---
kind: concept
slug: implicit-neural-representation
title: Implicit neural representation (INR)
summary: >-
  An implicit neural representation is an MLP that maps coordinates to a signal, such as colour, occupancy or signed distance, so the signal lives in the weights. Four choices matter: the nonlinearity (WIRE's Gabor wavelet), the supervision (iSDF, online from depth), a latent code shared across objects (NeuSE, Neural Scene Graphs) and time as the coordinate (StyleGAN-V).
tags: [neural-fields, activation-function, inverse-problems, implicit-representation, signed-distance-function, latent-code]
papers: ["2204.02296", "2301.05187", "2303.07308", "2403.07563", "2011.10379", "2112.14683", "2112.05124", "2103.03231"]
related: [neural-radiance-field, dynamic-neural-radiance-field, neural-implicit-surface, positional-encoding, fourier-features, euclidean-equivariant-network, simultaneous-localization-and-mapping, feed-forward-reconstruction, knowledge-distillation, video-generation, dense-correspondence, learning-from-demonstration, novel-view-synthesis]
status: draft
updated_at: 2026-11-01
---

## Overview

An implicit neural representation (INR) is a multilayer perceptron that takes
a coordinate, such as a pixel position or a 3D point, and returns the signal
value there. The signal is stored in the network weights rather than in a grid
of samples [arxiv:2301.05187 §3.1]. Each hidden layer applies a linear map and
a nonlinearity σ [arxiv:2301.05187 eq. 1]. The choice of σ largely decides
which frequencies the network fits first and how readily it also fits noise
[arxiv:2301.05187 §1] [arxiv:2301.05187 §3.2].

Fitting is done by gradient descent on a loss between the forward-modelled
output and the measurements, with Adam and an l2 loss and no other
regularisation in the WIRE paper [arxiv:2301.05187 §4].

A second design axis is the training signal. Where WIRE fits a fixed set of
measurements, iSDF fits a signed distance field (SDF) online, as posed depth
frames arrive, and it constrains each prediction with a bound taken from the
measured surface [arxiv:2204.02296 §4] [arxiv:2204.02296 §4.3]. Its MLP uses a
positional embedding and softplus layers rather than a wavelet
[arxiv:2204.02296 §4.1].

A third axis is conditioning. NeuSE uses one occupancy decoder for every
object of a category. The decoder takes a query point and a per-object latent
code, so the object's identity enters through the code rather than through a
separate set of weights [arxiv:2303.07308 eq. 3] [arxiv:2303.07308 §3.1].

A fourth axis is the coordinate frame and the coordinate itself. Neural Scene
Graphs gives each dynamic object its own field in that object's frame, so
objects can be moved or removed by editing the graph [arxiv:2011.10379 §3].
StyleGAN-V treats a video as a function of continuous time and generates a
frame at any real-valued time [arxiv:2112.14683 §3.1].

Two further axes appear in the newer cards. NDF trains an occupancy network
and then reads the activations of all its layers as a descriptor for every
point, which is used to match poses rather than to reconstruct a signal
[arxiv:2112.05124 §II-A] [arxiv:2112.05124 eq. 4]. DONeRF keeps a coordinate
MLP for radiance but gives the sample placement to a second network, so the
expensive network is evaluated at few points per ray [arxiv:2103.03231 §5.1].

## Variants

Eight variants have paper cards. WIRE and iSDF each fit one signal, and they
differ in the activation (WIRE) and the supervision (iSDF). NeuSE shares one
decoder across objects of a category and differs in its conditioning. GeFF
goes further: one feed-forward encoder produces the latent for a whole scene
from posed RGB-D frames, so a new scene needs no per-scene fitting. Neural
Scene Graphs builds a graph of fields for one driving scene, and StyleGAN-V
uses time as its coordinate for video. NDF uses a shape-conditioned descriptor
field for manipulation, and DONeRF uses a depth oracle to place the samples of
a radiance field. The
other activations that the WIRE paper compares against (sine, Gaussian, ReLU
with positional encoding, and a Gabor-filter multiplicative network) are named
as baselines there, but they have no card yet [arxiv:2301.05187 §4].

