PIN: Prolate Spheroidal Wave Function-based Implicit Neural Representations
Authors: Dhananjaya Jayasundara
Heng Zhao
Demetrio Labate
Vishal M. Patel
Venue: ICLR 2025 Poster (2025)
Keywords: Prolate Spheroidal Wave Functions
Implicit Neural Representations
MLPs
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Implicit Neural Representations (INRs) provide a continuous mapping between the coordinates of a signal and the corresponding values. As the performance of INRs heavily depends on the choice of nonlinear-activation functions, there has been a significant focus on encoding explicit signals within INRs using diverse activation functions. Despite recent advancements, existing INRs often encounter significant challenges, particularly at fine scales where they often introduce noise-like artifacts over smoother areas compromising the quality of the output. Moreover, they frequently struggle to generalize to unseen coordinates. These drawbacks highlight a critical area for further research and development to enhance the robustness and applicability of INRs across diverse scenarios. To address this challenge, we introduce the Prolate Spheroidal Wave Function-based Implicit Neural Representations (PIN), which exploits the optimal space-frequency domain concentration of Prolate Spheroidal Wave Functions (PSWFs) as the nonlinear mechanism in INRs. Our experimental results reveal that PIN excels not only in representing images and 3D shapes but also significantly outperforms existing methods in various vision tasks that require INR generalization, including image inpainting, novel view synthesis, edge detection, and image denoising.

Published as a conference paper at ICLR 2025
PIN: PROLATE SPHEROIDAL WAVE BUNDDION-BASED IMPLICIT NEURAL REPRESENTATIONS
Dhananjaya Jayasundara', Heng Zhao?.3, Demetrio Labate², Vishal M. Patel' Johns Hopkins University. UUniversity of Houston, *Data Science Platform, The Rockefeller University {vjayasul, vpatel3 363 j0ju.. edu, (hzhao258centl dlabatel). uh. edu
ABSTRACT
Implicit Neural Representations (INRs) provide a continuous mapping between the coordinates of a signal and the corresponding values. As the performance of INRs heavily depends on the choice of conlincar-activation functions, there has been a significant focus on encoding explicit signals within INRs using diverse activation functions. Despite recent advancements, existing INRs often encounter significant challenges, particularly at fine scales where they often introduce noise- like artifacts over smoother areas compromising the quality of the output. More- over, they frequently struggle to generalize to unseen coordinates. These draw- backs highlight a critical area for further research and development to enhance the robustness and applicability of INRs across diverse scenarios. To address this challenge, we introduce the Prolate Spheroidal Wave Function- Implicit Neural Representations (PIN), which exploits the optimal ppace-frequenyy do- main concentration of Prolate Spheroidal Wave Functions (PSWFs) as the nonlin- ear mechanism INRs. Our experimental results reveal that PIN excels not only in representing images and 3D shapes but also significantly outperforms existing methods in various vision tasks that require INR generalization, including image inpainting, novel view synthesis, edge detection, and image denoising.
INTRODUCTION
Discrete representations of images and shapes currently dominate computer vision tasks, with ben- efits derived from their convenience in storage and computation. Nonetheless, it is worth noting that memory consumption escalates exponentially with the dimensions and resolutions of the data. Moreover, the discrete formation limits the expressivity with the finite discrete grid size volume size. Implicit Neural Representation (INR) (Sitzmann et al. 2020); Tancik et al. (020) provides an alternative mechanism that continuously represents signals parameterized through Multi-layer Per- ceptions (MLPs). INRs are usually fully connected neural networks designed to learn a continuous mapping between coordinates of a signal the corresponding signal values, resulting in a compact representation of the signal with arbitrary resolution. Beyond representation (Chen et al.) 2023b Saragadam et al., 2022), INRs have demonstrated prominent advantages various vision tasks, in- cluding image reconstruction (Czerkawski et al. 2021 Saragadam et al., 2023). medical imaging (Shen et al. 2022] Sun et al. 2021), and novel view synthesis (Barron et al. 2021 Mildenhall al.
2020 Niemeyer et al. 20200.
Despite the evident advantages delineated above for INRs in vision computing tasks, properties of some inherent structures of INRs present obstacles to more widespread application in various vi- sion tasks. First, the expressivity of INRs is heavily influenced by the choice of activation function (Sitzmann et al. 2020 Ramasinghe & Lucey 2022. Saragadam et al. (023). The original INRs employing ReI.U as activation function were demonstrated to exhibit poor performance in signal representation tasks due the spectral bias of MLPs (Basri al.) 2020 Rahaman al.) 2019). which favors learning low-frequency and lacks the ability to represent fine details. Also, when re- constructing from sparse measurements, INRs tend to overfit to the coordinates used during training, i.e., the INR can only accurately represent the signal values of the trained coordinates, leading to blurry or noisy reconstructions.
"Corresponding Author

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Fourier features (Tancik al. 2020 or positional encoding (Mildenhall et al. 2020) is trans-
formation which maps low-dimensional coordinates to nigh-dimensional features, enabling MLP's
input layer to embed high frequencies and learning the high-frequent freeueegyynny content of signal (Tancik)
et al. 2020). Several alternative nonlinearities including sinusoids (Sitzmann et al. 2020 Gaussian
functions [Ramasinghe & Lucey (022), and Gabor Wavelets (Saragadam al.] 2023) have been
proposed as replacements for ReLUs along with positional embedding schemes, often leading to
notable enhancements in signal encoding capabilities. One key property of Gaussians and Gabor Wavelets is their good balance of joint space-frequeee energy concentration, which explains their efficient performance according to classical signal processing. Nevertheless, another limitation of current INR design is their high sensitivity to hyperparameter selection of parameters. Gaussian and Gabor wavelet nonlinearities require to set a predefined frequency or scale parameter for the INR to achieve competitive performance. How to determine best hyperparameters for the INR activation function and how to initialize the network is still heuristic, and highly dependent on the processing task.
