Thera: Aliasing-Free Arbitrary-Scale Super-Resolution with Neural Heat Fields
Authors: Alexander Becker
Rodrigo Caye Daudt
Dominik Narnhofer
Torben Peters
Nando Metzger
Jan Dirk Wegner
Konrad Schindler
Venue: Accepted by TMLR ()

Abstract: Recent approaches to arbitrary-scale single image super-resolution (ASR) use neural fields to represent continuous signals that can be sampled at arbitrary resolutions. However, point-wise queries of neural fields do not naturally match the point spread function (PSF) of pixels, which may cause aliasing in the super-resolved image. Existing methods attempt to mitigate this by approximating an integral version of the field at each scaling factor, compromising both fidelity and generalization. In this work, we introduce neural heat fields, a novel neural field formulation that inherently models a physically exact PSF. Our formulation enables analytically correct anti-aliasing at any desired output resolution, and -- unlike supersampling -- at no additional cost. Building on this foundation, we propose Thera, an end-to-end ASR method that substantially outperforms existing approaches, while being more parameter-efficient and offering strong theoretical guarantees. The project page is at https://therasr.github.io.

Published in Transactions on Machine Learning Research (10/2025)
Thera: Aliasing-Free Arbitrary-Scale Super-Resolution with Neural Heat Fields
Alexander Becker*
alezander. bww.:/deebbbekeeeeeeoooo....ooe etwz.ch
Photogrammetry and Remote Sensing, ETH Zurich
Rodrigo Caye Daudt
rudrip..caycdaudddeyyoo....y etha.ch
Photogrammetry and Remote Sensing, ETH Zurich
Dominik Narnhofer
dominik.narmooffr@@geoo..oo ethz.ch
Photogrammetry and Remote Sensing, ETH Zurich
Torben Peters
torben.peterss@geod.cuoo ettgg..tte..kk
Photogrammetry and Remote Sensing, ETH Zurich
Nando Metzger
mando.mettger@geeodbbu....t eth.ch
Photogrammetry and Remote Sensing, ETH Zurich
Jan Dirk Wegner
jondirk. wewgerkeeeeeeruu..cco
Department Mathematical Modeling and Machine Learning, University of Zurich
Konrad Schindler
schindler@ethz.ch
Photogrammetry and Remote Sensing, ETH Zarich
Reviewed OpenReview: https: httpp:://pperreveee openrevee.. net/ str/formm? id- fmqy9
Abstract
Recent approaches to arbitrary-scale single image super-resolution (ASR) use neural fields to represent continuous signals that can be sampled at arbitrary resolutions. However, point-wise queries of neural fields do not naturally match the point spread function (PSF) of pixels, which may cause aliasing in the super-resolved image. Existing methods attempt to mitigate this by approxinating compromising both fidelity and generalization. In this work, introduce neural heat fields, novel neural field formulation that inherently models physically exact PSF. Our formulation enables analytically correct anti-aliasing at any desired output resolution, and ullike supersampling at no additional cost. Building on this foundation, we propose Thera, an and-to-end ASR method that substantially outperforms existing approaches, while being more carrameter-effcciin and offering strong theoretical guarantees. The project page is at ftaps:://therasr.githaee gith... io
integral version of the field at each scaling factor,
1 Introduction
Over the years, learning-base based image super-resolution (SR) methods have achieved increasingly better results. However, unlike interpolation techniques that can resample images at any resolution, these methods typically require retraining for each scaling factor. Recently, arbitrary-scale SR (ASR) approaches have emerged, which allow users to specify any desired scaling factor without retraining, significantly increasing flexibility (Hu gt all 2019). Notably, with LIIF, Chen et al. (2021) pioncered the use of neural fields for single-image SR, exploiting their continuous representation to enable SR at arbitrary scaling factors. LIIF has since inspired
"Equal contribution.

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Neural heat field Bo v.f)
Auti-aliased signal for varions pixel grids
Learner 2D frequency bank
0.5
Input image
Phase shifts
Hy Hypereett wetwwwo work
Backboun
shift
E(sat) D activation
scale
6o03
01.9
Amplitudes
pixel
grid
Figure 1: We present Thera, the first method for arbitrary-scale super-resolution with a built- in physical observation model. Given an input image, a hypernetwork predicts the parameters of a specially designed neural heal field, inherently decomposing the image into sinusoidal components. The field's architecture automatically attenuates frequencies as a function of the scaling factor so as to match the output resolution at which the signal is ro-sampled.
several follow-ups which build upon the idea of using per-pixel neural fields (Lee & Jin 2022 Cao el al. 2023 Chen et al. 2023 Zhu et al.) 2025). This is not surprising: Neural fields are in many ways a natural match for mariable-rcsolutinn computer vision and graphics (Xic et al. [002]. By implicitly paramoterizing a target signal as a neural network that maps coordinates to signal value, they offer a compact representation, defined over continuous input domain, and are analytically differentiable.
While neural fields naturally model continuous functions, they do not easily allow for observations of such functions other than point-wise evaluations. For many tasks, however, integral observation models such as point spread functions (PSFs) are desirable. This is particularly true for neural fields-based ASR methods, which by nature do not commit to a fixed upscaling factor a priori but regress continuous representations with unbounded spectra that can be observed various sampling rates. If the Nyquist frequency corresponding to the desired sampling rate is lower than the highest frequency represented by the field, the sampling operation is prone to aliasing. This explains the initially counterintuitive relevance of anti-aliasing for super- resolution: When using neural fields, signals are first upsampled to infinile (continuous) resolution and then resampled at the desired resolution, and this latter operation must be done carefully. Incorporating
a physically plausible observation model is not trivial (Barron et al. 2021 2022 Lindell et al. 2022 Yang
et all 2022: Hu et al. 2023 Barron et al. 2023), but has the potential to avoid aliasing. For this reason, Chen
al al. 2021 | and successor works (Lee & Jin 2022 Cao et al. 2023 Chen et al. 2023 Zhu al. 2025 have
already taken first step towards learning sulti-scale representations, via cell encoding. Fundamentally,
these "lcarning-basee based anti-aliasing" approaches require the scaling factor (or, equivalently, the output pixel area) additional input to the neural field and learn an integrated (i.e., appropriately blurred and therefore anti-aliased) version of the field for each scaling factor; arguably wasting field capacity to approximate a relation that can be described exactly through Fourier theory.
In this work, we combine recent advances in implicit neural representations with ideas from classical signal theory to introduce newral heat fields, a novel type of neural field that guarantees anti-aliasing by construction. The key insight is that sinusoidal activation functions (Sitzmann of individual components depending the exact computation of Caussian-blureed versions of the field for any desired (isotropic) blur radius. When rastcrizing an image, the field can therefore be queried with a Gaussian PSF that matches the target resolution, effectively preventing aliasing. In practice, heat fields receive an additional input coordinate i, controlling the strength of the Gaussian blur applied to the signal. Unlike learning baased alti-aliasing, the resulting filtering operation is expressed analytically, rather than learned from data. In other words, previous approaches fit 3D field (I, y, and scale) while we only need to fit a 2D field (a and y), whereas the scale dimension is computed analytically, significantly reducing field complexity and data requirements. Notably,
al. 2020b) enable selective attenuation their spatial frequency, following Fourier theory. This allows for
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32.1
Thera Pr
32
PSNR (dB)
31.9
MSTT
31.8
(Ours)
Thera Air
Thera Plus
31.7
CinoSR
CUF
SRNO
CLIT
31.6
LIIF
31.5
MrlaSR
10
10
10
10
Model Size (M parameters)
Figure 2: Comparison of recent ASR methods, averaged over x x22, 3, 4} scales. We generally achieve higher performance at lower parameter counts. Our best model, Thera Pro, achieves highest overall performance by a large margin.
filtering with neural heat fields incurs no computational overhead: The querying cost the same for any width of the anti-aliasing filter kernel, including infinite and zero widths.
Building this, we then propose Thera an end-to-end ASR method that combines hypernetwork (Ha et all 2017) with a grid of local neural heat fields, offering theoretical guarantees with respect to multi- scale representation (see Figure D Empirically, Thera outperforms all competing ASR methods, often by a substantial margin, and is more parameter-efficient (see Figure 2 To the best of our knowledge, Thera is also the first neural field method to allow bandwidth control at test time.