### Wavelet activation (WIRE)

The nonlinearity is a complex Gabor wavelet with frequency ω0 and width s0.
Its complex exponential gives frequency selectivity, as in a sine activation,
and its Gaussian window gives spatial compactness, as in a Gaussian
activation [arxiv:2301.05187 §3.2] [arxiv:2301.05187 eq. 2]. Because the
window already keeps each layer local, the default uniform random
initialisation works for any ω0 and s0 [arxiv:2301.05187 §3.2]
[arxiv:2301.05187 app. A.1].

The paper also gives a real-valued form, sin(ω0 x) times a Gaussian window.
Setting s0 to 0 recovers a sine activation, and setting ω0 to 0 recovers a
Gaussian one [arxiv:2301.05187 §3.4]. For two or more spatial dimensions, the
paper adds one Gaussian window per extra coordinate to the same layer
[arxiv:2301.05187 §3.5] [arxiv:2301.05187 eq. 5]. Card:
[WIRE: Wavelet Implicit Neural Representations](/papers/2301.05187).

### Online signed distance field (iSDF)

The INR maps a 3D point to a signed distance, with the sign positive in front
of the surface [arxiv:2204.02296 §3] [arxiv:2204.02296 §4]. It is a randomly
initialised MLP with no pretraining and no ground-truth SDF [arxiv:2204.02296
§1] [arxiv:2204.02296 §4]. Each iteration samples points along rays from a few
posed depth frames: the two latest frames and three keyframes
[arxiv:2204.02296 §4.2].

The supervision is the distinctive part. For each sampled point, the bound is
the signed distance to the nearest measured surface point among all rays in
the batch [arxiv:2204.02296 eq. 4]. Points in free space take a one-sided loss
that is zero between 0 and the bound [arxiv:2204.02296 eq. 6]. Points near the
surface take an L1 term to the bound [arxiv:2204.02296 eq. 7]
[arxiv:2204.02296 eq. 8]. Cosine and Eikonal terms regularise the gradient
[arxiv:2204.02296 eq. 9] [arxiv:2204.02296 eq. 10]. The paper argues this
avoids the voxel grid and surface-fusion step of older real-time methods
[arxiv:2204.02296 §1] [arxiv:2204.02296 §2].

The paper assumes camera poses from an external tracker and does not do full
SLAM [arxiv:2204.02296 §4]. Card:
[iSDF: Real-Time Neural Signed Distance Fields for Robot Perception](/papers/2204.02296).

### Latent-conditioned occupancy decoder (NeuSE)

The decoder Φ maps a query point x and a latent code z to an occupancy value
in [0, 1] [arxiv:2303.07308 eq. 3] [arxiv:2303.07308 §3.1]. The code is not a
weight set. It is a latent point cloud of k = 512 points [arxiv:2303.07308 §5.1]
[arxiv:2303.07308 app. A1], produced by an encoder from a partial point cloud
of one object. The encoder is built so that the code transforms with the
object: rotating, translating or scaling the input transforms the code in the
same way [arxiv:2303.07308 eq. 1] [arxiv:2303.07308 eq. 2]
[arxiv:2303.07308 eq. 4].

That equivariance is what makes the codes useful for pose. Two codes of the
same object give a relative SE(3) transform in closed form, by Horn's method,
without rendering the object and fitting it [arxiv:2303.07308 §3.3]
[arxiv:2303.07308 eq. 6]. Ambiguous objects, such as upright bottles, are
trained with a Chamfer-distance loss on the aligned clouds, not a single
ground-truth pose [arxiv:2303.07308 §3.4] [arxiv:2303.07308 eq. 8].

The decoder is trained for each category separately, on simulated ShapeNet
objects rendered in PyBullet [arxiv:2303.07308 §5.1] [arxiv:2303.07308 app. A1].
Card: [NeuSE: Neural SE(3)-Equivariant Embedding for Long-Term Object-Based
Simultaneous Localization and Mapping](/papers/2303.07308).

### Generalizable scene feature field (GeFF)

GeFF is a scene-level, generalizable NeRF. A trainable encoder reads N posed
RGB-D frames and outputs N latent vectors, with no optimisation for the new
scene [arxiv:2403.07563 §3.1] [arxiv:2403.07563 §3.2]. Two decoders then read
any 3D point together with the latents. One gives a signed distance, used for
geometry and opacity along rays. The other gives a feature vector, distilled
from a frozen 2D vision-language teacher, which is used for open-vocabulary
queries [arxiv:2403.07563 §3.2] [arxiv:2403.07563 eq. 2] [arxiv:2403.07563
eq. 4]. The decoders are isotropic, so they do not take the viewing direction
[arxiv:2403.07563 §3.2].

