niiv: Fast Self-supervised Neural Implicit
Isotropic Volume Reconstruction
Jakob Troidl1 , Yiqing Liang2 , Johanna Beyer1 , Mojtaba Tavakoli3 , Johann
Danzl3 , Markus Hadwiger4 , Hanspeter Pfister1 , and James Tompkin2
1

Harvard University,2 Brown University,3 IST Austria,4 KAUST

Abstract. Three-dimensional (3D) microscopy data often is anisotropic
with significantly lower resolution (up to 8×) along the z axis than along
the xy axes. Computationally generating plausible isotropic resolution
from anisotropic imaging data would benefit the visual analysis of largescale volumes. This paper proposes niiv, a self-supervised method for
isotropic reconstruction of 3D microscopy data that can quickly produce
images at arbitrary (continuous) output resolutions. Within a neuralfield, the representation embeds a learned latent code that describes theimplicit higher-resolution isotropic image region. Under isotropic volume assumptions, we self-supervise this representation on low-/highresolution lateral image pairs to reconstruct an isotropic volume fromlow-resolution axial images. We evaluate our method on simulated andreal anisotropic electron (EM) and light microscopy (LM) data. Compared to a state-of-the-art diffusion-based method, niiv shows improved
reconstruction quality (+1 dB PSNR) and is over three orders of magnitude faster (2,000×) to infer. Specifically, niiv reconstructs a 1283 voxel
volume in 1/10th of a second, renderable at varying (continuous) highresolutions for display.
Keywords: Volume Reconstruction · Neural Fields · Self-Supervised

1

Introduction

3D imaging data is ubiquitous in scientific domains such as biology or materialsciences. However, many imaging modalities like 3D electron microscopy (EM) or
light microscopy (LM) have limited axial (z) resolution due to physical sectioningof tissue slices or optical limitations. Thus, resolution is typically much higher
in the lateral directions than in the axial direction (Fig. 1a).
Downstream tasks like interactive visual analysis would benefit from highresolution isotropic volumes, but these can be extremely costly or impossible
to obtain. Several computational methods attempt to generate isotropically resolved volumes [29,30,11,20]. Some approaches [29,30] require specific shape priors, such as exact point spread functions (PSFs), which are difficult to measure
in practice. Machine learning approaches like diffusion can model complex data
distributions but require copious training data and are slow to infer [11,20]—
requiring multiple minutes or even hours to reconstruct a small isotropic volume.

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Authors Suppressed Due to Excessive Length

Fig. 1. Training and reconstruction. (a) Lateral (xy) slices in anisotropic volumes
have a high spatial but low inter-frame resolution, while axial (xz/yz) slices have a
low spatial but high inter-frame resolution. (b) Thus, our model is trained to predicthigh-resolution lateral slices from artificially downsampled input. Our model infers
isotropic volumes at test time by predicting high-resolution axial (xz/yz) slices fromtheir low-resolution counterparts.

At the same time, anisotropic imaging volumes grow in size every year, containing terabytes [35,27] or even petabytes [23] of imaging data. Reconstructing
isotropic volumes is typically an offline postprocess after image acquisition, butwith increasing data sizes this becomes infeasible. For instance, Lee et al. [12]take up to three minutes to reconstruct a 1283 voxel volume. Instead, isotropic
volumes should be reconstructed on-demand and locally at interactive rates fromanisotropic data for visual inspection.
To this end, we propose a self-supervised method to reconstruct isotropic
volumes from anisotropic data. Building on recent advances in neural field representations [31,10,18,25], our model uses a super-resolution encoder [13] to relate a low-resolution axial slice to a plausible high-resolution image via a latent
space [2,3]. A multi-layer perceptron (MLP) decodes a set of latent codes intoa high-resolution axial slice sample, with bilinear latent interpolation creating
an output image at any pixel resolution (Fig. 2). Both encoder and decoder aretrained end-to-end on simulated anisotropic slices by downsampling isotropic lateral slices (Fig 1b). Inference is fast, requiring only 1/10th of a second to generate
a 128 × 128 × 128 volume. This allows interactive isotropic visual inspection.
For validation, we compare our approach against bilinear upsampling and two
current self-supervised methods: a neural field reconstruction without the superresolution encoder [25] and a diffusion-based model [12]. Our approach shows
quality improvements over both and quantitative improvements by +1 dB overthe diffusion model and +3 dB over the neural implicit baseline. We demonstrate
this through peak-signal-to-noise ratio (PSNR) computations in a sweep of afrequency-clipped Fourier domain, which offers a more robust metric than pixelwise PSNR to noise in the training data. For computational efficiency, niiv
is 2,000× faster than the diffusion model and 1,000× faster than the SIREN
baseline considering volume-specific pertaining times.