In summary, our main contributions are:
1. We introduce neural heat fields, which represent a signal with a built-in, principled Gaussian obser- vation model, and therefore allow anti-aliasing with minimal overhead.
We use neural heat fields to build Thera, a novel method for ASR that offers theoretically guaranteed multi-scale capabilities, delivers state-of-the-art performance and is more parameter efficient than prior art.
2 Related Work
2.1 Neural Fields
A neural field, also called an implicit neural representation, is neural network trained map coordinates onto values of some physical quantity. Recently, neural fields have been used for parameterizing various
types of visual data, including images Karras et. al.] 2021 Sitzmann et. al. 2020b Tancik et. al.] 2020:
Chen et al. 2021 Lee & Jin 2022 de Lutio et al.] 2019 Wu et al. 2023). 3D scenes (e.g., represented
as signed distance fields (Park et al.] 2019
Sitzmann et al. 2020b a Williams et al. 2022 Wu et al.
2023), occupancy fields (Mescheder et al. 2019
Peng et al. 2020), LiDAR fields (Huang et al.
2023 view- ), or digital 2022 Giebenhain et al.
dependent radiance fields (Mildenhall et al. 2021
Barron al. 2021
2022
2023 Wu et al.
2023
humans (Yenamandra et al.) [2021; Zheng al. 2022 Cao al. 2022; Xiu al.
2023 Frequently, it is desirable to impose
prior over the space of learnable implicit representations.
A common approach for such conditioning is encoder- based inference (Xie et al.
2022). where a parametric
encoder maps input obsorvations to a set of latent codes 2, which are often local (Chen et al. 2021 Loc &
Jin 2022 Vasconcelos et al., 2023: Cao el. al. 2023 Chen el al.] 2023). The encoded latent variables z are then used to condition the neural field, for instance by concatenating z to the coordinate inputs or through
a more expressive hypernetwork (Ha al. 2017), mapping latent codes z neural field paramcters 0. An
early example of this approach, which is gaining popularity (Xie et al. 2022). was proposed in Sitzmann
all (020bb)
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2.2 Arbitrary-Scale Super- Resol-Res
ASR is the sub-field of single-image SR in which the desired SR scaling factor can be chosen at inference time to be (thooretically) any positive number, allowing maximum flexibility, such that of interpolation methods. The first work along this line is MetaSR (Hu et al. 2019), which infers the parameters of a convolutional upsampling layer using hypernetwork conditioned on the desired scaling factor. An influential successor work is LIIF (Chen et al., 2021), in which the high-resolution image is implicitly described by local neural fields. These fields are conditioned via concatenation, with features extracted from the low-rcsolution input image. The continuous ure of the neural fields allows for sampling larget pixels al arbitrary locations and thus also arbitrary resolution.
Most subsequent work has since been built upon the LIIF framework. For example, UltraSR Xu et al. 2021 improves the modeling of high-frequency textures with periodic positional encodings of the coordinate space, as is common practice for e.g., neural radiance fields (Mildenhall et al. 2021 Barron et al. 2021 2022;
2023 LTE [ee & Jin 2022) makes learning higher frequencies more explicit by effectively implementing a learnable coordinate transformation into 2D Fourier space, prior to a forward pass through an MLP. Vasconcelos et al.] (2023) use neural fields CUF to parameterize continuous upsampling filters, which enables arbitrary-scale upsampling. More recently, methods like CiaoSR (Cao et al. 2023 CLIT (Chen gt al. 2023 and most recently MSIT (Zhu et al., 2025) have integrated (multi-scale) attention mechanisms, improving reconstruction quality. In parallel line of research, Wei & Zhang (2023 propose SRNO, an attention-based neural operator that learns a continuous mapping between low- and high-rosolution function
spaces.
Another line of work employs generative models such as denoising diffusion for SR (Saharia et al. 2022 Gao et al., (023)) While most mothods minimize per-pixel errors (cssontially predicting the minimum mean square error estimate), generalive models are trained to produce more realistic-looking outputs by predicting one of many plausible high-resolution images. However, since specifie ground truth details not exactly recovered, such models typically report worse distortion metrics (like PSNR and SSIM) compared to pixel-based methods, c.f. Blau & Michaeli (2018); Delbracio & Milanfar (003)). In this paper, we adopt a pixel-based objective to preserve fidelity to the ground truth, which is important for many downstream applications (e.g., face or license plate recognition).
2.3 Anti-Aliasing in Neural Fields
Early in the recent development of implicit neural representations, concerns regarding aliasing were raised. Barron et al. (2021) proposed integrating a positional encoding with Gaussian weights, which reduced aliasing in NeRF (Mildenhall et al.] 2021). Improvements were later proposed for unbounded scenes (Barron et al.,
2022) and to improve efficiency Hu et al. 2023). Barron et al. (2023) tackle anti-aliasing within the Instant-NGP Mulleer et al.) 2022) approach. Recent work has succeeded in limiting the bandwidth using multiplicative filter networks (Lindell et al., 2022), polynomial neural fields (Yang et al. 2022) or cascaded training (Shabanov et al., 2024), although these works are restricted to discrete, pre-defined band limits (and thus resolutions) and have not tackled super-resolution tasks. These methods are not good fit for ASR because they do not allow for continuons anti-aliasing, nor bandwidth control at test time. To perform scale-dependent filtering, most fields-based ASR methods instead explicitly provide the scale as input to the field, attempting to learn an appropriate observation model from data. While this approach may work reasonably well in in-distribution settings, it seeks learn a model from data that can be described exactly with a differential equation, ultimately sacrificing fidelity and generalization.
In contrast, in this paper explore way to directly integrate a physics-informed observation model into
the neural field representation.
3 Method
In this section we introduce Thera, a novel neural fields-based ASR method that guarantees analytical anti- aliasing at any desired output resolution at no additional cost. First, we present neural heat fields, special
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Field
(2)
(511)
plase shifts
paramiter estimation
Global, learned freq. bank W;
thermal activation E(s (
We linear coefficients
preediets
Local heat field (rontimuous)
Backbous
*t
Tasterize
Ingut innge
Pixel-wise
weighted & summ
Sampled output image
field parameters
W
W
Figure 3: Overview of Thera. hypernetwork estimates parameters (61, W2) (Ad) of pixel-wise, local neural heat fields. The phase shifts by operate on globally learned components, before thermal activations scale each component depending their frequency and the desired scaling factor. The components are then linearly combined using coefficients W2, resulting in appropriately-blure continuous local neural field. This field is then rasterized at the appropriate sampling rate (resolution) to yield part of the final output image (red square).
type of neural field that inherently achieves anti-aliasing by implicitly attenuating figh-froyuency components function of time coordinate. Next, propose mechanism for learning prior grid of neural heat fields, enabling them to represent multi-scale output image conditioned on a lower-resolution input image. Finally, we show that our formulation allows us to impose a regularizer on the underlying, continuous signal itself something that, to the best our knowledge is not possible previous methods.
3.1 Neural Heat Fields for Analytical Anti-Alias Anti-Aliasing
Let x E R2 denote the spatial coordinates of a continuous image function f(x). Aliasing occurs when this continuous signal is sampled at a rate that does not adequately capture its highest frequency components, resulting overlapping spectral replicas in the Fourier domain. One must therefore apply low-pass filter g(x) whose cut-off frequency is aligned with the Nyquist frequency of the sampling rate, then sample the band-limited signal f ( gx).. The key of method is that, if signal decomposed into sinusoidal components, such filtering
be done simply by re-scaling each component by factor that depends their frequency as well as time coordinate, which we call t. The time coordinate acts as third, continuous input to the neural field and controls the amount of re-scaling, and therefore the Gaussian blur applied to the signal. This perfectly mimics how high-frequency components decay faster than low-frequency ones in the analytical solution to the heat equation. The detailed derivation can be found in Appendix [ The behavior described above is naturally accomplished by parameterizing the field &b as a two-layer perceptron,
b V/ (W)) k,, b2, ((x,t) We E (W
(1)
with parameters 0 = (W1, W2, b,, by). Intuitively, W1 serves as a frequency bank, with its components acting as the basis functions that compose the signal (xx 0), and phase shifts encoded by b, The matrix W2, with one row per output channel, contains initial magnitudes global bias of & per channel. Finally, we introduce the thermal activation function E(-), which models the aforementioned decay of sinusoidal components (implied by Wi) over time:
these components, and b2 is the
{ v, k,,) sin(z) exp(- 2 kt).