The training signal combines two sources. Novel views with ground truth
supervise the rendered depth and features, and the SDF is trained with the
iSDF loss plus an Eikonal term [arxiv:2403.07563 §3.2] [arxiv:2403.07563 eq. 6].
The paper trains the feature head to match the teacher's features on those
novel views, not on the input views the encoder sees [arxiv:2403.07563 §3.2].
Pre-training is on ScanNet for 50 epochs; the paper reports six days on eight
GPUs [arxiv:2403.07563 §4.1]. Card:
[Learning Generalizable Feature Fields for Mobile Manipulation](/papers/2403.07563).

Where the NeuSE latent encodes one object, the GeFF latents cover a whole
scene. Both let one decoder read a latent at arbitrary points. NeuSE's encoder
is designed to be SE(3)-equivariant [arxiv:2303.07308 eq. 1], and the GeFF
paper does not make an equivariance claim for its latents [arxiv:2403.07563
§3.1].

### Object-graph radiance fields (Neural Scene Graphs)

The scene is a directed acyclic graph with a camera, one static background
field and one radiance field per dynamic object [arxiv:2011.10379 eq. 1]
[arxiv:2011.10379 §3]. Each object field is defined in its own frame. A
world point is mapped into that frame using the object's box size, so points
inside the box fall in [-1, 1] [arxiv:2011.10379 §3.2] [arxiv:2011.10379 eq. 8].
Objects of one class share weights and differ through a 256-dimensional
latent vector [arxiv:2011.10379 §3.2] [arxiv:2011.10379 eq. 6].

The object density depends only on the object-frame point and its latent. The
colour also depends on the view direction and on the object's global position,
which keeps the density consistent across poses [arxiv:2011.10379 §3.2]
[arxiv:2011.10379 eq. 7] [arxiv:2011.10379 eq. 9]. The background is sampled
on a few planes parallel to the camera image plane, not ray-marched through a
volume [arxiv:2011.10379 §4.1]. Edges between nodes carry rigid transforms or
scales, so moving or removing an object is an edit to the graph
[arxiv:2011.10379 §3]. The training data are driving videos plus object
tracks, which come from manual labels or from a tracker [arxiv:2011.10379 §5.1].
Card: [Neural Scene Graphs for Dynamic Scenes](/papers/2011.10379).

### Time-coordinate video generator (StyleGAN-V)

StyleGAN-V is a GAN, not a per-signal fit, but it uses the coordinate idea
along the time axis. A video is modelled as x(t) for continuous t ≥ 0, and
training uses sparse subsamples of it [arxiv:2112.14683 §3]. A content code
sets the look of a clip and is shared by all its frames [arxiv:2112.14683
§3.1]. A motion code is computed for any query time from a grid of noise
codes, using wave-based time embeddings that are acyclic and can be queried
at real-valued times [arxiv:2112.14683 §3.1] [arxiv:2112.14683 eq. 4]. The
frames are generated independently, so generation is not autoregressive
[arxiv:2112.14683 §3.1]. The synthesis network is unchanged except for its
constant input, which takes the motion code [arxiv:2112.14683 fig. 5].
Card: [StyleGAN-V](/papers/2112.14683).

### Descriptor field for pose matching (NDF)

NDF's core network is an occupancy decoder conditioned on one object's point
cloud. A point's descriptor is the concatenation of the activations of every
layer of that network [arxiv:2112.05124 eq. 4] [arxiv:2112.05124 §II-A]. The
encoder and the decoder are made SE(3)-equivariant with Vector Neurons, so
transforming the cloud and the query point by the same rigid motion moves the
descriptor field with them [arxiv:2112.05124 eq. 5] [arxiv:2112.05124 eq. 7].
Training is 3D reconstruction from partial clouds, with no keypoint or
correspondence labels [arxiv:2112.05124 §II-A].