Numerous recent studies have explored the inherent properties of INRs through theoretical analyses and experimental investigations advance the understand of the mechanisms behind their success and limitations. The Neural Tangent Kernel (NTK) (Jacot et al. 2018) was used to reveal that stan- dard MLP converges very slowly to -reeurree signals in low-dimensional coordinate-bll baaded INRs, owing to the rapid frequency fall-off of its corresponding kernels. When utilizing Fourier fea- tures (Tancik et al., 2020) or sinusoidal activation functions (Sitzmann et al. 2020) or their recently introduced variants (Liu et al. 2024, Shi et al. 2024; Kazerouni et al. 20224) which aim to achieve superior representation through variable periodic activation functions, sinusoid adjustments based on deep prior knowledge, and Fourier reparameterization, respectively. the Neural Tangent Kernel (NTK) is transformed into a stationary (shift-invariant) kernel, enabling control over the range of learned frequencies (Tancik et al.] 2020). The work by Yūce et al. (2022) provided a theoretical analysis of the expressivity and inductive bias of INRs, as well as the imperfect recovery resulting from the inadequacy of input frequencies to properly capture the frequency of the signal. Exper- imentally, Saragadam et al.] (2023) investigated the first layer output of INR, demonstrating that the spatial and frequency compactness of the activation function, well the multi-dimensional non-linearity, provide more accurate representations for natural images. [Roddenberry et al. (2023) demonstrated that the output of an INR with a generic wavelet activation in the first layer can be expressed in terms of the same wavelets.
In this paper, motivated by these recent theoretical and experimental findings, we propose the em- ployment of a novel activation function for INRs: the Prolate Spheroidal Wave Function (PSWF). PSWFs are designed to maximize the spatial and frequency domain energy concentration, and have previously been used in many vision applications (Lindquist & Wager] 2008 Wendt et al. 2010 Brown & DC, 1968) demonstrating that having an optimal balance of energy concentration both spatial and frequency domains is critical for efficient signal representation and approximation (Ra-) masinghe & Lucey] 2022] Saragadam et al., 2023). Our results below showcase that emplying PSWFs as the activation function for INRs endows this architecture with greater expressivity and generalizability as compared to sinusoidal, Gaussian, and Gabor functions which are currently used in state-of offte----hhe-rrr ahe art INR implementations. We attribute this improved performance the high energy compactification of PSWF atoms and their flexibility, namely the ability to easily tune their hyper- parameters to various lasks. Extensive numerical experiments demonstrate that our new INR model excels not only in representation tasks but also in more challenging reconstruction tasks, e.g., image inpainting, where exisitng INRs often perform poorly.
2 RELATED WORKS
Implicit Neural Representation. INRs have emerged in recent years showcasing remarkable per- formances not only in signal representation tasks but also in many inverse vision applications, e.g., unordered signal representation such as 3D shape representation (Park et al. 2019 Mescheerr et al.)
2019) or novel-view synthesis (Barron et al. 2021 Mildenhall et al. 2020 Niemeyer et al. 2020) The INR represents signals by parameterizing the mappings between input coordinates and signal values using an MLP. The MLP usage here differs from the standard approach in that the inputs are low-dimensional coordinates instead of high- digh-diiensional pixels. In such cases, the ReLU activation function performs poorly due to its spectral bias. Recent works have improved the expressivity
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of INRs by incorporating coordinate transformation introducing special type of non- con-linearity for the MLP, resulting in enhanced performance in signal representation tasks. Unlike conventional neural networks, the training of INRs is always conducted case-by-case basis, which hinders their widespread adoption in vision computing tasks. Several works have proposed accelerating training through multi-scale methods (Saragadam al., 2022), local blocks (Reiser et al. 2021
Mixture-of-Expert (Wang et al., 2022). adaptive coordinate (Martel et al. 2021).
Expressivity of INRs. Despite the widespread adoption of INRs in various tasks, a thorough the- oretical understanding is still limited. Fourier analysis was used to explore the spectral bias of
standard MLPs with ReLU (Rahaman et al. 2019 Cao et al. 2019 Basri et al., 2020). i.e. low
frequency is learning fast and robust to noise. The use of Fourier Feature Network (FFN) (Tancikk
et al. 2020) or periodic non-linearity (SIREN) (Sizzmann et al., 2020p can alleviate spectral bias by
transforming the NTK to a stationary (shift-invarian) kernel. Building upon the basic property of trigonometric function, Yúce et al. (2022) demonstrated that the expressivity of FFN and SIREN essentially share the same expressive power, which is characterized by the linear combination of sinusoidal functions at integer hamonics. They also identified the reason for the failure recovery as resulting from the uncovered spectral components, which leads to severe artifacts. For non-periodic activations, such as wavelet (Saragadam et al. (oddeeneedyy et al. (2023) derived a bound of functions that can be represented by INR.
3 METHODOLOGY
3.1 FORMULATION OF AN INR
An INR encodes the mapping between input coordinates R² and corresponding signal values frr E Rº, denoted as R' + RO, and the mapping which is 9. is parameterized by fully connected neural network 48 RR' + R where 0 represents the parameters of the neural network. For instance, when representing an image, the INR maps pixel coordinates r = (zz, 3/: to the corresponding RGB values (rr = (R., Gi, B.). The fully connected network t typically an
layer MLP, each layer given by:
20 (())
21 = o
b,)), L 1
27 WLZL-1 Py.
where y is the position encoding, is the nonlinear activation function. WR. by are weights and biases of the k'th layer. 2a E R.. is the input coordinate and zz. E ROO is the output of final layer.