niiv: Fast Self-supervised Neural Implicit Isotropic Volume Reconstruction

2

3

Related Work

Isotropic Volume Reconstruction. Recent self-supervised approaches [20][12,14] train 2D diffusion models to learn the distribution of high-resolution
lateral images. During reconstruction, they use low-resolution axial slices as priors for the backward diffusion process to predict missing volume information.
While diffusion models achieve high-quality results, their usability is limitedby compute-intensive training and time-consuming inference. In contrast, ourapproach improves reconstruction quality while inferencing three orders of magnitude faster than the diffusion baseline [12]. Recently, Zhang et al. [32] also
use arbitrary scale superresolution neural representations for anisotropic volume reconstruction. However, they focus on MRI imaging requiring different
degradation models. Additionally, they do not discuss model inference speeds
and evaluation strategies in the presence of noisy ground truth data. Deng etal. [4] learn a degradation model that generates realistic low-resolution, highresolution training pairs. These pairs are used to train a reconstruction model
like Iso-Net-2 [29]. Both models are trained independently, leading to complex
training setups. On the other hand, supervised approaches [9] show high-quality
results but are difficult to use in practice since isotropic volumes are required attraining time. Other methods use video transformers [8], optical flow field interpolation [1], or standard Conv-Nets [29,34], which are often limited to a specificoutput pixel resolution, whereas our model can be decoded at any resolution.
Neural Implicit Super-Resolution. Encoding spatial information throughneural networks [19,31] has proven to be useful in areas, such as inverse graphics [18,25], shape representation [26], video encoding [10], and super resolution [2,3] [17]. Our approach builds upon local implicit image functions (LIIF) [2],which allows the sampling of images at arbitrary resolution while retaining highquality visual details. Chen et al. [3] extend LIIF for video super-resolution.
However, their supervised approach requires high-resolution and high-frame rate
training videos and can thus not be directly transferred to self-supervised isotropic
reconstruction. Finally, Kim et al. [10] propose a neural video encoding withsuper-resolution capabilities. However, they optimize a single model per video,
making their approach unsuitable for fast reconstruction of unseen volumes.

3

Methodology

Problem Statement. Given a sampling of a volume V with isotropic xy axes
and anisotropic z axis, we aim to learn a model g that reconstructs an isotropicz sampling of a volume Ṽ purely by self-supervision. We assume that the volume sampling contains a physical medium whose distribution of material canbe effectively modeled by observing local statistics of the xy samplings. Given
the low-resolution anisotropic slice Iix×z as input, g must reconstruct a plausible
high-resolution isotropic xz slice I˜ix × αz , where i denotes a slice from the input
volume sampling and α is the axial anisotropy factor (e.g., 8). Then, Ṽ is constructed by stacking predicted high-resolution slices Ṽ = {I˜0 , ..., I˜n }. The same
approach applies equally to both xz and yz slices.

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Authors Suppressed Due to Excessive Length

Fig. 2. Model overview. Inspired by Chen et al. [2,3], a super-resolution encoder [13]first embeds the anisotropic input slice into latent codes that are placed within a 2D
spatially-referenced grid M . We predict a pixel value in the output by querying M at
its coordinate c and concatenating the resulting latent code (blue) with the positionallyencoded c (yellow) and the value at c in the input (beige). The decoding function fθ ,parametrized by an MLP, predicts the respective pixel value (orange).

Model Architecture. Our hybrid neural field [31] uses latent codes within
a 2D space to encode the high-resolution slice. First, a convolutional neuralnetwork super-resolution encoder Eϕ with parameters ϕ embeds a low-resolution
axial slice I ∈ Rx×z into a 2D referenced d-dimensional latent grid M ∈ Rx×z×d ,of latent codes l. Finally, a MLP decoder fθ with parameters θ (Fig. 2) produces
reconstructed image intensities at an output pixel coordinate c:˜fθ ([lc , γ(c), v]) = I(c).