A rigid body, such as a gripper or a rack, is represented by query points
placed in its box, and its pose descriptor stacks the descriptors of those
points [arxiv:2112.05124 §II-B] [arxiv:2112.05124 eq. 10]. In the few-shot
pick-and-place procedure, the pick and relative-place descriptors are averaged
over the demonstrations. A test pose is found by minimising the L1 distance
between the candidate's pose descriptor and that average
[arxiv:2112.05124 §II-C] [arxiv:2112.05124 eq. 11]. Card:
[Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](/papers/2112.05124).

### Oracle-guided sampling of a radiance field (DONeRF)

DONeRF keeps a coordinate MLP that returns colour and opacity, but it changes
how the samples on each ray are chosen. A depth oracle is evaluated once per
ray. It scores depth classes on a logarithmic axis, and the scores are read as
a piecewise-constant density whose inverse CDF sets the sample depths
[arxiv:2103.03231 §5.1] [arxiv:2103.03231 §5.2]. A shading MLP then evaluates
only those few samples [arxiv:2103.03231 §3].

The oracle is trained as a classifier against the depth of the first surface,
so training needs a reference depth for each pixel [arxiv:2103.03231 §5.1]
[arxiv:2103.03231 §6.1]. Its targets are spread across neighbouring rays and
depth classes, which makes the oracle keep a little extra empty space
[arxiv:2103.03231 eq. 7] [arxiv:2103.03231 eq. 8] [arxiv:2103.03231 §5.1].
Without a reference depth, the paper takes depth maps from a dense NeRF
instead [arxiv:2103.03231 §6.1]. Card:
[DONeRF: Towards Real-Time Rendering of Compact Neural Radiance Fields using Depth Oracle Networks](/papers/2103.03231).

## When to use which

The papers' results are author-reported. Each row names the constraint and
the evidence for it.