Here, |M| - (W)) denotes the row-wise Euclidean norm of W1, representing the magnitudes of the implied wave numbers (frequencies). Interestingly, Equation I constitutes the solution of the isotropic heat equation 04 के = k 7 as derived in Appendix A We therefore refer to this MLP as a neural heal field.
There an idcal bijection between the desired sampling rate fa and At t no filtering takes place, implying a continuous signal (f, + oo). A low-pass filtered version of the signal observed for To obtain desired level of anti-aliasing, only need to compute the corresponding value of t. The
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relationship between the cut-off frequency of the filter and t is controlled by a global diffusivity constant K; which defines how fast. components of different frequencies decay over time in the underlying PDE model. We can freely set A to any positive number, but for simplicity, and without loss of generality, we set K in our theoretical derivations such that the native resolution of the (observed, discrete) signal D corresponds to = 1. Assuming equal sampling rate La along both coordinate axes, the optimal value of x then evaluates to
In(4)
R
2f2x²
(3)
To subsample the signal D by factor S, the field d should be sampled at
S=
(4)
where S is the subsampling rate, i.e., the inverse of the scaling factor. In other words, the equation above defines the correct value for according to the scaling factor at which the field should be sampled. For a derivation of these values and demonstration of the filtering mechanism of neural heat fields, see Section
3.2 Learning
Resolutio Prior
The multi-scale signal representation inherent in neural heat fields is natural match for ASR. Still, two challenges must be addressed. First, our formulation restricts the choice of architecture to MLPs with single hidden layer. Second, while it theorelically guaranlees the downsampling operation (1 >>>>> 1), the upsampling operation (0 < t < 1) remains ill-posed. To narrow down the (infinite) solution space to a unique result, a prior must either be defined in an unsupervised fashion or learned from data. Our solution to both challenges is to condition local fields with a hypernetwork V R xHxC + RWxHXN First, a standard backbone, as
used previous work (Chen et al.] 2021 Lee Jin 2022: Vasconcelos et al. 2023 Cao et al. 2023 Chen
et a.l 2023), extracts image features from the low-resolution input image. Then, the hypernetwork maps
these features to the N parameters of each local neural heat field. As originally proposed by LIIF Chen
et a 2021) and adopted by recent ASR methods (Lee & Jin 2022 Cao el al. 2023 Vasconcelos et al. 2023
Chen et al. 2023 each local field spans the area of one pixel of the low-resolution input. It is important that even though the fields themselves model only local part of the image, the hypernetwork informs them with contextual features collected over a large receptive field.
During training, the local fields are supervised with values of high-rcsolution target pixels at the appropriate spatial coordinates x and time index (ensuring that the signal is correctly blurred for the target resolu- tion), and the entire architecture is optimized end-to-end. In practice, we directly optimize single global frequency bank WT, rather than having the hypernetwork predict a separate W1 for each low-resolution pixel. Not only does this better fit the idea to represent the signal with a single, consistent basis, it also reduces the total parameter count.
The described scheme, which we call Thera, depicted in Figure 3. It allows for arbitrary-scalo super-resolution, combining the multi-scale signal representation within neural heat fields with the expres- sivity of proven feature extraction backbones for SR and image restoration. As the entire network is trained end-to-end, the feature extractor can learn super-resolution priors for whole range of resolutions covered by the training data. E.g., network trained with scaling factors up to x4 will encode priors that enable us to observe the field at <> 1. that allows the network to adapt to different downsampling operators. Finally, we set the bias terms for the three color channels of every local field & (i.e., b2) to the RGB values of the associated low-resolution pixel. Thus, the hypernetwork only predicts field-wise phase shifts b and amplitudes W2.
16- By training on multiple resolutions, we can also make k a trainable parameter
3.3 Total Variation at t=0
To allow Thera to better generalize to higher, out-of-domain scaling factors, we can place an unsupervised regularizer at 0. Note that this is a prior on the continuous signal itself something that, to the best of our knowledge, sets Thera apart from all previous methods. In our implementation takes the form of total variation (TV) loss term, well known promote piece-wise constant signals that describe natural
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images well (Chugunov et al. 2024). We use an el variant of TV, Lrv(O(x, 0)) Fx [x
Given our continuous signal representation, ( 0) can be computed analytically by automatic differentia- tion, rather than falling back to a neighborhood approximation in most previous work (Rudin et al., 1992) We further motivate this approach in Figure B which demonstrates that our method faithfully recovers the gradients of super-resolved images.
3.4
Implementation and Training
Thera is implemented in JAX (Bradbury et al. 2018). Similar to prior work (Chen et al. 2021 Lee & Jin
Cao et al. 2023 Zhu et al,, 2025),, randomly sample scaling factor r ~ (1.2, 4) for each image during training, then randomly crop area of size (48r)² pixels as the larget patch, from which the source is generated by bicubic downsampling to size 482, As corresponding targets, 48 random pixels are sampled from the target patch. We train with standard augmentations (random flipping, rotation, and resizing), using the Adam optimizer (Kingma Ba 2015) with a batch size of 16 for x 106 iterations, with initial learning rate 10-1, 81 = 0.9, 32 = 0.999 and c 10-8. The learning rate is decayed to zero according to a cosine annealing schedule (Loshchilov & Hutter 2016 We use MAE as reconstruction loss, to which the TV loss from Eq. 5 is added with weight of 10 Like previous work [imofte et al., 2016 Lim et al. 2017 Vasconcelos et al. 2023). we employ geometric self-ensembling (GSE.) instead of the local self-ensembling introduced in LII (Chen et al., 2021). In GSE, the results for four rotated versions of the input are averaged at test time. Including reflections did not improve performance.
2022
4 Results
Throughout this section we evaluate three variants of our method, which differ solely in the size of the hypernetwork and the number of field parameters:
Thera Aér: A tiny version with the number of globally shared components in W1 set to 32, and the hypernetwork being single 1 x 1 convolution that maps features to field parameters. This version adds only 8,256 parameters on top of the backbone.
Thera Plus: A balanced version that employs an efficient ConvNeXII-based (Liu al. 2022 hyper- network. Its parameter count of x1.41 M matches that of recent medium-sized competitors like Cao ct al.
2023
Thera Pro: The strongest version uses a high-capacity, attention-based hypernetwork. Its added parameter count is s.4.63 M, still less than the most recent competitor (Zhu et al.) 2025) and much smaller than Chen et al. (2023), both attention-based
Datasets and metrics. Following previous work, our models are trained with the DIV2K Agustsson & Timolte 2017) training set, consisting of 800 high-resolution RGB images of diverse scenes. We re- port evaluation metrics on the official DIV2K validation split as well as on standard benchmark datasets: Set5 (Bevilacqua et al. 2012). Set14 (Zeyde et al., 2012), BSDS100 (Martin al.] (001), Urban100 (Huang et ala, 22015), and Mangal09 Matsuu al. 2017). Following prior work, we use peak signal-to-noise ratio (PSNR, in decibels) as the main evaluation metric and compute it in RGB space for DIV2K and the
luminance (Y) channel of the YCbCr representation for benchmark datasets. Additional quantitative results are given in Appendix C Not all numbers could be computed for competing methods for which code or checkpoints were not publicly shared (see Appendix Hp.
Backbones. We combine cach of the three variants of our method with two standard backbones for super- resolution and image restoration, done in previous work: (i) EDSR-bascling (Lim et al., 2017) (1.22 M parameters) and (ii RDN (Zhang et al. 2018 (22.0 M parameters).
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Table 1: Quantitative comparison of peak signal-to-noise ratio (PSNR, in dB) obtained by various methods on the held-out DIV2K validation set. The highest PSNR value per backbone and scaling factor is bold and the second highest is underlined.
Backbone
Num of add. params.
In-distribution
fut-of-distribbuton
(params.)
Method
x2
x3
x6
x12
x18
x24
x
Bicubic
31.01
28.22
26.66
24.82
22.27
21.00
20.19
19.59
Sine interpol.