(1)

Here, c is encoded using a 2-band frequency basis γ(c), and v is the pixel value
obtained by simply bilinearly interpolating I at c. As M accepts continuous
input, we can query M at any 2D coordinate c = (hc , wc ) to retrieve a latentcode lc ∈ Rd , making it possible to sample arbitrary pixel resolutions. Thus,
niiv can adapt flexibly to the requirements of interactive display across devices,
unlike other approaches [9,29]. Note that Eϕ and fθ are shared between allvolumes in the respective training and test dataset.
Simulating axial degradation. During training, we use artificially degraded xy images Idxy as model inputs and supervise outputs with the respectivexyhigh-resolution xy slices Igt. We apply a function d that aims to simulate degraxy
dation along the z axis such that d(Igt) = Idxy . Here, we define d as an average
pooling operator. In principle, other degradation models [4,12,7] can be applied
based on the specific application and imaging domain.
Model Training. We minimize a loss function L by end-to-end optimizationof the encoder and decoder parameters ϕ and θ. Studies show that perceptual
image quality metrics are superior to simple pixel-wise metrics for image superresolution tasks [6]. Perceptual metrics [6,5,33] use features of deep pretrainedmodels like VGG [24] as a basis for comparison. Yet, the mean absolute error(MAE) excels at image denoising tasks [6]. Thus, we combine the perceptual
Deep Image Structure and Texture Similarity (DISTS) [5] metric and the MAE
xyxy
between predicted Ipred
and isotropic slices Igt. Our loss function L is therefore

niiv: Fast Self-supervised Neural Implicit Isotropic Volume Reconstruction

5

Fig. 3. Fourier space separates incorrect from correctly reconstructed frequencies. (a) In the spatial domain, reconstruction errors are randomly distributed forthe baseline and our reconstruction. (b) Visualizing errors in the Fourier domain (rightcolumn) separates erroneous (bright) and correctly reconstructed frequencies (dark).
(c) By cropping the Fourier domain’s high frequencies, the PSNR is not perturbed byfrequencies corresponding to noise, yielding more informative PSNR values.

xy
xy
xy
xy
L = w1 DISTS(Ipred, Igt) + w2 MAE(Ipred
, Igt
),

(2)

where wi are the respective weights. We find w1 = 1 and w2 = 30 to consistently
achieve good results.

4

Evaluating Reconstructions with Noisy Ground Truth

Evaluating niiv brings challenges, as only noisy ground truth exists, leading
to uninformative PSNR values (Fig. 3a). Thus, a method that perfectly reconstructs lateral slices is overfitting to the noise; this is especially problematic in
low-data regimes with powerful data-fitting models [12]. We wish to assess thereconstruction of biological structures despite the noise. Prior research has addressed this problem by downscaling the data to diminish noise [9] at the cost of
sacrificing resolution. We propose calculating the PSNR in the Fourier domain
where it is easier to separate high-frequency components of signals [7] such as
noise (Fig. 3b), where we vary a cutoff frequency fcutoff (Fig. 3c) across a range

6Authors Suppressed Due to Excessive Lengthof values to observe the quality across frequencies. For example, if a method
only achieves a greater PSNR than another at high fcutoff but not at low fcutoff ,then it is likely overfitting the noise. Given Parseval’s theorem [21] and the unitary nature of the Fourier transform F, we can directly compute the PSNR in
the frequency domain (see derivation in supplement), sidestepping the inverse
transformation to the spatial domain.

PSNR(I, Ipred ) = PSNR(FI, FIpred ).

(3)

We now incorporate the clipping operation in the Fourier domain, denoted bya low-pass operator L that discards frequencies above fcutoff . Given the above
relationship, we receive
PSNR(F −1 L FI, F −1 L FIpred )) = PSNR(L FI, L FIpred ).

(4)That is, we can transform both images into the Fourier domain, apply the clipping operation via L, and then compute the PSNR directly in Fourier space.

5

Experiments

5.1

Data and Implementation Details

We demonstrate the effectiveness of our approach on the publicly available
FlyEM Hemibrain [22], FAFB [35] EM datasets and also ablate against LM
approaches like LICONN [28] (Fig 5b). While the Hemibrain contains the central brain region of Drosophila melanogaster imaged at isotropic 8 × 8 × 8 nm
pixel resolution, we downsample the data to 8× anisotropy along the z -axis
through average pooling. FAFB shows the entire brain of a female adult fruit fly
at naturally 5× anisotropic 8 × 8 × 40 nm pixel resolution. We randomly sample
400 subvolumes (1283 pixels in the Hemibrain and 1303 pixels in FAFB) andseparate them into training (N = 350) and test datasets (N = 50). All metrics
Table 1. Quantitative comparison on the Hemibrain data. We report clipped
Fourier PSNR (CF PSNR) with fcutoff = 25, regular PSNR, and SSIM. We differentiate between volume-specific pre-training time and inference time (Pre/Infer). We
consistently outperform baselines for 8× reconstruction of 1283 volumes. The highest
scores are highlighted in red, the second and third scores in orange and yellow, respectively.
Hemibrain Data CF PSNR ↑ PSNR ↑ SSIM ↑ Time (Pre/Infer) ↓ VRAM (GB)Nearest
23.16
20.42
0.50
–
–
Bilinear
23.53
20.92
0.51
–
–
SIREN [25]
19.55
18.24
0.34
(150s/0.003s)
3.9
Diffusion EM [12]
23.24
20.65
0.56
(-/264s)
3.9
Ours
24.91
21.56
0.56
(-/0.113s)
3.4

niiv: Fast Self-supervised Neural Implicit Isotropic Volume Reconstruction

7

Fig. 4. Qualitative comparison of simulated (a) and real (b) anisotropic data.
We achieve up to three orders of magnitude faster inference compared to diffusion baselines [12] and other neural implicit approaches like SIREN [25] while also reconstructing
smaller structures with higher fidelity (see arrows).

