---
kind: concept
slug: frequency-regularization
title: Frequency regularization
summary: >-
  Hide the high-frequency bands of a sinusoidal input encoding at the start of training and reveal them step by step. Shown for few-shot NeRF: FreeNeRF reports 19.92 object PSNR on DTU with 3 views and 1.02-1.04x the baseline training time, author-reported. Main trade-off: the schedule end T, which the paper sets differently for 9 views in two places.
tags: [nerf, few-shot, positional-encoding, regularization]
papers: ["2303.07418"]
related: [neural-radiance-field, novel-view-synthesis, few-shot-learning]
status: draft
updated_at: 2026-10-18
---

## Overview

A neural network fed raw coordinates through a sinusoidal encoding can learn
high-frequency noise before it has learned the coarse shape. With few training
images this is the failure that matters: the model fits the training views and
invents geometry, such as blobs near the camera, in views it has not seen
[arxiv:2303.07418 §3.2] [fig. 1]. Frequency regularization fixes the timing,
not the model. It gates the encoding so that only the low frequencies are
visible at first and the higher ones are revealed as training proceeds
[arxiv:2303.07418 §3.2] [eq. 4]. The invariant is that the full encoding is
visible at the end of the schedule, so the final model has the same inputs as
a plain NeRF; only the order in which they are learned changes. For the
general problem, see [Few-shot learning](/concepts/few-shot-learning).

## Variants

### Linearly revealed mask on the NeRF positional encoding

Multiply every entry of the encoded input (the raw coordinates plus their
sine/cosine bands) by a visibility weight. The weight is 1 for the bands that
are fully visible at step $t$, a fractional value for the next three entries,
and 0 for the rest. The visible prefix grows linearly with $t$, at $L/T$
entries per step, so the encoding is fully visible at the schedule end $T$
[arxiv:2303.07418 §3.2] [eq. 5]. The same kind of mask is applied to the
view-direction encoding [arxiv:2303.07418 §4.1]. Card:
[FreeNeRF: Improving Few-shot Neural Rendering with Free Frequency Regularization](/papers/2303.07418).

- Blender, 8 views, PSNR: 24.259 for FreeNeRF against 23.147 for DietNeRF and
  13.931 for the reproduced NeRF baseline [arxiv:2303.07418 tab. 1].
- DTU, object PSNR, 3 / 6 / 9 views: 19.92 / 23.25 / 25.38, against 18.50 /
  22.18 / 24.88 for the reproduced RegNeRF baseline [arxiv:2303.07418 tab. 2].
- LLFF, PSNR, 3 views: 19.63, against 18.84 for the reproduced RegNeRF baseline
  [arxiv:2303.07418 tab. 3].
- Training-time multiplier over each baseline: 1.02x (Blender), 1.04x (DTU and
  LLFF) [arxiv:2303.07418 tab. 4]. These are wall-clock figures on the authors'
  setup.
- The paper pairs this mask with a separate penalty on density near the camera
  [arxiv:2303.07418 §3.3]. That penalty is not frequency regularization, and
  this page does not cover it.

The paper distinguishes this approach from the coarse-to-fine frequency
schedules in Nerfies and BARF, saying it targets few-shot failure modes
[arxiv:2303.07418 §3.2]. Neither has a card in this wiki, so they are not
described here.

## When to use which

| Situation | Use | Why |
|---|---|---|
| NeRF-style model with a positional encoding, 3 to 9 input views, no budget for pre-training | The linearly revealed mask | The paper reports gains over per-scene regularized baselines, and says the method needs neither pre-training nor extra supervision [arxiv:2303.07418 §1] [tab. 2] [tab. 3] |
| Training budget is tight | The mask, with its reported 1.02-1.04x overhead | Compared with 1.69x (DTU) and 1.98x (LLFF) for RegNeRF [arxiv:2303.07418 tab. 4] |
| More than 9 input views | Not evaluated | The experiments cover 3, 6 and 9 views [arxiv:2303.07418 §4.1] |
| Model has no sinusoidal input encoding | Not applicable | The mask acts on the encoded entries only [arxiv:2303.07418 eq. 4] |
| Want more high-frequency detail with 9 views, at some PSNR cost | Shorter schedule | The paper reports that a shorter schedule shows more high-frequency detail at 9 views [arxiv:2303.07418 app. A.1] |