3.2 WHY PSWFS?
In the recent INR literature, periodic functions such as the sinusoidal activation function and its analogous Fourier embedding have shown to yield better approximations than ReL.U, particularly for reconstructing high-frequency components. Functions with better spatial localization, such as Gaussian (Ramasinghe & Lucey 2022) or Gabor wavelets (Saragadam et al., 2022) have shown higher accuracy in image and shape representation tasks, benefiting from their analytic properties. Even though INRs with these activations work competitively in image and shape representation problems, they work poorly for reconstruction tasks from sparse or noisy measurements, such as image inpainting and denoising. Furthermore, when encoding the signals, existing INRs have a tendency to lose the balance between smoother and finer detailed areas in favor of finer details. A reason for these limitations could be the inadequate energy compactification of INRs when employ- ing Gabor wavelets or Gaussians. This problem is demonstrated by the great sensitivity of INRs to the selection of scale and frequency parameters for Gaussians and Gabor wavelets (Saragadam al., 2023). Signal representation can be improved by adjusting these parameters for each signal to make such activations more compact in space. However, fine tuning those parameters from sig- nal to signal extremely tardy, and inefficient task. In addition to that, when INRs are trained on partial data and evaluated the entire dataset, they fail to generalize, even with different parameter settings. The recovered -askedd areas tend to be purely noisy. Similarly, in image denoising, the INRs tend to learn the noise, making it difficult to remove effectively. These observations motivate us to explore alternate activation functions with better space- frequencr compactificatio and lead to PSWFs. Figure shows the space-frequent trade-off of the existing activations along with PSWF.
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PSWF
Gabor
Gaussian
Sinusoid
Amplitude
Spetial Distance
Magnitude
0 :
Frequency
Figure 1: Space-Frequency Tradeoff of Activations: The top row illustrates how the values of activation functions change with spatial distance, while the bottom row demonstrates how their magnitudes vary in the Fourier domain with frequency. When activation function compresses in the spatial domain, it tends to have a broader spectrum the frequency domain, and vice versa. This phenomenon is known the space-frequency tradeoff of functions. PSWFs recognized for their optimality in preserving the highest energy across both domains.
3.3 PROLATE SPHEROIDAL WAVE FUNCTIONS
PSWFs originated from question posed by Shannon MMore & Cada 2004) in the 1960s: To what extent are functions, which are confined to a finite bandwidth, also concentrated in the time domain? Slepian (Slepian & Pollak [961 [Slepian 1964; Landau & Pollak 1961b 1962). discov- ered that such functions are associated with the solutions to Sturn-Liouville problem arising from the Helmholz equation the prolate sphere (hence the name). In seminal sequence of papers, Slepian, Pollak and Landau at Bell Labs introduced an integral formulation for this energy concen- tration problem which led to the following equation (Landau & Pollak [961a] Slepian & Pollak 1961 where the PSWFs vin (c,t)
f) are the eigenfunction solution of the integral operator problem
Un (c,t) sin M(x f) dt (c, xc).(c). π - t)
where (c) are corresponding eigenvalues, is the bandwidth parameter and c tn S2 is the cut-off frequency of A, (c). The PSWFs form an orthonormal basis of the space 32-bandlimited functions with the fundamental property of being maximally concentrated in time and frequency domain. This energy concentration property has been proven to very advantageous in various signal processing tasks (Gosse 2013 Hu et al. 2014 Lindquist & Wager 2008), most notably in sampling problems where PSWFs have been employed for image reconstruction from sparse samples (Khare & George 2003 Hogan al. 2010 Lindquist (003). The success of PSWFs in sampling applica- tions of particular interest here, as suggests their potential for efficiently representing complex information. In our results section below, we confirm that leveraging excellent expressivity proper- ties with robustness in the sparse and possibly noisy sampling setting; PSWFs leads to a remarkably effective choice of activation function for INRs.
In an INR model that employs polynomial activation functions and compactly supported wavelets as template functions in the first layer, [Roddenberry et al. (2023) demonstrates that the Fourier transform of the INR output can be characterized by the convolutions of the Fourier transforms of the first-layer atoms with themselves. This observation can be extended to PSWFs which have not compact support but have rapid space decay. They are generalization of Legendre polynomials and can represented by the expansion:
(n(c = Σ Pt),()).
where (t(t) is normalized version of Legendre polynomial of order is where the coefficients B, can be calculated by recurrence relation shown in Moore & Cada (2004).
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As shown in Roddenberry et al. (2023) and Yúce et al. (002) the support of scaled and shifted versions of the template in the first layer is preserved, implying that the output at given coordinate relies solely the template in the first layer whose support contains this coordinate. For PSWFs indeed true that the Fourier support is preserved. As a result, the Fourier output at given coordinate is dependent only upon the PSWFs in the first layer whose Fourier support contains
SPACE-FREQUENCY LOCALIZATION
Strong localization in both space and Fourier domains is highly desirable property in classical signal processing which is known to be critical for robust signal approximations and reconstructions (Donoho et al. 1998). nutshell, localization entails that the energy of a signal is highly localized, so that complex information can be efficiently decomposed into well localized components. Emerg- ing evidence shows that apecunrry localization of the activation function is equally important in INRs.