are reported on entire volumes rather than individual images. Our method is implemented in PyTorch, and all experiments were performed on a single NVIDIA
RTX 3090 Ti GPU. All experiments use the EDSR [13] super-resolution encoder
without upsampling modules, 16 residual blocks, and 64-dimensional output features. The MLP is 5 layers deep, each 256 neurons wide. We train our modelfor 1500 epochs using the Adam optimizer and a learning rate of 5 × 10−5 . Notethat we train separate models for the Hemibrain, FAFB, and LICONN data.
5.2

Qualitative and Quantitative Comparison

To showcase our method’s suitability for interactive isotropic reconstruction, we
capped the GPU memory usage at 4 GB for all methods, reflecting a mid-tierlaptop’s typical capacity. Within this constraint, our approach significantly outperforms the diffusion baseline, delivering inference speeds up to three ordersof magnitude faster (0.11 vs. 264 seconds) for an anisotropy reconstruction taskwith anisotropy α = 8 on 1283 volumes (Hemibrain). The advantage is due toDiffusion-EM’s slow iterative inference process and the need to enforce frameby-frame consistency for the probabilistic reconstruction process by conditioning
each slice inference on a latent code retrieved from the previous slice, prohibitingbatch processing. We also outperform the neural implicit SIREN [25] baseline
as it requires separate pretraining for each subvolume, leading to costly inference on unseen data (Table 1). Comparing reconstruction quality, in contrast
to the baselines, our model can reconstruct fine details (Fig. 4a) with slice-byslice consistency (see supplementary video) and sharper edges (Fig. 4b). While8

Authors Suppressed Due to Excessive Length

Fig. 5. Ablation studies. (a) While the unclipped PSNR in the Fourier domain showsno significant difference between our approach and the baselines (∆0.6), the difference
becomes more evident for thresholds fcutoff ∈ [14, 40] (∆1.63). (b) We also test niiv on
recent expansion light microscopy data [28] with simulated 8× anisotropy. fcutoff = 25.

the diffusion results visually look sharp, small details are often reconstructed
incorrectly, explaining the lower metric scores. Diffusion EM also fails for volume sizes, not in {2i } (Fig. 4b). Also, SIREN reconstruction results look blurry,
making it unsuitable for isotropic reconstruction.
5.3

Ablation Studies

Fourier PSNR Threshold. We tested the effect of the Fourier clipping threshold on the PSNR (Fig. 5a). If no clipping threshold is applied, the PSNR valuesof our method and the baselines are low and close together due to the random
image noise in the ground truth data. However, the black box (Fig. 5a) indicatesa clipping window in Fourier space where image quality differences are moreaccurately represented through the PSNR given noisy GT images.
Data Modality. We use a recent, near isotropic voxel-size (9.7 × 9.7 × 13nm) expansion LM dataset [28] of a mammalian hippocampus and simulate 8×
anistropy using average pooling as a degradation model (Fig. 5b). Next, wetrain on 350 randomly sampled volumes and reconstruct 50 unseen 1283 voxel
test volumes at isotropic voxel size. Fig. 5b shows input and GT images and also
compares our results with nearest- and bilinear interpolation. We find that niiv
produces sharper images compared to bilinear interpolation also for LM data.

6

Conclusions and Future WorkInteractive isotropic rendering of anisotropic data is useful for large-scale datavisual inspection tasks. To this end, we demonstrate that neural fields andencoder-based superresolution representations are promising for fast and flexible self-supervised volume reconstruction. We propose three avenues for future

niiv: Fast Self-supervised Neural Implicit Isotropic Volume Reconstruction

9work. First, developing more accurate physics-based axial degradation models(Sec. 3) to improve the simulation of anisotropic xy slices during training. Denget al. [4] take a first step in that direction. Second, integrating machine-learning
elements like our approach into low-power Web-based image-rendering tools suchas neuroglancer [15] or Viv [16] would rapidly deploy these advances. Third, future work should investigate if latent image representations express higher-level
semantically-interpretable morphological features, as these could be useful indownstream tasks like tissue classification (e.g., neuron typing).

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