| Situation | Use | Why |
|---|---|---|
| Noisy measurements, denoising or inverse problems | WIRE | The authors report the highest PSNR among the compared INRs for denoising and CT, and say the fit does not overfit the noise as quickly [arxiv:2301.05187 §4.2] [arxiv:2301.05187 §3.3] |
| Few views, for example NeRF with 25 or 100 training images | WIRE | +0.2 dB at 25 views and +0.4 dB at 100 views over the other nonlinearities or SIREN on the synthetic drums scene [arxiv:2301.05187 §4.3] [arxiv:2301.05187 app. A.5]. One scene only |
| Sensitivity to the frequency and width settings | WIRE | Less sensitive to ω0 and s0 than SIREN or Gaussian activations [arxiv:2301.05187 §3.4] [arxiv:2301.05187 fig. 3] |
| Fewer training epochs to a target PSNR | WIRE | The authors report an order of magnitude fewer epochs than other INRs on denoising [arxiv:2301.05187 §3.3] |
| Complex-valued weights not available in your framework | WIRE, real form | The real form sin(ω0 x) times a Gaussian window is given as an alternative [arxiv:2301.05187 §3.4] |
| Wall-clock time on very large signals | Not established | The paper reports 1.6 minutes for a 43.2 dB Kodak fit [arxiv:2301.05187 §4.1] but gives no timing against other INRs |
| Online mapping from a posed depth camera, with a queryable SDF while exploring | iSDF | The authors report SDF error below 6 cm at the end of six main sequences and about 33 ms per training iteration on their hardware [arxiv:2204.02296 §6.2] [arxiv:2204.02296 §6.3] |
| Planner needs collision cost and gradients from a compact field | iSDF | The authors report the lowest collision-cost error among the compared methods, and a 1 MB network plus about 20 keyframes, against 5 MB for Voxblox and 30 MB for KinectFusion+ [arxiv:2204.02296 §6.2] [arxiv:2204.02296 §6.3] |
| Regions not seen by the camera need an estimate | iSDF, qualitatively | The paper shows filled-in regions and the back of objects, with no numerical measure [arxiv:2204.02296 §6.1] |
| No camera poses available | Neither as stated | iSDF assumes an external tracker [arxiv:2204.02296 §4] and the paper does not evaluate tracking |
| Object-level SLAM in a scene where objects are added, removed or moved between visits, with a trained category | NeuSE | The authors report full-trajectory ATE of 0.047 m for the multi-category set on a planar layout, against 0.116 m for a mug-only model, and 100% precision and recall for change detection on the 4-Round sequence, against 87.5% recall for PMT [arxiv:2303.07308 tab. 1] [arxiv:2303.07308 tab. 7] |
| Add object constraints to an existing SLAM system | NeuSE | The authors report the 4-Round full ATE falling from 0.128 m to 0.093 m with ORB-SLAM3 plus NeuSE [arxiv:2303.07308 tab. 5] |
| Objects of a category never seen in training, or open-set use | iSDF, with a caveat | iSDF has no category pretraining [arxiv:2204.02296 §4], while NeuSE is trained per category [arxiv:2303.07308 app. A1] and names this as its main limit for open-set use [arxiv:2303.07308 §9] |
| Depth noise of about 1 cm | Not established from these papers | At 1 cm the paper reports that NeuSE fails to recover a valid trajectory, with ATE of 37.4 m [arxiv:2303.07308 tab. 4] [arxiv:2303.07308 §7.1] |
| Open-vocabulary text queries over a whole room, on a robot, with no per-scene optimisation | GeFF | The authors report 0.39 s onboard latency on an AGX Orin and an overall success of 62.4%, against 43.2% for LERF, which takes about 2 h offline per scene [arxiv:2403.07563 tab. 1] |
| Part-level queries, such as a mug handle or a cart bar | GeFF, with low expectations | The authors report 44.4%, 66.7% and 80.0% for the three part queries, against 36.1%, 20.0% and 70.0% for LERF [arxiv:2403.07563 tab. 1]. The part set is small, with 4 instances per category [arxiv:2403.07563 §4.1] |
| Objects added after the initial scan | GeFF, with a drop | The authors report 4/9, 6/9 and 8/9 successes with added objects, against 7/9, 7/9 and 8/9 without, across three scenes [arxiv:2403.07563 tab. 3]. LERF was not run with changes |
| Compact latents that still give usable geometry at a 512-point budget | GeFF | The authors report a depth L2 error of 0.012, against 0.040 and 0.061 for two downsample-and-upsample baselines [arxiv:2403.07563 tab. 4] [arxiv:2403.07563 §4.2 A6] |
| Reproducing GeFF from the paper alone | Not established | No official code was found, and the learning rate, batch size, loss weights and latent width are not stated [arxiv:2403.07563 §3.2] [arxiv:2403.07563 §4.1] |
| Dynamic driving scene where objects must be moved, removed or recombined, with object tracks available | Neural Scene Graphs | The authors report PSNR 26.66 against 23.34 for NeRF on three KITTI scenes, and 25.11 against 18.25 for novel composition, each scene trained on its own sequence [arxiv:2011.10379 tab. 1] [arxiv:2011.10379 §5.2] |
| Dynamic scene with no object tracks or boxes | Not this method | The graph places each object with a box and track, from labels or a tracker [arxiv:2011.10379 §5.1] |
| Manipulation from a few demonstrations, with object poses that vary and no keypoint labels | NDF | The authors report overall success of 0.88 for an upright mug with 10 demonstrations, against 0.45 for Dense Object Nets, and 0.58 for mugs in arbitrary poses against 0.17 [arxiv:2112.05124 tab. I] |
| One demonstration only | NDF, with a drop | The authors report 0.46 overall success with one demonstration, against 0.32 for Dense Object Nets, but this count is tested on the upright mug only [arxiv:2112.05124 tab. IV] |
| Quantitative evidence on a real robot | Not established | The paper reports qualitative real-robot runs and a video, with no real-robot numbers [arxiv:2112.05124 §IV-C] |
| Real-time radiance-field rendering with a few samples per ray, and reference depth for training | DONeRF | The authors report mean PSNR 31.14 dB at 4 samples per ray, against 30.52 dB for NeRF at 256 samples per ray, over six Blender scenes [arxiv:2103.03231 tab. 2]. At 800×800 with 2 samples the authors report about 51 ms per frame on one RTX 3090, with shading the largest stage [arxiv:2103.03231 §6.1] |
| Training depth is unavailable | DONeRF, with a loss | Averaged over the scenes, the variant with depth maps from a dense NeRF scores 0.8 to 1.5 dB below the variant with ground truth, for 2 to 16 samples per ray [arxiv:2103.03231 tab. 2] |
| Dynamic scene, or a scene with many views and changing content | Not established for either | DONeRF is tested on static synthetic Blender scenes [arxiv:2103.03231 §1] [arxiv:2103.03231 §6.2]. NDF is tested on simulated tabletop objects with a static environment [arxiv:2112.05124 §III] [arxiv:2112.05124 §VI] |
| Speed comparison against NeRF in FLOP per pixel | DONeRF, with a caveat | The FLOP column gives about 15 to 78 times fewer operations than NeRF, but the paper reports no NeRF wall-clock time [arxiv:2103.03231 tab. 2] [arxiv:2103.03231 §6.3] |
| Frame rate or clip length chosen at generation time, with no fixed frame count | StyleGAN-V | Frames are generated at any time from the motion codes [arxiv:2112.14683 §3.1] [arxiv:2112.14683 §4.3]. The authors report FVD16 of 47.41 on FaceForensics, against 62.5 for DIGAN and 124.7 for MoCoGAN [arxiv:2112.14683 tab. 1] |
| Image quality close to StyleGAN2 is needed for each frame | StyleGAN-V, with a cost | The authors report FID 9.44 against 8.42 for StyleGAN2 on FaceForensics at 256², about 10% worse [arxiv:2112.14683 §4.3] |
| Clips of unknown length, with a per-clip time gap | StyleGAN-V, with caution | The paper samples a window of up to 1024 frames and gaps up to 32 frames [arxiv:2112.14683 §3.3] [arxiv:2112.14683 app. B]; longer or higher-rate clips are shown only as samples [arxiv:2112.14683 §4.3] |