For this reason, the selection of PSWFs in the PIN design has major impact to explain their ex- cellent performnance. The visualization in figure illlutraats the behavior of PSWFs across space and Fourier domains. In the spatial domain, we can observe how PSWF is highly localized within specific region, rapidly tapering off away from the center. This localization is mirrored in the fre- quency domain, where the function confined within a finite bandwidth. In section 5 we prove that the output of PIN, (r) is polynomial of PSWF with degree of K 1-1 Moreover, the Fourier transform of oo(r)) (r) is band-limited and with rapid space decay which contributes to the advantage in INRs.
LOCALIZATION AND EXPRESSIVITY PROPERTIES OF PIN
Our INR architectures can be decomposed into mapping function (r) Rd - To also called positional encoding, followed by an MLP with weights Wa E I biases bμ E T and activation function P applied element wise. By denoting as 24 the post-activation functions, the INR computes the output function Dy z1 form equation I Our PIN adopts the PSWFs, denoted below with the symbol v as the activation function (p v) We choose v to be the PSWF order 0.
Theorem Let Po(r) : RRd + be an To layer PIN, with weights W. TRTxxT=-11, biases by E Te in k-th layer, assume the activation function vb is PSWF which can be approximated with polynomial of degree at most K, that is (xxx) EK amx"
Then, D (r) can be expressed as polynomial of ob (W (1) (r) + be) with degree at most K L-1
RL-1
@o(r) = ( (o) "
II de.
m= 81
=m
Theorem 1 shows that the Fourier transform of (o is -rd-r convolution of Fourier and the convolution of band-limites functions is
transforms of PSWFs v. Since v is band-limited, then (r) is also band-limited. Additionally, since convolution increases regularity, (r(r) has high-order of regularity in the Fourier domain implying that &o has very rapid decay in space, i... it is highly localized.
ADAPTIVE ACTIVATION FUNCTION PARAMETERS LEARNING
We have argued above that the optimal spatial and frequency concentration of PSWF positively im- pacts the performance of the INR model. Another critical advantage of this approach is its flexibility to determine the best hyperparameters of the PSWF activation.
As noted above, the parameters of activation functions in WIRE and GAUSS are chosen for differ- ent tasks and different signals using a grid search (Saragadam et al. 2023). However, the activation function's parameters resulting from such a grid search are local. i.e., they are only optimal for the
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signal that has been used in the grid search, while they are suboptimal for other signals, affecting the expressivity of the INR model. Even if those parameters are designed to be learnable, the initial- izatiin still impacts the parameters of the learned activation function. For instance, the parameters in the Gabor wavelet vb (z;w, = e c |sxr" are and s, representing the frequency and scale respectively; in Gaussian function vb (I; 8) = e (sm)², the parameter & represents the scale. As the parameters of these functions appear in exponent, it is difficult to learn wider frequency or scale distributions.
The situation is different for our PSWF activations as we do not use explicit formulation, but use a numerical approximation, so there are no explicit parameters to control. Rather, we apply an indirect way to control the amplitude, frequency, and height characteristics of the PSWF. That is, for the predefined PSWF function (x(z), we instantiate the learnable PWSF activation function by (xx) (w(w)) + b. In this case, the amplitude, frequency, and height of the PSWF activation function are indirectly controlled by T, w, and b, respectively. The construction PSWFs as an activation, detailed explanations and ablations adaptive activation function parameter learning are provided in section A.I in Appendix.
EXPERIMENTS
7.1
IMAGE REPRESENTATION
Image representation, commonly referred to as image regression, is a measure that evaluates the expressive capabilities of INRs. A superior INR should not only represent the original signal but also preserve a substantial amount of the structural information, texture, color, and contrast of the image. For assessing the effectiveness of INRs in image representation tasks, the Kodak Lossless True Color Image Dataset (Franzen 1999) was employed. This dataset was chosen to analyze the performance of the PIN across a comprehensive set of images rather than limiting the evaluation to single example. Comprising 24 images with varied spatial and frequency content, the dataset presents a robust challenge, requiring the PIN to maintain high fidelity in representation. PIN was subsequently trained on each image in the dataset. The left panel of figure 2 showcases the 15th image from the dataset alongside the image representations generated by various INRs. On the right of the same figure displays the variation in PSNR across the dataset for different INRs, where each image is indexed along the circumference of the radar plot. The PSNR values for PIN are highlighted in black.
The child image on the left of figure 2 showcases the PIN's color accuracy and its capability of extracting highly varying contents of the image into the weights and biases of an INR compared to existing INRs. A closer look at the decoded representations of the existing baselines reveals that INRs which are only "requency-compact like SIREN, tend towards a low-pass representation of the explicit representation. failing to capture intricate details. On the other hand, INRs that are both space-frequency compact but with rapid decay both domains showcase the ability to capture fast-varying components of images into the INR. However, when they focus more on finer details, they fail to balance smooth regions, introducing some additional noise. As PSWFs are the optimal representation in both domains, PIN focuses on both finer and smooth regions, encoding the signal optimally with minimal distortion. Due to PIN's ability to effectively focus on both high and oow-frequency components, it outperforms the existing baselines across all images in the dataset, consistently delivering a PSNR of at least 30 dB. Additional image representation results including thorough evaluation on DIV2K dataset (Agustsson & Timofte, 2017), and learning curves provided in the supplementary material.