34.52
30.89
28.98
26.75
23.76
22.25
21.29
20.62
MetaSR
0.45
34.64
30.93
28.92
26.61
23.55
22.03
21.06
20.37
LIIF
0.35 M
34.67
30.96
29.00
26.75
23.71
22.17
21.18
20.48
LTE
0.49 M
34.72
31.02
29.04
26.81
23.78
22.23
21.24
20.53
CUF
0.30 M
34.79
31.07
29.09
26.82
23.78
22.24
EDSR- baseline
CiaoSR
1.43 M
34.88
31.12
29.19
26.92
23.85
22.30
21.29
20.44
(1.22 M)
CLIT
15.7 M
34.81
31.12
29.15
26.92
23.83
22.29
21.26
20.53
SRNO
0.80 M
34.85
31.11
29.16
26.90
23.84
22.29
21.27
20.56
MSIT
4.83 M
34.95
31.23
29.22
26.94
23.83
22.27
21.26
20.54
Thera Air (ours)
.008 M
34.75
31.09
29. 19.10
26.84
23.80
22.26
21.26
20.56
Thera Plus (ours)
1.41
34.89
31.22
29.24
26.96
23.89
22.34
21.32
Thera Pro (ours)
4.63 M
35.19
31.50
29.51
27.19
24.09
22.51
21.48
20.73
+ Sinc interpol.
34.59
31.03
29.12
26.89
23.87
22.34
21.30
20.68
MetaSR
0.45 M
35.00
31.27
29.25
26.88
23.73
22.18
21.17
20.47
LIIF
0.35 M
34.99
31.26
29.27
26.99
23.89
22.34
21.31
20.59
LTE
0.49 M
35.04
31.32
29.33
27.04
23.95
22.40
21.36
20.64
CUF
0.30
35.11
31.39
29.39
27.09
23.99
22.42
RDN (22.0 M)
CiaoSR
1.43 M
35.13
31.39
29. 13.13
27.13
21.03
22.45
21.41
20.55
CLIT
15.7
35.10
31.39
29.39
27.12
21.01
22.45
31.38
20.64
SRNO
0.80 M
35.16
31.42
29.42
27.12
24.03
22.46
21.41
20.68
MSIT
4.83 M
35.16
31.42
29.42
27.11
23.99
22.42
21.37
20.65
Thera Air (ours)
.008 M
35.06
31.42
29.43
27.13
21.04
22.48
21.44
20.71
Thera Plus (ours)
1.41 M
35.00
31.40
29.44
27.16
24.06
22.49
21.45
20.71
Thera Pro (ours)
4.63 M
35.25
31.56
29.57
27.25
24.14
22.56
21.52
20.77
Table 2: Results on common benchmark datasets for in-distribution scale factors with an RDN (Zhang et al, 2018) backbone. The numbers represent PSNR in dB, calculated on the luminance (Y) channol the YCbCr representation following previous work.
Se15
Se114 x3
B100
Urlmu100 x3
Manga108 109
Method
x
x3
x4
x2
x4
x22
x3
x4
x:2
x4
X2
x3
x4
MetaSR
38.22
31.63
32.35
33.98
30.54
28.78
32.33
29.26
27.71
32.92
28.82
26.55
LIIP
38.17
34.68
32.50
30.53
32.32
29.26
27.74
28.82
20.26
LTE
38.23
34.72
32.61
34.09
30.58
28.55 28.88
32.36
29.30
27.77
33.04
28.97
26.81
39. 28
34.32
31.30
CUP
38.28
31.80
32.63
34.08
30.65
28. 92
32.39
29.33
27.80
33.16
29.05
26.87
CinaSR
38.29
31.85
32.66
34.22
30.65
28. 93
32.41
29.34
27.83
33.30
29.17
27.11
39.51
34.57
31.57
CLIT
38.26
34.80
32.69
34.21
30.66
28. 91
32.39
29.34
27.82
33.13
29.04
26.91
SRNO
38.32
3181
32.69
34.27
30.71
28.97
32.15 13
29.37
27.83
33.33
29.11
26.98
39.52
34.67
31.61
MSIT
35.31
32.72
34,26
30.70
42
29.35
27.81
34.62
31.58
Theru Air
38.18
34.75
32.60
34.13
30.70
28.95
32.33
29.29
27.80
33.16
29.14
26.91
39.03
34.57
31.66
Thera Plus
38.11
31.67
32.56
31.20
30.67
28.97
32.26
29.28
27.81
33.11
29.15
26.97
38. 69
34. 12
31.65
Thera Pro
38.36
34.88
32.79
34.43
30.85
29.08
32.46
29.39
27.87
33.63
29.58
27.26
39.62
34.98
31.98
4.1 Resouttesol Performance
Quantitative results. We first evaluate the three variants of our method on the held-out DIV2K validation set, following the selup described above. Table D shows PSNR values for all tested methods, for both in- distribution (x2 to x4) and diitriiidittiittiinn (x6 to x30) scaling factors. Thera Pro outperforms all competing methods at all scaling factors, often by substantial margin (e.g., 29.51 us. 29. on EDSR x4), even though its parameter overhead on top of the backbone lower compared the second- best method minimal MSIT (Zhu et al. 2025 and less than third of CLIT CChen al. 2023). Interestingly,
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Low-res input
LUF
SRNO
MSIT
Thera Pro (ours)
GT
DIV2K
Set14
Urban100
COMICS
COMICS
COMICS
COMICS
COMICS
Manga 09
Mangal 109
Figure Qualitative examples for representative scale factor, with an RDN 7Zhan al. backbone for all mothods. Best viewed zoomed in.
2018
variant Thera Air with only about 8000 parameters on top of the backbone- - performs on par with or better than methods of much higher parameter count. This supports our claim that hard-wiring theoretically principled sampling model, which rules out signal aliasing, enables better generalization and higher-fidelity reconstruction. For comparison with conventional interpolation, we also report numbers obtained with Lanezos (sinc) resampling top of the respective x4 backbone. This baseline is consistently outperformed by dedicated ASR mothods, indicating that the latter do learn scale-specific priors
Like earlier work, we further report the performance of Thera on five popular benchmark datasets with an RDN backbone in Table 2 Our method again outperforms all competing methods in all settings, often substantially (e.g., 29.58 es. 29.14 on Urban100 x3). We hypothesize that Thena' hird-wired PSF also beneficial when generalizing unseen datasets. Once again we observe that the performance of Thera Air is often comparable to that of methods with orders of magnitude higher parameter counts.
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ReL.U-based
Thera (ours)
Y
Figure 5: Thera reconstructs signal & and its gradient V more faithfully than ReLU-based competitor (Chen et al. 2021 Due to its natural, Fourier-inspired representation, Thera is also infinitely differentiable, while Resee competitors approximate the signal as a percewise-linea function with mull higher deriva- tives (last row).
Qualitative results. Upon visual inspection see Figure for examples - we observe that Thera produces results that are both perceptually convincing and more correct, particularly in the presence of repeating structures. Neural heat fields enable Thera to reproduce high level of detail without suffering from aliasing, matter the sampling scale (see also Figure in Appendix B
Fidelity of the Signal and its Derivatives. Neural fields with periodic activation functions have been shown to be superior when comes to fitting high-resolution, natural signals, and to correctly recovering their derivatives (Sitzmann et al.) 2020b). We observe similar effects for Thera, whose thermal activations at 0= can be seen as special case of periodic activations, cf. Figure 5 In fact, due to the use of thermal activations and unlike all prior work based on multi layer ReLtvactivaeed fields Thera is infinitely differentiable.
4.2 Ablation Studies
In Table 3 we ablate individual components and design choices of our method to understand their contribu- tions to overall performance. The comparisons use Thera Plus with EDSR backbone, and are representative of all variants.
Single scale training. We run three experiments using a single scale (x2, x3, x4) to test how this affects scale generalization. was fixed at the theoretically derived value for these experiments, as multi-scale superior performance of single-
training is required to optimize it. As expected, we observe equal scale training when tested at the training scale (marked in yellow in Table but significant drop compared to the default multi-scale version when generalizing to other scaling factors.
Trainable k. Fixing at the thcoretically derived value (Equation 3) leads small drop performance. This suggests that there remain effects that are not accounted for by our proposed observation model, albeit very minor.
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Table 3: Ablation study using Thera Plus (w/ EDSR-baseline)
In-distribution
Out-of-distrbbution
Experiment
x2
x3
x6
x18
x30
Thera Plus
34.89
31.22
29.24
26.96
22.34
20.61
x2 only, fixed 35.00
30.96
28.76
26.34
21.87
20.25
x3 only, fixed
34.70
31.25
29.25
26.90
22.23
20.53
x4 only, fixed
34.36
31.18
29.25
26.98
22.31
20.58
Fixed & No GSE
34.85
31.22
29.24
26.95
22.29
20.56
34.81
31.15
29.18
26.91
22.29
20.57
No TV prior
34.89
31.23
29.24
26.87
20.42
18.68
ReLU instead &
34.80
31.11
29.13
26.87
22.28
20.57
Predicted comps.