Schedule end T: the paper's default is 90% of training for 3 views and 70% for
6 views [arxiv:2303.07418 §4.1]. For 9 views the paper gives two values, see
Contradictions. The authors report that a longer schedule gives higher PSNR
but lower LPIPS, and that fine-tuning the trained model can recover LPIPS
[arxiv:2303.07418 §4.3]. The per-T results are only in a figure, so the page
does not quote them [arxiv:2303.07418 fig. 7].

Attribution is limited. The paper's ablation removes the occlusion penalty
[arxiv:2303.07418 tab. 5], and the text does not give numbers for a run without
frequency regularization; the paper shows that run only in a figure
[arxiv:2303.07418 fig. 7]. So the page cannot say how much of the gain comes
from the mask alone.

## Implementation sketch

The core is one mask multiplied into the encoded input. The paper summarizes
the schedule as a single line of code [arxiv:2303.07418 §3.2].

```pseudocode
# Visibility weight for each encoded entry at step t  [arxiv:2303.07418 eq. 5]
# Inputs: step t, schedule end T, encoded band count L
# Output: list of L + 3 weights in [0, 1]
def visibility(t, T, L):
    # Progress along the schedule, in encoded entries  [arxiv:2303.07418 eq. 5]
    progress = t * L / T
    fraction = progress - floor(progress)
    weights = []
    for i in range(1, L + 4):
        # Entries up to progress + 3 are fully visible  [arxiv:2303.07418 eq. 5]
        if i <= progress + 3:
            weights.append(1.0)
        # The next three entries are faded in  [arxiv:2303.07418 eq. 5]
        elif i <= progress + 6:
            weights.append(fraction)
        # Higher frequencies stay hidden  [arxiv:2303.07418 eq. 5]
        else:
            weights.append(0.0)
    return weights
```

```pseudocode
# Masked encoding of one input vector  [arxiv:2303.07418 eq. 4]
# Inputs: raw vector x, step t, schedule end T
# Output: encoded vector of length L + 3
def masked_encoding(x, t, T, L):
    # Raw input followed by its sine/cosine bands  [arxiv:2303.07418 eq. 2]
    encoded = concat(x, positional_encoding(x, L))
    # Scale each entry by its visibility weight  [arxiv:2303.07418 eq. 4]
    return encoded * visibility(t, T, L)
```

Where implementations may differ:

- The mask applies to view directions too, but the paper does not give the
  direction encoding length [arxiv:2303.07418 §4.1].
- The paper states the mask with $L$ as the encoded length of the coordinates.
  The paper card records how the released code's ramp differs slightly from
  eq. 5; see [the paper card](/papers/2303.07418).
- The paper's photometric loss form is not given in its text, so the loss in
  a training step must come from the baseline you build on.

## Contradictions

- **Schedule end for 9 views.** The main text gives 20% of training
  [arxiv:2303.07418 §4.1]. The appendix gives 50% [arxiv:2303.07418 app. A.1].
  The two values are not reconciled in the paper, so check it before picking
  one. The paper's 9-view comparison reports an object PSNR of 25.59 at a 90%
  schedule and 25.38 at a 50% schedule [arxiv:2303.07418 app. A.1].

## Papers

- [FreeNeRF: Improving Few-shot Neural Rendering with Free Frequency Regularization](/papers/2303.07418): introduces the linearly revealed positional-encoding mask, with an occlusion penalty on near-camera density as a second regularizer.