7.2 SOLVING THE WIDE FREQUENCY SPECTRUM CHALLENGE VIA PIN
Even though INRs are proficient at converting explicit signals to implicit representations, their per- formance is impeded when dealing with signals that encompass wide frequency spectrum. Specif- ically, while INRs that incorporate Fourier feature encodings (Landgraf et al. 2022) excel at cap- turing fine details through high-frequency components, they simultaneously introduce undesirable noisy elements in areas that should remain smooth. Notably, according to the findings in (Landgraf et al. 2022), SIREN (Sitzmann al. 2020) also exhibits this limitation, struggling to maintain fi-

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36.00 00 db
31.81 81 d6
0.903
0.799
Ground Truth
PIN
WIRE
30, 63 de
28. da
0.872
SEAAE
SIREN
GAUSS
ReLU+PE
Figure 2: Image representation capacity of PIN: The left image shows an instance of PIN's rep- resentation capacity, and the right image shows the PIN's representation capacity evaluated all Kodak lossless true color images. PIN stands out as the INR that achieves the highest PSNR and SSIM metrics, indicating minimal distortion and maximum preservation of structural information.
delity in smoother regions while effectively representing finer textures, and (Landgraf et al. 2022) presents hierarchical representational approach to resolve this issue. Further, rapid decaying space- frequency non-linearities such as Gabor Wavelet (Saragadam et al. 2022) and GAUSS (Ramasinghe) fast varying components of images and capturing them ef-
& Lucey 2022) are good at focusing ficiently into weights and biases, but when they focus on intricate details, they lose their focus on smooth regions and introduce some form of noisy component (see figure 3 However, as PSWFs known for being optimally space and band-limited limited, they excel in the preservation of energy concen- tration. This characteristic allows them to adeptly represent finer details without encountering this issue. Therefore, PIN achieves an optimal representation that maintains high fidelity across different textural features. This effect is illustrated in figure 3 where the efficacy of PSWF's in balancing de- tail and smoothness is visually demonstrated compared to the existing INRs. Furthermore, this issue can be seen in the decoded representations of the child image in figure 2 as well. Therefore. from the presented visual results, it can be clearly seen that without the need for any additional architectural changes to INRs, PIN can resolve this wide frequency spectrum challenge.
25. I0
26,47 de
26.15
22.65
Ground Truth
PIN
WIRE
SIREN
GAUSS
ReLU+PE
Figure 3: Wide-Frequency spectrum challenge:
Existing INRs often attempt emphasize finer details while not focusing much on smoother regions. This can be attributed to their nonlinearities' fpace-freuencyy trade-offs, In contrast, PIN exhibits optimal energy concentration for a given band, enabling them to effectively encode both fine details and smooth regions.
7.3 OCCUPANCY FIELD REPRESENTATION
When comes to representing three dimensional occupancy fields. the topic of image representation, as explored in figure extends into the realm of three- vire-dimensional spaces. This extension allows for an examination of the capability of INRs depict three-ding dimensional signed distance fields. this context, the transformation occurs from a tttree-dimensions domain to binary signal space, marked by the values or O. Where, a value of signifies the presence of the signal within predefined area, whereas value of 0 denotes its absence from that area. For this experiment two occupancy volumes namely Asian Dragon, and Armadillo, which are shown in figure 4 were obtained from Stanford grid of
3D shape dataset (Stanford University Computer Graphics Laboratory), and sampled 512 x 512 x 512, assigning a value to each voxel inside the volume and a value of to those outside. The methodology for this study was derived from the work mentioned in Saragadam et al.

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The obtained results shown in figure 4 It can be observed that in both the instances PIN effectively encode the rapidly changing contrast, when examining the performance of WIRE and SIREN, it is evident that these architectures struggle to incorporate the quickly varying components of the structure into their weights and biases, resulting in representation that primarily captures lower frequency components. While GAUSS activation function achieves performance metrics similar to PIN, a detailed examination of the three- dimensional statues decoded from Gauss reveals distortions some uniform areas in both instances. Conversely, PIN encodes the structure accurately without introducing any additional artifacts.
2023
structure into its neural representation. In
GT
PIN
WIRE
SIREN
GAUSS
ReLU+PE PE
LML
Asian Dragon
Armadillo
Figure 4: Occupancy fields representation capacity of PIN: PIN stands out as the INR that exhibits the highest similarity the ground truth occupancy field. Unlike other INRs, the PIN approach does not converge to a low-pass representation; rather, it accurately preserves high-frequency components throughout the training process. This characteristic enables PIN to faithfully capture intricate de- tails, making it particularly adept for tasks requiring accurate representation of complex occupancy patterns.
7.4
IMAGE INPAINTING
INRs are often employed to learn continuous functions from discretized signals. Once an INR is adequately trained, it should demonstrate excellent generalization capabilities even with limited training samples. A notable application computer vision is image inpainting. In this approach, during the training phase, INR receives partial data from image. In the testing phase, the INR tasked with reconstructing the missing positions of the image. Image inpainting is a measure to determine if the learned underlying representation has overfitted to the provided training data.
We employ two testing strategies: one involves training with 70% of the data sampled randomly, and the other uses predefined mask that obscures the image with varying font sizes. The top row of the figure shows inpainting results, corresponding to 70% random sampling, whereas the bottom of the same figure shows the text-masked inapinting results. In both instances the second column represents the masked image. As can be clearly seen from the results, PIN is the only architecture that maintains the highest PSNR value in both instances, indicating superior recovery capabilities with minimal distortion to the original image. Additionally, it is the architecture that produced the most visually appealing reconstruction comparison with the ground truth.
When comparing the performance of various INRs, it is evident that WIRE exhibits signs of over- fitting to the training data and has not generalized well. SIREN demonstrates commendable gener- alization capabilities within the context of INRs, yet it is constrained by its need for highly specific weight initialization and frequency parameters. Therefore, PIN is particularly effective for image inpainting, producing visually appealing results when reconstructing highly obscured images. This underscores its robust generalization ability compared to existing compactly supported INRs like WIRE and GAUSS, and sets benchmark that surpasses current state-of-the-art methods.