34.90
31.23
29.25
26.97
22.34
20.61
Geometric self-ensemble. In line with previous work Timmofte et al. 2016 Lim et al. 2017 Vasconcelos gt all 2023) we see a notable performance boost with geometric self-ensembling. Note, though, an appli- cation prioritizes inference speed over quality this add-on can be disabled at test-time without re- re-training the network.
Total variation prior. The regularizer has negligible effect for in-domain scaling factors, but performance degrades significantly without it for out-of-distributio scales.
Thermal activations. We replace thermal activations (Fquation 2 with standard ReLU activations. What remains is only the hypernetwork controlling the parameters of the local fields. consistent loss in performance shows the impact of the proposed thermal activation underlying our multi-scale representation.
Shared components. Predicting W1 along with b and W2 leads to negligible gains. This comes at the cost of doubling the amount of field parameters that the hypernetwork predicts. Thus, Thera uses shared, global frequency bank.
4.3 Limitations and Future Work
Neural heat fields as introduced in this paper, and by extension Thera, come with relatively strict archi- tectural requirements that currently only allow for single hidden layer in the neural field. While this can be beneficial from a computational standpoint, it limits hierarchical feature learning and potentially makes modeling of complex non-linear relations harder than necessary. Nonetheless, as our experiments show, the current neural heat field architecture does easily have enough capacity to model local, subpixel information for the scaling factor range discussed in this paper. We have compensatod for the relatively less expres- sive fields with higher-capacity hypernetwork, and we speculate that there may be ways to extend the tignal-theoretic guarantees of Thera to multi-layer architecturos in future work. This could result in even higher parameter efficiency, and potentially better generalization. We also expect that more advanced priors than TV could be even more effective at regularizing 4. Priors at t = 0, made possible by Thera, have the potential to regularize the continnous signal itself, and therefore improve SR quality for all scaling factors.
5 Conclusion
We have developed novel paradigm for arbitrary-scale super-resolution by combining traditional signals theory with modern implicit neural eepresentations Our proposed neural heat fields implicitly describe an image as a combination of sinusoidal components, which can be selectively modulated according to their frequency to perform Gaussian filtering (anti-aliasing) between scales analytically, with negligible overhead. Our experimental evaluation shows that Thera, our ASR method based on neural heat fields, consistently outperforms competing methods. At. the same time, it is more paffiente--fffcient and offers theoretical
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guarantecs w.r.t. aliasing. We believe that Thera-style representations could benefit other computer vision tasks and hope to inspire further research into neural methods that integrate physically meaningful and theoretically grounded observation models.
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A Theory
A.1 Preliminaries
As was described in Section 3.1 the idea underlying our neural heat field with thermal activations is formulate neural field $(x,t), t), with being the 2-dimensional spatial coordinates (21, such that + follows the heat equation:
a
= k V2
(6)
0t
0z3
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The reason for this is that the analytical solution to the (isotropic) heat equation be modeled convolution of the initial state (xx,00 0) with Gaussian kernel
1
x2
g(x, t)
exp
(7)
4тkt
4kt
By fitting the data (image 1) at ((x,1 1), we are assuming a Gaussian point spread function (PSF) with the shape
PSF(x)
1
exp
2
22
(8)
4m
In this formulation, we attempt to recover a "pure" signal at f or higher sampling rates t 1 given an observation at f 1. Note that
meaningless, pure signal,
if < 0 i if ift=1
((x, t)
ill ill-posed,
(9)
I,
well-posed,
ift
The ill-posed problem for 0 < < 1 is the interesting case, where this formulation relates to super-resolution. The super-resolution algorithm should somehow condition the solution space to find the appropriate solution in this domain.
For all the formulations here, we define the image 7 to correspond to the coordinates 21,42 E [-0.5, 0.5).
A.2 Thermal Diffusivity Coefficient
To use the above formulations, we need to compute the thermal diffusivity coefficient k. One way to do so is to match the cut-off frequency the filter in Equation 7 at t to the well-known Nyquist frequency given by the image's sampling rale. We take the cut-off frequency of the Gaussian filter defined in Equation to be the frequency whose amplitude is halved, which
(n(4) 2n 2kk
fc = VI(4) of
(10)
2k/
For the signal compressed into the domain [-0.5, 0.5,, we can compute the Nyquist frequency to be
N
fNNqqust 2
(11)
where the number samples along a given dimension. This formulation assumes even sampling over 21 and Ø2. To extend this formulation to non-square images, would be necessary change the shape of the signal's domain in order maintain even sampling in all spatial dimensions.
If we solve for fo /Nyquist at f = 1, we get
In(4)
(12)
222
For our proposed Thera formulation, we want & to contain a single pixel at t t= 1. This is the pixel from the low-resolution input which will become SR² pixels for super-resolution with a scaling factor SR. Therefore we initialize with
In(4) K
(13)
2T
Note that the exact value of will depend on the characteristics of the system that is being modeled and the anti-aliasing filter that was used (or assumed). Lower values of allow for sharper signals to be represented at any given value of f, but also more prone to aliasing.
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Image domain
Original
x1/4 AA
x1/8 AA
x1/16 AA
Heat field
$(x,11 1)
4xx, 42)
(xx 82)
4 162)
Figure 6: Comparison between theoretical anti-aliasio filter (top row) and anti-aliasing with neural heat fields (bottom row), which is computed without convolutions over-sampling. The heat field was supervised at (xx, 1) only.
Finally, we would like highlight that Equation I0 specífic to the case where x is 2-dimensional. The theoretically ideal value of the only part of our formulation that does not directly apply when using our field's formulation in spaces with numbers of spatial dimensions other than 2. Nonetheless, computing for other cases would be simple matter of repeating the steps above with the formulas for Gaussian filter with the appropriate number of dimensions.
A.3 Relationship Between t and
Assuming that the field is learned appropriately, we still need to know at what time t we should sample from to obtain the correct (aliasing- free)) signal for diffcrent sampling rate. If we define S to be the subsampling rate (i.e., if our base image has = 128 and we want. subsample down to N 64, we have S 2) we need to find such that fNyqulat scales by I/S. Using Equation 10 and Equation I1 we can easily find the
quadratic relationship
t= S2
(14)
For instance, if want to upsample the image by factor of 2, should use t = 0.5² 0.25. Thus, 0 < < refers to super-resolution, while / >> 1 refers to downsampling. This is intuitive: As L grows, the image becomes blurrier (the Gaussian kernel gets wider), which corresponds to stronger low-pass filters and therefore lower sampling rates.
Figure shows an example where we fit a neural heat field = to the image. After training, any low-pass filtered version of the image can be generated by setting according to Equation 4 We emphasize that: (i) Computing these filtered images requires no over-sampling or convolutions; (ti) The computational cost does not depend the size of the blur kernel or on t; (ti)) Given ((x,to), the filtered versions Φ(x, t) are known for any 2 lg-
In Figure 7 we show four local neural heat fields in which aliasing would occur without the anti-aliasing mechanism modeled by thermal activations. Note that aliasing is not always as obvious to the eye as Moiré patterns: in the cases shown the figure, would simply mean that sampling the center location at would be representative the pixel's footprint. Figure further illustrates how such biur equivalent to scale-appropriate anti-aliasing filter.
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a)
b)
c)
d)
Figure Example situations where aliasing would occur without the suppression of high frequencies modeled by neural heat fields. Sampling the center location at t = 0 would not be representative of the pixel's footprint.
0.0
0.3
0.3
Point sample
Blur sample
PSF sample
Figure 8: Blurring image before point-wise sample is equivalent to observing with PSF equivalent to the blur kernel.
A.4 Other Filters
In theory, the formulation presented in Section 3.1 allows us to use any low-pass filter we want, since we can modulate different components freely. Gaussian filters are an obvious choice since they are often used as anti-aliasing filters, and since they are fully defined by single parameter, the standard deviation, Initial explorations of sharp low-pass filter that completely removes components above JNyquist led to performance reduction, likely due to the associated effect gradients during training, It remains open question whether more complex filters (e.g., Butterworth) would improve the current formulation in any way. For quantitative evaluations, this is unlikely, since the downsampling operations use Gaussian anti-aliasing but in real-world applications or other scenarios, this may be desirable.