Caveats for the rows above:

- Most inverse-problem results use one image or one volume, so generalisation
  to other signals is not shown [arxiv:2301.05187 §4.2] [arxiv:2301.05187 app. A.4].
- The implicit-bias argument rests on an empirical NTK study on 64 by 64
  images, and the authors argue rather than prove it [arxiv:2301.05187 §3.3].
- Settings were picked to give the fastest approximation on the test signal,
  with no validation split stated [arxiv:2301.05187 §4.1].
- The iSDF evaluation uses simulated ReplicaCAD and handheld ScanNet sequences,
  and the paper describes no robot-platform experiment [arxiv:2204.02296 §5.1].
- The iSDF results are read from text and a figure. The authors say the
  baselines are plotted in figure 8 only [arxiv:2204.02296 §6.2].
- The iSDF timing is for one optimisation iteration on one hardware setup. Frame
  rate and tracking are not evaluated [arxiv:2204.02296 §5.2] [arxiv:2204.02296 §6.3].
- The released iSDF configs use a different bound from the one the paper uses in
  its experiments. The [iSDF card](/papers/2204.02296) lists the differences
  between paper and code.
- NeuSE is trained only in simulation, on five object categories [arxiv:2303.07308 §5.1].
  Its real-world tests add two objects of an untrained size, and the paper
  does not treat those runs as comparable to the controlled sequences
  [arxiv:2303.07308 §7.2].
- The NeuSE timings (6 ms of network inference per object, about 4 fps for the
  full localisation system in the crowded controlled setting) are on the
  authors' laptop, with sequential processing [arxiv:2303.07308 §7].
- GeFF's main comparison mixes timings. LERF and ConceptGraphs were run offline
  with all frames batched, and ConceptGraph-Online ran on the robot with
  dropped frames [arxiv:2403.07563 §4.1]. The 62.4% versus 43.2% gap therefore
  compares different operating modes as well as different methods.
- GeFF's robot results come from one platform in four rooms, with 3 trials per
  object in the scene-change cells and no confidence intervals [arxiv:2403.07563
  §4.1] [arxiv:2403.07563 §4.2 A4].
- Neural Scene Graphs reports numbers only for three KITTI scenes. The
  qualitative results, the tracker comparison and the 3D detection are described
  in words, with no figures in the text [arxiv:2011.10379 §5.2] [arxiv:2011.10379 fig. 9].
- Neural Scene Graphs uses a different box scale for some object labels and a
  different latent penalty from the paper's formula. The
  [Neural Scene Graphs card](/papers/2011.10379) lists these differences.
- StyleGAN-V's FVD numbers come from the authors' own re-trained baselines and
  FVD protocol, so they are not comparable with numbers from other papers
  [arxiv:2112.14683 §4.1] [arxiv:2112.14683 app. C].
- StyleGAN-V's frame count is a setting. Table 3 has k = 2 with a lower
  SkyTimelapse FVD16 than the default k = 3, while k = 3 is lowest on
  FaceForensics [arxiv:2112.14683 tab. 3].
- NDF's numbers are simulation results in PyBullet, with 10 demonstrations per
  task and one model shared by mug, bowl and bottle [arxiv:2112.05124 §III].
  The authors' released code differs from the paper in several places, such as
  descriptor normalisation and input scale. The [NDF card](/papers/2112.05124)
  lists them.
- DONeRF's baseline NeRF is a PyTorch port with the same layout as DONeRF's
  shading network, not the original NeRF network, and it uses a different
  batch size [arxiv:2103.03231 app. E] [arxiv:2103.03231 §6.1]. The
  comparison matches capacity, not the full training recipe.
- The DONeRF official code is non-commercial, so the [DONeRF card](/papers/2103.03231)
  is written from the paper alone and no code details are used here.