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Susper Masked Emage
6.300
ELEZE
Ground Truth
PIN
WIRE
SIREN
GAUSS
RoLU+PE
Figure 5: Image Inpainting Capabillities of PIN: Among the existing INRs, PIN stands out as the top image inpainting performer, excelling not only in achieving the highest PSNR in both instances, but also generating the inpainting results most visually similar the ground truth. PIN's superior metrics highlight its proficiency in restoring images with minimal distortion while preserving max- imum structural information.
7.5 NEURAL RADIANCE FIELDS
The core area benefiting from INRs is NeRFs (Mildenhall et al. 2020). In NeRF, scene is encoded into an INR by feeding the viewer's coordinates (z, y, 2) and the viewing directions (0, ø) into the INR. The INR is tasked with predicting the corresponding location's 's color and density. When the INR is well-trained, it gains the ability to generate novel views from different spatial locations and viewing angles, those are not included in the training data. This remarkable capability is achieved by training a 3D implicit function using spatial coordinates (z, y, 2) and viewing directions (0, Φ) allowing for the synthesis of highly realistic images and scenes. For this experiment, we used Vanilla NeRF architecture consisting of two fully connected blocks, each containing four layers, and the drums dataset with 100 training images and 200 testing images. One of the obtained novel views for each INR is presented in figure PIN's remarkable performance can be attributed to its optimal energy concentration within specified frequency band, distinguishing it from other methods and reinforcing its superiority. Additional novel views are provided in the supplementary material.
Grownd Trush
PIN 25. dB
WIRE 23.12 dB
SIREN 23.94 dB
GAUSS 25.21 dB
Figure 6: Novel View Synthesis Capabilities of PIN: PIN achieves the highest PSNR in novel views performance as compared to existing INRs. While existing INRs tend to produce blurry novel views in feature-dense areas, PIN's ability to preserve maximum energy in a given state allows it to encode intricate details into its weights and biases, and produce novel views that exhibit the highest similarity to the ground truth.
7.6
ABLATION STUDY
7.6.1 NETWORK HYPERPARAMETERS
An ablation study was conducted to evaluate the effectiveness of PINs with various hyperparameters selected during the training environment. For this experiment, we used the child image shown in figure [ This hyperparameter tuning included the variation of PSNR with the number of hidden neurons, while keeping the number of hidden layers constant at 3 (shown in the left figure of fig- ure 7.. Additionally, the study examined the variation of PSNR with the number of hidden layers, while maintaining the number of hidden neurons at 300 (shown in the middle figure of figure 7 Lastly, the variation of PSNR with the learning rate was analyzed, with the number of hidden layers kept at 3 and the number of hidden neurons 300 (shown in the right figure of figure 7)
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275
23.0
PSNNNEE
273
585
350
320
258
363
453
Number Layars
Learning Rate
2001
Figure 7: Hyperparameter Turning of PIN: PIN demonstrates a sharp linear increase in PSNR with the addition more hidden neurons and layers compared to existing INRs. Instead of be- coming unstable with higher learning rates, PIN stabilizes its PSNR, maintaining nearly constant performance.
In general, increasing the number of hidden neurons or hidden layers in an INR enhances its ca- pacity due to the higher number learnable parameters. As shown in the left and middle plots of figure 7 PIN exhibits approximately linear increase, with higher gradient compared other INRs, in PSNR with the addition of more hidden neurons and hidden layers. However, as the learn- ing rate increases, PIN demonstrates PSNR saturation effects instead of tending toward instability, as illustrated in the right plot of figure 7
7.7 THEORETICAL ANALYSIS, ABLATION STUDIES, AND ADDITIONAL EXPERIMENTS
In the appendix, we provide a comprehensive theoretical analysis of PIN's forward propagation, alongside combined theoretical and experimental investigation into why existing baseline space- frequency compact INRs fall short compared to PIN. This analysis helps to elucidate the superior performance of PIN in capturing complex details. The appendix also includes the complete numer- ical implementation of PIN, where we demonstrate its robustness to variations in parameters and provide detailed examination of how different weight initialization strategies impact its perfor- mance. Furthermore, we showcase PIN's learning curves and extensive experimental results to sup- port the findings presented in the main sections. In addition these results, the appendix features further analysis of PIN's capabilities through set of additional experiments. These experiments explore PIN's ability to handle high-frequency encoding, perform effective image denoising, and detect edges with precision.
8 CONCLUSION
Implicit Neural Representations (INRs) have emerged as very promising framework in computer vision and image processing, yet their performance very sensitive on the choice of activation func- tions. Empirical results indicate that well-localized activation functions with good space-frequeee concentration are more effective than sinusoidal functions which suffer from poor spatial decay. For instance, current INRs often focus more on encoding intricate details into weights and biases of the INR, and do handle well smoother regions. Further, they underperform reconstruct- ing sparse or noisy data due to overfitting. To overcome these limitations, propose the use of PSWF as activation function for INRs. PSWFs maximize joint space-frequency domain concentra- tion, erucial feature for enhancing INR capabilities. Our numerical experiments demonstrate that INRs with PSWF activations significantly improve expressivity and generalizability, boosting per- formance across various tasks including image and 3D shape representation, novel view synthesis, reconstruction from sparse or noisy measurements, and edge detection.
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9 ACKNOWLEDGMENT
This work is supported by the Intelligence Advanced Research Projects Activity (IARPA) via De- partment of Interior/ Interior Business Center (DOI/IBC) contract number 140D0423C0076. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions con- tained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, DOI/IBC, or the U.S. Government.