A.5 Initialization of Components
We have noticed during our experiments that the initialization of the components, W1 in Equation | is important. Sitzmann et al. (2020b) made similar observations when periodic activation functions were first used for neural fields. The final distribution of frequencies [p(W))] did not change much during training. Thus, we choose initialize W, such that
((v(w:))) [p(wi)]
(15)
up to given maximum frequency, allotting more components to higher frequencies. See code for more details.
B Continuous Upsampling Example
In Figure g
provide practical showcase of the continuous upsampling capabilities of method.
18

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Published in Transactions on Machine Learning Research (10/2025)
x1.0
x1.6
x2.4
x3.1
x3.9
x4.5
x5.3
Showcase of multiscale upsampling using Thera Pro with RDN (Zhang et al. 2018 backbone, shown with non-integer scaling factors.
Figure 9:
C
Additional Quantitative Results
C.1
Further Metrics
Table shows SSIM metrics on the DIV2K validation set, which complement the PSNR values reported Table We use the SSIM implementation from torchmetrics (Detlefsen et al. 2022 We observe strong performance Thera, although overall there is relatively little variance across all methods using this metric.
Table 4: SSTM scores (higher is better) for several methods and scaling factors evaluated on the (hold-out) DIV2K validation set. We use the RDN backbone for models. Some methods did not provide checkpoints at the time of writing, see Appendix
In-distribution
OOD
x3
x6
x8
LITF
.934
.866
.804
.704
.636
LTE
.936
.868
.807
.707
.639
CiaoSR
.938
.871
.814
.718
.651
SRNO
.937
.873
.819
.738
.685
MSIT
.942
882
.829
.751
.700
Thera Pro
.943
.884
.833
.757
.706
Thera Pro (std.)
+
=4.0e-5
48.9e-5
+9.7e-5
+5.3e-5
Table 5: Evaluation of GMSD
(lower is better) on DIV2K for various methods.
Backbone
Model
In-distribution
Out-of-distribution
x2
x3
x4
x6
x12
x18
x24
x30
LIIF
0.0065
0.0354
0.0667
0.1192
0.1976
0.2319
0.2516
0.2645
LTE
0.0064
0.0351
0.0664
0.1187
0.1970
0.2316
0.2515
0.2645
Ciao SRNO
0.0063
0.0340
0.0641
0.1159
0.1942
0.2295
0.2497
0.2635
EDSR-b.
0.0063
0.0344
0.0648
0.1165
0.1947
0.2296
0.2499
0.2631
MSST
0.0061
0.0334
0.0636
0.1156
0.1944
0.2298
0.2502
0.2634
Thera Pro
0.0057
0.0314
0.0596
0.1094
0.1878
0.2242
0.2451
0.2591
Thera Pro (std.)
+.0e--5
=4.le-5
-7.6e-5
+1.6-5
+7.1e-5
5
#1.2e-4
#1.6e 4
LIIF
0.0060
0.0330
0.0628
0.1145
0.1934
0.2285
0.2489
0.2622
LTE
0.0058
0.0327
0.0622
0.1135
0.1925
0.2280
0.2487
0.2621
Ciao
0.0058
0.0321
0.0609
0.1114
0.1897
0.2258
0.2470
0.2614
RDN
SRNO
0.0057
0.0321
0.0611
0.1118
0.1906
0.2266
0.2477
0.2613
MSIT
0.0057
0.0321
0.0610
0.1122
0.1920
0.2288
0.2501
0.2641
Thera Pro
0.0055
0.0309
0.0588
0.1083
0.1869
0.2235
0.2446
0.2588
Thera Pro (std.)
+7.9e-6
=1.4--5
=4.0e-5
+3.1e-5
#3.7e-5
+2.0e-5
+6.3e-5
+9.0e-5
19

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Published in Transactions on Machine Learning Research (10/2025)
Table 6: PSNR (Y channel) on common benchmark datasets for out-of-distribution scale factors, with an RDN (Zhang et al. 2018 backbone. For some methods, code and/or checkpoints were not publicly available,
see Appendix H
Set5
Set14
B100
Urban100
Manga109
Method
x6
x
x6
x8
x6
x
x6
x8
x6
x8
MetaSR
29.04
29.96
26.51
24.97
25.90
24.83
23.99
22.59
LIIF
29.15
27.14
26.64
25.15
25.98
24.91
24.20
22.79
27.33
25.04
LTE
29.32
27.26
26.71
25.16
26.01
24.95
24.28
22.88
27.49
25.12
CUF
29.27
26.74
26.03
24.32
CiaoSR
29.46
27.36
26.79
25.28
26.07
25.00
24.58
23.13
27.70
25.40
CLIT
29.39
27.34
26.83
25.35
26.07
25.00
24.43
23.03
SRNO
29.38
27.28
26.76
25.26
26.04
24.99
24.43
23.02
27.66
25.31
MSIT
29.34
27.29
26.75
25.26
26.05
24.98
24.43
22.99
27.61
25.26
Thero Air
29.31
27.25
26.76
25.27
26.05
24.99
24.39
23.00
27.70
25.30
Thero Plus
29.31
27.29
26.80
25.32
26.07
25.00
24.45
23.04
27.80
25.41
Thera Pro
29.51
27.34
26.90
25.38
26.12
25.04
24.70
23.24
27.94
25.49
Table reports Gradient Magnitude Similarity Deviation (Xue et al. 2013)
(GMSD) metrics for various methods. Thera reaches significantly lower GMSD for all backbone-scale combinations, indicating more laithful gradient structure. Note that some methods did not provide cole/checkpoints and can therefore not H..
be re-evaluated (see Section
Furthermore, in Table 6 we show quantitative evaluations which are out of distribution both in terms of data (benchmark datasets) and in terms of scaling factors (above x4).
C.2 Parameter Efficiency
we compare the number of additional parameters and PSNR values for various methods and individual upsampling factors the DIV2K validation set.
In Figure 10
C.3
Error Bars
Table reports standard deviations for PSNR observed over = training runs that were initialized with different random seeds, complementing the main table. For SSIM and GMSD, we have reported standard deviations alongside the respective metrics in Tables And Standard deviations are in all cases significantly lower than performance improvements over competing moth
x2 Scaling Factor
x3 Scaling Factor
x4 Scaling Factor
35.2
31.5
29.5
PSNNR (dB)
35.1
31.4
PSNR (dB
29.4
34.8
34.9
35
There Air
(Owes)
Thera Flas
MSIT
PSNR
31.3
31.2
(Gum)
ClnSR
x
MSIT
CLIT
29.3
29.1
29.2
(Ouna)
Thcra Plus
CUT
CooSR
MSIT
31
34.7
29
LTE
10-2
10-1
100
10-2
10-1
100
101
10-2
10-1
100
10
Model Size (M parameters)
Model Size (M parameters)
Model Size (M parameters)
Figure 10: Comparison of ASR methods for different scaling factors (x2, x3, and x4) on DIV2K. Thera consistently achieves better performance at lower parameter counts across all scaling factors.
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Table 7: Standard deviation of PSNR (in dB) on the DIV2K validation set over N =3 runs.
In-distribution
Out-of-distribution
Backbone
Method
x2
x3
x4
x6
x12
x18
x24
x30
Thera Air
0.0137
0.0143
0.0149
0.0107
0.0084
0.0055
0.0040
0.0040
EDSR-b,
Thera Plus
0.0131
0.0122
0.0114
0.0094
0.0075
0.0049
0.0060
0.0022
Thera Pro 0.0063
0.0088
0.0062
0.0034
0.0054
0.0049
0.0056
0.0034
Thera Air 0.0187
0.0152
0.0113
0.0045
0.0036
0.0054
0.0022
0.0034
RDN
Thera Plus
0.0045
0.0018
0.0011
0.0011
0.0009
0.0013
0.0026
0.0024
Thera Pro
0.0025
0.0009
0.0038
0.0024
0.0021
0.0036
0.0021
0.0012
D Analysis of Learned Components & Kappa
Figure 11 shows statistical analysis of the frequency components learned by Thera-Pro with RDN backbone for the DIV2K training data. Components are distributed uniformly across directions, with fewer components at. low frequencies and progressively more components at higher frequencies. This provides more representational capacity for the reconstruction of fine details, analogous to the typical distribution of froquency components for other decompositions such as the Fouricr or discrete cosine transforms.