## Implementation sketch

The WIRE activation and its forward pass, following [arxiv:2301.05187 §3.1]
and [arxiv:2301.05187 §3.2]. Complex arithmetic keeps phase through the
hidden layers, and only the real part of the output is the signal.

```pseudocode
# Gabor activation, complex-valued  [arxiv:2301.05187 eq. 2]
# Input:  pre-activation x (N, H)
# Output: activation (N, H), complex

def gabor(x, omega_0, scale_0):
    # e^{j w0 x} times the Gaussian window e^{-|s0 x|^2}
    return exp(1j * omega_0 * x - abs(scale_0 * x) ** 2)
```

```pseudocode
# Forward pass of a WIRE-style INR  [arxiv:2301.05187 §3.1] [§3.2]
# Input:  coords (N, D) in [-1, 1]
# Output: signal (N, C), real

def inr_forward(coords, first, hidden, readout, omega_0, scale_0):
    # First layer: real weights, output is complex  [§3.2]
    h = gabor(first(coords), omega_0, scale_0)
    # Hidden layers: complex weights and activations  [eq. 1]
    for layer in hidden:
        h = gabor(layer(h), omega_0, scale_0)
    # Read out and keep the real part as the signal  [§3.2]
    return real(readout(h))
```

Starting values the paper gives for images: ω0 = 20 and s0 = 10 for WIRE, and
three hidden layers [arxiv:2301.05187 §4.1]. The learning rate is not stated
for images. The authors' code sets some values differently from the paper; see
the [WIRE card](/papers/2301.05187).

The batch bound used by iSDF for one sampled point, following
[arxiv:2204.02296 §4.3] and [arxiv:2204.02296 eq. 4]. The sign is positive in
front of the surface.

```pseudocode
# Batch bound for one sampled point  [arxiv:2204.02296 eq. 4]
# Input:  sample x (3,), measured surface points P (K, 3), sign s in {-1, 1}
# Output: scalar bound on the predicted signed distance

def batch_bound(x, P, s):
    # Distance from x to the nearest measured surface point in the batch
    return s * min(norm(x - p) for p in P)
```

The paper gives this as the bound for the free-space loss, zero for a
prediction between 0 and the bound [arxiv:2204.02296 eq. 6]. The truncation
distance is not stated in the paper, so an implementer must choose it
[arxiv:2204.02296 §4.4].

The NeuSE decoder query, following [arxiv:2303.07308 eq. 3] and
[arxiv:2303.07308 eq. 5]. The latent is centred, and the query is shifted by
the latent's centre so that the decoder sees a translation-free input.

```pseudocode
# Occupancy query of a latent-conditioned decoder  [arxiv:2303.07308 eq. 5]
# Input:  query points x (M, 3), latent z (k, 3) from the encoder
# Output: occupancy in [0, 1], (M,)

def occupancy(x, z, decoder):
    # Centre of the latent code  [eq. 5]
    z_bar = mean(z, axis=0)                            # (3,)
    # Decoder sees the centred latent and the shifted query  [eq. 5]
    # The decoder ends in a sigmoid  [arxiv:2303.07308 tab. A2]
    return decoder(x - z_bar, z - z_bar)               # (M,)
```

## Contradictions

None across papers: the eight cards use different activations, supervision,
conditioning and domains, and no paper compares directly against another of
the eight.
Within WIRE and iSDF, the authors' released code differs from the paper for some settings.
The [WIRE card](/papers/2301.05187) and the [iSDF card](/papers/2204.02296) list
the differences. NeuSE and GeFF have no public code, so their cards record no
code differences.