REFERENCES
Eirikur Agustsson and Radu Timofte. Ntire 2017 challenge on single image super-resolution: Dataset and study. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, July 2017.
Jonathan T Barron, Ben Mildenhall, Matthew Tancik, Peter Hedman, Ricardo Martin-Bruall and Pratul P Srinivasan. Mip-nerf: multiscale representation for anti-aliasing neural radiance fields. In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pp. 5835-5844. IEEE, 2021.
Ronen Basri, Meirav Galun, Amnon Geifiman, David Jacobs, Yoni Kasten, and Shira Kritchman. Frequency bias in neural networks for input of non-uniform density. In International Conference on Machine Learning, pp. 685-694, PMLR, 2020.
Henry A Brown and NAVAL RESEARCH LAB WASHINGTON DC. Image Enhancement with Prolate Spheroidal Wave Functions. Naval Research Laboratory, 1968.
Yuan Cao, Zhiying Fang, Yue Wu, Ding-Xuan Zhou, and Quanquan Gu. Towards understanding the spectral bias of deep learning. arXiv preprint arXiv: 19/2.01 198, 2019.
Anpei Chen, Zexiang Xu, Xinyue Wei, Siyu Tang, Hao Su, and Andreas Geiger. Factor fields: A unified framework for neural fields and beyond. arXiv preprint arXiv:2302.012266 2023a.
Zhang Chen, Zhong Li, Liangchen Song, Lele Chen, Jingyi Yu, Junsong Yuan, and Yi Xu. Neurbf: A neural fields representation with adaptive radial basis functions. In CVPR, pp. 4182-4194, 2023b.
Mikolaj Czerkawski, Javier Cardona, Robert Atkinson, Craig Michie, Ivan Andonovic, Carmine Clemente, and Christos Tachtatzis. Neural knitworks: Patched neural implicit representation networks. arXiv preprint arXiv:2109. 2021.
David L. Donoho, Martin Vetterli, Ronald A. DeVore, and Ingrid Daubechies. Data compression and harmonic analysis. IEEE transactions on information theory, 446))2445524776 1998.
Rich Franzen. Kodak lossless true color image suite. httpps::/ro...uaagappaaae////aa 1999. Accessed on May 21, 2024.
Laurent Gosse. Compressed sensing with preconditioning for sparse recovery with subsampled matrices of slepian prolate functions. Annali dell' Universita' di Ferrara, 59:81-116, 2013.
Jeffrey A Hogan, Scott Izu, and Joseph Lakey. Sampling approximations for and bandlimit- ing. Sampling Theory in Signal and Image Processing, 9:91-117, 2010.
Nan Hu, Xu Xu, and Zhongfu Ye. Doa estimation for wideband signals based on sparse signal reconstruction using prolate spheroidal wave functions. Signal processing, 96:395-400, 2014.
Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kemnl:: Convergence and gen- cralization in neural networks. Advances in neural information processing systems, 31, 2018.
Amirhossein Kazerouni, Reza Azad, Alireza Hosseini, Dorit Merhof, and Ulas Bagci. Incode: Implicit neural conditioning with prior knowledge embeddings. In Proceedings of the IEEE/CVF Winter Conference Applications of Computer Vision, pp. 1298-1307 2024.
11

---

Published as a conference paper at ICLR 2025
Kedar Khare and Nicholas George. Sampling theory approach to prolate spheroidal wavefunctions. Journal of Physics A: Mathematical and General, 36(39):10011, 2003.
Henry J Landau and Henry O Pollak. Prolate spheroidal wave functions, fourier analysis and uncer- tainty-ii. Bell System Technical Journal, 40(1):65-84, 1961a.
Henry J Landau and Henry O Pollak. Prolate spheroidal wave functions, fourier analysis and uncer- tainty-ii. Bell System Technical Journal, 40(1):65 1961b.
Henry J Landau and Henry O Pollak. Prolate spheroidal wave functions, fourier analysis and uncer- tainty-iit the dimension of the space of essentially time-and band-limited signals. Bell System Technical Journal, 1194)11295-1336, 1962.
Zoe Landgraf, Alexander Sorkine Hormung, and Ricardo Silveira Cabral. Pins: progressive implicit networks for multi-scale neural representations. arXiv preprint arXiv:2202.04713. 2022.
Martin A Lindquist. Optimal data acquisition in fmri using prolate spheroidal wave functions. International Journal of Imaging Systems and Technology, 13(2):126-132, 2003.
Martin Lindquist and Tor D Wager. Spatial smoothing in fmri using prolate spheroidal wave functions. Human Brain Mapping, 29(1 (9(1)):176-1288 2008.
Zhen Liu, Hao Zhu, Qi Zhang, Jingde Fu, Weibing Deng, Zhan Ma, Yanwen Guo, and Xun Cao. Finer: Flexible spectral-bias tuning in implicit neural representation by variable-periodic activa- tion functions. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2713-2722, 2024.
Julien NP Martel, David B Lindell, Connor Z. Lin, Eric R Chan, Marco Monteiro, and Gordon Wet- zstein. Acorn: adaptive coordinate networks for neural scene representation. ACM Transactions on Graphics (TOG), 40(4):1-13, 2021.
Lars Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Oc- cupancy networks: Learning reconstruction in function space. In Proceedings ofe IEEE/CVF conference on computer vision and pattern recognition, pp. 4464-444477 2019.
B Mildenhall, PP Srinivasan, M Tancik, JT Barron, R Ramamoorthi, and R Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In European conference on computer vision, 2020.