To investigate whether these components form a generalizable basis, we fit heat fields to images from various datasets other than the training data: Urban100, Manga109, and highly dustr--iistii subset from the fastMRI (Zbontar et al. 2018) medical imaging datasct (singlo-coil knee validation split, with intensitics scaled to [0.1).. In all experiments, we fix the frequency bank to the one shown in Figure 1 and oplimize only the scale and shift parameters with the AdamW optimizer for 600 iterations per image, with learning rate 0.001. We use local fields that cover patches of N x N pixels, with N E {3, 4, 6, 12, 18]. Table shows that the pre-trained components reconstruct all datasets with negligible error at smaller local field sizes (in-distribution scales), and still achieve very low errors at larger field sizes up to 18 18 pixels (out-of- distribution scales).
These numbers are not surprising, the frequency bank was optimized to fit local fields and can act ower-parametrized dictionary for smaller field sizes (e.g., 4 x 4 GT pixels per field). Notably, it still works relatively well for large OOD sampling factors. Also, there is no obvious difference between the values for natural images and those for MRI images, which suggests that the frequency bank is a general-purpose basis, similar to Fourier or DCT components.
Furthermore, Table D reports final, converged values of the thermal diffusivity coefficient k. We observe that is very similar across runs (a 0.003), but deviates from the theoretically derived value (for Gaussian downsampling model, Equation 3 by a factor of s 1/2. This deviation suggests that the cutoff frequency of the anti-aliasing used in the cubic Mittchel--Netraalli filter used to downsample images during training does not exactly match the Gaussian one used in the theoretical derivations, being more lenient towards aliasing than the theoretical model (anti-aliasing filters are tuned empirically to balance aliasing against loss details). The result indicates that Thera is indeed able to tune 6 as necessary to approximate the downsampling characteristics seen in the training data, in a repeatable manner.
Table 8: Reconstruction MSE when fitting heat field grids of various sizes to out-of-distribution datasets.
Local field size
3x3
4x4
6x6
12x12
18x18
Sct5
2c-10
6c-11
1c-11
1.37c-6
2.16c-4
Urban100
40-9
9c-10
2c-10
2.83e-6
7.54c-4
Manga109
26-8
4e-9
7e-10
1.41e-6
3.01e-4
fastMRI
3e-11
le-11
le-11
8.83e-6
9.23e-4
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Published in Transactions on Machine Learning Research (10/2025)
Table 9: Converged values of x for multiple runs.
Backbone
Model
Converged x
Thera Air
0.0295
EDSR-b.
Thera Plus
0.0336
Thera Pro
0.0266
Thera Air
0.0300
RDN
Thera Plus
0.0360
Thera Pro
0.0293
Theoretical Gaussian
0.0702
90
135
45
180
-0/2
Direction [rads]
225
315
270
Spatial frequency
Figure 11: Statistical distribution of the converged frequency bank of the Thera Pro run with RDN backbone. Left: Polar scatter plot of spatial frequency and angular direction of components. Right: Corre- sponding marginal distributions.
E Hypernetwork Architecture
In our implementation, Thera Air uses feature refinement blocks top of the backbone except x convolution mapping pixel-wise features into field parameters, done in SIREN (Sitzmann al. 2020b). Thera Plus uses 6 ConvNeXt [Liu et al. 2022 blocks with d = 64 followed by 7 ConvNeXt: blocks with d = 96 and 3 ConvNeXt blocks with d = 128, prior to the final mapping layer. Projection blocks are added between blocks with different d, which consist of a layer normalization and a convolution operation. For Them Pro, two windowed transformer- with 7 and 6 layers respectively, and 6 attention hcads in cach layer.
blooks (Tiang et al. 2021 Liu et al. 2021
blocks are used
There are 128 field parameters for Thera Air and 2048 for the larger variants Thera Plus and Thera Pro. These numbers are computed follows: Let be the number of components used (32, 512, and 512 respectively), then we need field parameters indicating phase shift and 3c field parameters for the lincar mapping between components and RGB channels, resulting in a lotal of 4c field parameters produced by the hypernetwork. The total parameter count of the hypernetwork is 8,192, 1.41 M, and 4.63 M for the three variants, respectively. To this we have add the parameters of the components themselves (i.e., the mapping from coordinate space thermal activation arguments defined by W1) that are defined in global, learnable frequency bank, adding further 2c parameters. We highlight that these do not come from the hypernetwork.
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Table 10: Comparison of runtime and VRAM footprint various methods with an EDSR-baseline backbone.
Method
Add. params.
Time
VRAM
MctaSR
0.45 M
3.72 ms
834 MiB
LIIF
0.35 M
13.8 ms
604 MiB
LTE
0.49 M
16.7 ms
574 MiB
CiaoSR
1.43 M
37.5 mss
3894 MiB
CLIT
15.7 M
86.5 ms
9402 MiB
SRNO
0.80 M
11.2 ms
2702 MiB
MSIT
4.83 M
92.2 ms
9388 MiB
Thera Air
.008 M
3.56 ms
322 MiB
Thera Plus
1.41 M
14.2 ms
664 MiB
Thera Pro
4.63 M
19.44 ms
686 MiB
F Computational Complexity
Table 10 reports inference time and VRAM requirements of different Thera variants as well comparison methods, on the 48 48 pixels standard input patch size and scaling factor (i.e., 192x 192 output size). Each Thera variant improves compute time and memory efficiency compared to methods with similar higher parameter counts, with particularly pronounced gains over Transformer-base based competitors (e.g., Thera Pro uses less than 1/4 of the time and 1/13 of the VRAM footprint MSIT, at similar parameter count). All tests were performed on an NVIDIA GeForce RTX 3090 Ti GPU.
G Real-World Optical Zoom Data
To evaluate the effectiveness of approach real-world continuous optical zoom data, conducted experiments using the COZ dataset (uu et al. 2024). We introduce Thera++, which combines two compo- nents: (1) Thera Plus (EDSR) trained on the COZ dataset, and (2) lightweight spatial transformer network (STN) (Jaderberg et al. 2015) that estimates just parameters per image to correct domain-specific affine distortions. Despite its name, the STN is not a transformer network in the modern sense that employs attention mechanisms; rathor, an image model block that explicitly allows the spatial manipulation of data within a convolutional neural network.
The STN component follows standard architecture, consisting of a simple convolutional localization network with adaptive pooling to handle variable input sizes. The network processes the input image along with the scale factor and outputs six parameters of a 2D alfine transformation ion (cy-translation, anpio sy-scale, rotation and shear). With approximately 10K parameters, this lightweight component efficiently corrects for geometric distortions while adding minimal computational overhead.
Table 11: Results (PSNR dB) on the COZ test set.
In-distribution
Out-of-distribution
Method
x2
x3
x4
x5
x6
MetaSR
28.1 70
26.55
25.17
24.31
23.25
LIFF
28.72
26.61
25.16
24.32
23.23
LTE
28.67
26.55
25.15
24.37
23.26
SRNO
28.73
26.59
25.15
24.31
23.25
LIT
28.74
26.58
25.15
24.35
23.19
LMT
28.86
26.66
25.22
24.39
23.29
+
29.06
26.84
25.45
24.49
23.46
23

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Published in Transactions on Machine Learning Research (10/2025)
+ addresses important limitation of the COZ dataset. When looking at the data it becomes obvious that COZ samples are not perfectly aligned, resulting in zy-jitter between images. Additionally, dynamic objects like people, dust, leaves, and moving shadows appear inconsistently across images of the same scone, significantly increasing the noise level. These challenges make super-rosolttion particularly difficult for this dataset. However, + outperforms previous stale-of-the-arl methods, including LMI, across all scaling factors, highlighting its applicability under real-world imaging conditions, see Table 11
Reproducibility of Existing Methods
We encountered challenges attempting recrcate the results reported for some of the competing mothods, which explains why some are missing or differ from the originally reported numbers. Details are provided below.
CUF (Vasconcelos et al., 2023) At the time of writing, there were no public code or checkpoints available for CUF. Therefore, we could not generate numbers for datasets and scaling factors not reported in the original paper. We have denoted those missing values with =_" in the tables. Furthermore, we could not create any qualitative samples using CUF.