Within GeFF, the card's sources disagree on the teacher. The method names
MaskCLIP as the teacher [arxiv:2403.07563 §3.2], while the training details
name a ViT-L CLIP model [arxiv:2403.07563 §4.1]. The paper does not say whether
MaskCLIP is applied to a ViT-L CLIP backbone, so both are reported here.

Within NeuSE, the text and the table disagree on one reduction. The text gives
a 65% ATE improvement on part of one trajectory [arxiv:2303.07308 §7.1]. Table 3
gives full-trajectory reductions against the point-cloud baseline of about 58%
(planar) and 66% (non-planar) [arxiv:2303.07308 tab. 3]. Both figures are
stated here as the paper gives them, without choosing between them.

Within StyleGAN-V, the text and Table 1 disagree on how far the method leads.
The text says it "significantly outperforms" the others on almost all
benchmarks [arxiv:2112.14683 §4.1]. Table 1 supports the lowest FVD16 on three
of four benchmarks, and VideoGPT is lowest on RainbowJelly [arxiv:2112.14683
tab. 1]. The abstract's average-improvement figure is not shown in the sections
read here [arxiv:2112.14683 abstract]. Both sides are stated as the paper gives them.

Within DONeRF, the paper gives four speed-up ranges for the FLOP reduction
against NeRF: up to 48 times in the abstract [arxiv:2103.03231 abstract], 24
to 98 times in the introduction [arxiv:2103.03231 §1], 15 to 78 times in
§6.3 [arxiv:2103.03231 §6.3], and 48 to 78 times in the caption of tab. 2
[arxiv:2103.03231 tab. 2]. All four are stated here as the paper gives them.

Within NDF, the introduction says overall success is above 85% [arxiv:2112.05124 §I].
Table I gives 0.87 to 0.91 for upright objects, which fits that claim, and 0.58
to 0.78 for arbitrary poses, which does not [arxiv:2112.05124 tab. I].

## Papers

- [WIRE: Wavelet Implicit Neural Representations](/papers/2301.05187): a complex Gabor wavelet as the INR activation, with results on images, occupancy, denoising, super-resolution, CT and NeRF.
- [iSDF: Real-Time Neural Signed Distance Fields for Robot Perception](/papers/2204.02296): an MLP trained online from posed depth images, supervised by a batch bound from the nearest measured surface point.
- [NeuSE: Neural SE(3)-Equivariant Embedding for Long-Term Object-Based Simultaneous Localization and Mapping](/papers/2303.07308): a per-category occupancy decoder conditioned on an SE(3)-equivariant latent point cloud, used for object association, pose constraints and change detection in SLAM.
- [Learning Generalizable Feature Fields for Mobile Manipulation](/papers/2403.07563): a feed-forward scene encoder with SDF and distilled language-feature decoders, used for navigation and part-level grasping on a legged robot.
- [Neural Scene Graphs for Dynamic Scenes](/papers/2011.10379): one static background field and one field per tracked object in a scene graph, with class-shared weights and per-object latents, so objects can be edited and re-rendered.
- [StyleGAN-V: A Continuous Video Generator with the Price, Image Quality and Perks of StyleGAN2](/papers/2112.14683): a StyleGAN2-based video GAN whose motion codes are a continuous function of time, so frames are generated at any real-valued time.
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](/papers/2112.05124): an equivariant descriptor field from occupancy-network activations, used to match demonstrated poses for few-shot pick-and-place.
- [DONeRF: Towards Real-Time Rendering of Compact Neural Radiance Fields using Depth Oracle Networks](/papers/2103.03231): a depth oracle that places a few samples per ray for a shading MLP, reaching NeRF-level PSNR with few evaluations.