64\n",
    "\n",
    "# Planck beam at Nside=2048 is 5 arcmin\n",
    "fwhm_in = np.radians(5 / 60)\n",
    "\n",
    "m_input = np.zeros(hp.nside2npix(nside_in))\n",
    "\n",
    "# fwhm_out is auto-computed to preserve the ratio\n",
    "# fwhm_out = fwhm_in * (resol_out / resol_in)\n",
    "expected_fwhm_out = fwhm_in * (\n",
    "    hp.nside2resol(nside_out) / hp.nside2resol(nside_in)\n",
    ")\n",
    "\n",
    "# This should equal 160 arcmin (Planck Nside=64 beam)\n",
    "print(f\"Nside_in = {nside_in}, Nside_out = {nside_out}\")\n",
    "print(f\"  fwhm_in  = {np.degrees(fwhm_in)*60:.1f} arcmin\")\n",
    "print(f\"  expected fwhm_out = {np.degrees(expected_fwhm_out)*60:.1f} arcmin \u2190 matches Planck Nside=64\")\n",
    "\n",
    "# Verify by running the function (it will work even on zeros)\n",
    "m_out = hp.harmonic_ud_grade(\n",
    "    m_input,\n",
    "    nside_out=nside_out,\n",
    "    fwhm_in=fwhm_in,\n",
    "    use_pixel_weights=False,\n",
    "    pixwin=True,\n",
    ")\n",
    "print(f\"  \u2713 harmonic_ud_grade accepted the call with default fwhm_out\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "492dd1e7",
   "metadata": {},
   "source": [
    "## 6. Performance\n",
    "\n",
    "`harmonic_ud_grade` is more expensive than `ud_grade` because it\n",
    "requires a full spherical-harmonic transform (SHT) of the input\n",
    "map.  Below we benchmark both methods at `nside_in = 512` \u2192\n",
    "`nside_out = 128` (10 iterations each)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "27b0842d",
   "metadata": {},
   "outputs": [],
   "source": [
    "nside_in = 512\n",
    "nside_out = 128\n",
    "np.random.seed(42)\n",
    "# healpy synfast/synalm uses the global RNG\n",
    "m = hp.synfast(np.ones(3 * nside_in), nside_in, new=True)\n",
    "\n",
    "# ud_grade\n",
    "t0 = time.perf_counter()\n",
    "for _ in range(10):\n",
    "    _ = hp.ud_grade(m, nside_out=nside_out)\n",
    "t_ud = (time.perf_counter() - t0) / 10\n",
    "\n",
    "# harmonic_ud_grade\n",
    "t0 = time.perf_counter()\n",
    "for _ in range(10):\n",
    "    _ = hp.harmonic_ud_grade(\n",
    "        m,\n",
    "        nside_out=nside_out,\n",
    "        fwhm_in=0,\n",
    "        use_pixel_weights=False,\n",
    "        pixwin=True,\n",
    "        fwhm_out=0,\n",
    "    )\n",
    "t_harm = (time.perf_counter() - t0) / 10\n",
    "\n",
    "print(f\"ud_grade:           {t_ud*1000:.1f} ms\")\n",
    "print(f\"harmonic_ud_grade:  {t_harm*1000:.1f} ms\")\n",
    "print(f\"slowdown:           {t_harm/t_ud:.1f}x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37ee7008",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "The table below summarises the key differences between the two\n",
    "downgrading methods demonstrated in this notebook.\n",
    "\n",
    "| Feature | `ud_grade` | `harmonic_ud_grade` |\n",
    "|---------|------------|---------------------|\n",
    "| **Domain** | Pixel space (sub-pixel averaging) | Spherical-harmonic space (SHT) |\n",
    "| **Aliasing** | Heavy leakage at all $\\ell$ | Suppressed to numerical noise floor |\n",
    "| **Spectrum fidelity** | Corrupted \u2014 aliased power adds positive bias | Correct within the output band |\n",
    "| **Pixel-window handling** | Ignored | Deconvolved/re-applied (Planck 2015 XVI Eq. 1) |\n",
    "| **Beam scaling** | Not handled | Auto-scales FWHM to preserve the beam/pixel ratio |\n",
    "| **Custom beam** | Not supported | `beam_window_in` / `beam_window_out` arrays |\n",
    "| **Reconvolution** | Not applicable | `nside_out == nside_in` changes beam at same pixelisation |\n",
    "| **Gibbs ringing** | None \u2014 preserves positivity | Ringing around compact sources |\n",
    "| **Polarisation (diffuse)** | Aliased Q/U | Correct Q/U (separate T/E transfer) |\n",
    "| **Polarisation (point sources)** | Preserves compact Q/U | Gibbs ringing in Q/U |\n",
    "| **Typical speed** | Fast (\u2248 ms) | ~5\u201315\u00d7 slower (dominated by SHT) |\n",
    "\n",
    "**Recommendation:**\n",
    "- Use **`harmonic_ud_grade`** whenever scientific accuracy matters:\n",
    "  power-spectrum estimation, component separation, map-level\n",
    "  comparisons, or any analysis involving noisy or beam-deconvolved\n",
    "  data.  For non-Gaussian beams, provide `beam_window_in` /\n",
    "  `beam_window_out` arrays.\n",
    "- Use **`ud_grade`** when working with point-source maps, binary\n",
    "  masks, hit-count maps, or when speed is critical and aliasing\n",
    "  artefacts are acceptable.\n",
    "- For **mixed-signal maps** (diffuse + point sources), consider\n",
    "  separating the components and downgrading each with the\n",
    "  appropriate method."
   ]
  }
 ],
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