CLIT (Chen et al., 2023). For CLIT, at the time of writing, there is public code, but checkpoints. We have made bona fide attempt reproduce the models, but due to the cascaded training schedule and the large model size, the training process would require excessive amounts of compute: over month using 8x Nvidia GeForce RTX 3090 GPUs. Unfortunately, the authors did not respond to our requests for the trained chockpoints used in their paper.
CiaoSR. (Cao et al., 2023). We found that the official CiaoSR implementation border cropping prior to evaluating on DIV2K deviated slightly from all other methods. We thus adapted the evaluation code to match the competition and enable meaningful comparison. All DIV2K numbers were re-computed with the corrected code, resulting in slight deviations from those reported in the original paper.
MSIT (Zhu et al., 2025). The numbers reported in the paper were achieved by training on a roughly 3x larger training set, comprising not only DIV2K but also Flickr2K. For a fair comparison with all other ASR methods used in our paper, we re-trained MSIT on DIV2K alone. We used the authors' official code and configuration files and trained for two stages, each comprising 1050 epochs. For the second (RiM) training stage we used scaling factors between x1 and x4. Performance generally drops result of the smaller training set, in line with the ablation experiments reported for MSIT, where the models were also trained only with DIV2K.
24


Reviews:
[{"review_id": "yINERYRLXM", "reviewer": "Reviewer_HwDs", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a novel approach to overcoming aliasing in single-image any scale superresolution (ASR). First they introduce \"neural heat fields\", a new type of neural field grounded in signal processing theory that garuntees anti-aliasing by design. These neural heat fields analytically model the physical point spread function (PSF) of images. The neural heat field are then incorporated in a novel end-to-end ASR method called \"Thera\" that combines a hypernetwork with a grid of local neural fields. Experiments show that compared to other ASR methods Thera outperforms while being more parameter-efficient.\n\n### Strengths\n- The paper incorporates a few principled ideas from traditional image processing such as neural heat fields and TV regularization. This makes their method more interpretable and could provide some insights into how we design general neural fields.\n- The authors demonstrate 3 variants of Thera making it customizable for a range of computing resources.\n- The results are very convincing showing that Thera acheives a good or better PSNR than many other methods across a range of datasets. \n\n### Weaknesses\n- By design, neural heat fields only allow for a single hidden layer MLP representation. While this is sufficient for the tasks studied, it may limit the expressive capacity of the model compared to deeper neural fields. The authors note this and compensate with a heavier hypernetwork, but this restriction could hinder extension to more complex settings.\n- While the authors make $\\kappa$ trainable, its role and learned values are analyzed. Understanding how $\\kappa$ adapts in practice could provide insight into whether the model is truly learning a physically meaningful observation model or just tuning a free parameter.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes all claims made in the paper are accurate and convincing. The theoretical basis for neural fields is well explained and convincing. Moreover, the authors provide extensive experiments and compare across many other ASR methods backing up their claims that Thera is a significantly better model for ASR and that it offers significant savings in terms of parameters.", "audience_yn": "Yes", "audience_explanation": "Neural fields are very popular in the TMLR and machine learning community so I see this paper being of interest to much of TMLR's audience.", "requested_changes": "- Provide more discussion or visualization of the learned $\\kappa$ values across datasets and scales. This would clarify whether the parameter is learning meaningful physical structure.\n- This is more of a suggestions, but it would be interesting to see whether this approach can be useful for other neural field tasks such as computed tomography reconstruction with few samples.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "gIUrg3R4nb", "reviewer": "Reviewer_ZWRi", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Thera: Aliasing-Free Arbitrary-Scale Super-Resolution\nwith Neural Heat Fields\n\nThis paper works on arbitrary-scale super-resolution (SR) with neural heat fields. Previous methods approximate the integral version of the field at each scaling factor. This work introduces a neural heat field that can achieve analytically correct anti-aliasing at any scaling factor.\nThis paper introduces an interesting formulation of the problem with neural heat fields by using a two-layer perception to parameterize the field. For each pixel, the phase shift b1 and matrix W2 are modeled by the output of a hypernetwork with image prior.\nThe prior hyper-network can be modeled with different architectures, the authors introduced three kinds of architectures with different capacities, all exhibiting competitive performance.\nThe authors also conducted quantitative evaluations to show the effectiveness of the proposed method and ablative studies to verify the desigin choices.\n\nStrengths:\n\n1. The proposed method is interesting and verified by the experiments. Although it limits the network to a two-layer perception, the decoupling of frequency bank, phase shift and global bias makes sense.\n2. The authors presented multiple architectures for the prior hyper-network, with different number of parameters and capacities, allowing it for real-world use and different purposes.\n3. The Thera Air has a simple and efficient architecture, while still have competitive performance compared to previous methods. The authors also conducted ablative studies to verify the design choices.\n\nLimitation:\n\n1. No error bars in the quantitative evaluations, making it hard to understand the marginal performance gain on some of the results. \n2. The author argued that previous methods approximate an integral version of the field at each scaling factor. It would be beneficial to lay out what approximations are used in the previous methods and how the proposed method is different in a general framework in the appendix. This will be helpful for the readers in a broader community without referring to other papers.\n3. Writing can be improved in some sections. For example, the authors introduced the time coordinate $t$ in section 3.1 abruptly without a clear explanation. In the reader's understanding, it's linked with the scaling factor, but it was not directly explained in the main paper.\n4. Although the qualitative studies show the effectiveness of the proposed method, it is still a bit off compared to the ground truth. One might wonder how denoising diffusion models would perform in this case. Would a better prior be more helpful in this case?\n\nOverall, the paper is presented in a good manner, and the proposed method is supported by its experiments. The reviewer believes the strengths outweigh the limitations once the questions and limitations are addressed.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors performed a quantitative study on multiple datasets against previous methods, and the Thera Pro model achieved the best overall performance.", "audience_yn": "Yes", "audience_explanation": "The proposed method would be interesting to the single-image super-resolution community. The formulation of the continuous representation from point spread functions would also be helpful to other communities, such as neuromorphic computing.", "requested_changes": "Most of them have been covered in the limitations part of the review. The error bars and comprehensive analysis on the approximation would be preferred. Section 3.1 could possibly be improved for a smooth introduction. The discussions and analyses about diffusion models would be interesting to readers, too.", "broader_impact_concerns": "The reviewer did not notice significant impact concerns of the proposed method as long as the training data of the proposed method follows the rules and regulations on intellectual properties and privacy.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "hBHiRSBkpf", "reviewer": "Reviewer_jQXV", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a bunch of anti-aliasing techniques for arbitrary super resolution tasks. These include 1) a learnable Fourier feature mapping, where the frequency 'bank' is learnable; 2) multi-scaled training data is used for training the prior model; 3) band-limited activation function; 4) TV regularization to encourage smoothness. \n\nThe paper is clear in general, and there are some valid justification behind how to set the hyperparameters especially in the newly proposed activation function.", "claims_and_evidence_yn": "No", "claims_explanation": "I am concerned with several issues in the paper:\n\n1. What are the finally learned frequency bank? Are they separated with each other in frequency domain? Are they highly dependent on the training data, and how generalizable is the learned frequency bank, can they be used for other reconstruction tasks?\n\n2. Does lower number of model parameters directly reflect on the reduced inference time of the model? Please justify more on this, and more practically, in terms of the running time/complexity.\n\n3. What are the exactly values of $\\kappa$ and $t$ used for each upsampling task? If $\\kappa$ is trainable, then what is the trained value, is it consistent with the justification made on the value of $\\kappa$? The selection of $t$ in practical experiments is not clear to me.\n\n4. The TV regularization can smooth the image, besides PSNR, for some results, can the authors provide SSIM and GMSD to evaluate the final reconstructed image as well?\n\n5. There are several other approaches can be included for comparison in Figure 5, for example SIREN and its variants.", "audience_yn": "Yes", "audience_explanation": "The newly proposed activation function has been empirically proved to have reasonable performance in ASR task. The discussion on solving aliasing in INR type approach is valuable to the community.", "requested_changes": "The concerns are the corresponding requested changes or explanation are in previous sections.", "broader_impact_concerns": "Can this approach be modified a bit and adapt to other reconstruction or inverse problems? Can some of the ideas be useful to the other inverse problems?", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)