Ian C Moore and Michael Cada. Prolate spheroidal wave functions, an introduction to the slepian series and its properties. Applied and Computational Harmonic Analysis, 16(3):208-230, 2004.
Michael Niemeyer, Lars Mescheder, Michael Oechsle, and Andreas Geiger. Differentiable volumet- ric rendering: Learning implicit 3d representations without 3d supervision. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3504-330 3515,,515, 2020.
Andrei Osipov, Vladimir Rokhlin, Hong Xiao, al. Prolate spheroidal wave functions of order zero Springer Ser. Appl. Math. Sci, 187, 2013.
Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In Proceedings of the JEEE/CVF conference on computer vision and pattern recognition, pp. 165-174, 2019.
Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks. In International Confer- ence on Machine Learning, pp. 5301-5310. PMLR, 2019.
Sameera Ramasinghe and Simon Lucey. Beyond periodicity: Towards a unifying framework for acti- vations in coordinate-mlps. In European Conference on Computer Vision, pp. 142-158. Springer,
2022.
Christian Reiser, Songyou Peng, Yiyi Liao, and Andreas Geiger. Kilonerf: Speeding up neural radiance fields with thousands of tiny mlps. In Proceedings of the IEEE/CVF International Con- ference Computer Vision, pp. 14335-14345, 2021.
12

---

Published as a conference paper at ICLR 2025
T Mitchell Roddenbemy, Vishwanath Saragadam, Maarten V de Hoop, and Richard G Bara- niuk. Implicit neural representations and the algebra of complex wavelets. arXiv preprint arXiv:2310.00545. 2023.
Vishwanath Saragadam, Jasper Tan, Guha Balakrishnan, Richard Baraniuk, and Ashok Veer- araghavan. Miner: Multiscale implicit neural representation. In ECCV, pp. 318-333. Springer, 2022.
Vishwanath Saragadam, Daniel LeJeune, Jasper Tan, Guha Balakrishnan, Ashok Veeraraghavan, and Richard G Baraniuk. Wire: Wavelet implicit neural representations. In CVPR, pp. 18507-18516, 2023.
Liyue Shen, John Pauly, and Lei Xing. Nerp: implicit neural representation learning with prior embedding for sparsely sampled image reconstruction. IEEE Transactions Neural Networks and Learning Systems, 2022.
Kexuan Shi, Xingyu Zhou, and Shuhang Gu. Improved implicit neural representation with fourier reparameterized training. arXiv preprint arXiv:2401.07444 2024.
Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Im- plicit neural representations with periodic activation functions. Neur/PS, $3:7462-7473, 2020.
David Slepian. Prolate spheroidal wave functions, fourier analysis and uncertainty ivv: extensions to many dimensions; generalized prolate spheroidal functions. Bell System Technical Journal, 43 (6):3009-3057, 1964.
David Slepian and Henry Pollak. Prolate spheroidal wave functions, fourier analysis and uncer- tainty- i. Bell System Techmical Journal, 40(1):43-66 43, 1961.
Stanford University Computer Graphics Laboratory. The stanford 3d scanning repository, https: //graphics stanford. wet/ 3Dscannr//s Accessed: 2024-07-15.
Yu Sun, Jiaming Liu, Mingyang Xie, Brendt Wohlberg, and Ulugbek S Kamilov. Coil: Coordinate- based internal learning for tomographic imaging. IEEE Transactions Computational Imaging, 7:1400-1412, 2021.
Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keit Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in Neural Information Process- ing Systems, 33:7537-7547, 2020.
Peihao Wang, Zhiwen Fan, Tianlong Chen, and Zhangyang Wang. Neural implicit dictionary learn- ing via mixture-of-expert training. In International Conference Machine Learning, pp. 22613- 22624. PMLR, 2022.
Herwig Wendt, Maarten V de Hoop, Fredrik Andersson, and Anton Duchkov. Multi-scale struc- tured imaging using wave packets and prolate spheroidal wave functions. In SEG International Exposition and Anmual Meeting, pp. SEG-2010. SEG, 2010.
Gizem Yüie,, Guillermo Ortiz-liménez, Beril Besbinar, and Pascal Frossard. A structured dictionary perspective on implicit neural representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 19228 19228-19238, 2022.
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A APPENDIX
A.1 ANALYSIS
A.1.1 THEORETICAL ANALYSIS
Let us examine the forward propagation of the PIN.
First, we use positional encoding ((r) IR - RTo to map input coordinates to higher dimensional Fourier embedding. The 1-st layer pre-activation function the j-th node is:
where W1 E RRTixb is linear weights, hence, the 1tst layer j-th node after PSWF activation func- tion is:
2 = (v + 661) = 2 (W1 (r)))(()= + (61)
Notice that &b is bandlimited.
To examine the effect of functional composition in the subsequent layers, we will use the expansion of PSWFs into Legendre polynomials:
(xx) = Σe Bg (x)),,
k=0
where Pk is the Legendre polynomial of degree We assume below that the polynomial expansion of db is approximated with polynomial of degree most K:
(5x)
Oma m
The 2-nd layer pre-activation function at the j-th node is:
Using the polynomial expansion for the PSWF activation function, we have that the 2-nd layer j-th node after PSWF activation function is of the form:
(
(1)
db ( db (v1) + +b0))
22
Σ
W
W
1b (Wi((r) + b) + bb))
a'm
K
(t)
((+) (r) +b6)
Om
W
m==
II de, (((w(0>(r) + +bi)) Ps
Σ
m=0di+ --
K
II ae, (L(W(( + ba)) 8s
Σ
Σ
(1)
m=
E1
m
Eq. shows that the postactivation output of the j-th node in the second layer a polynomial of W ( ())r))))) + b.) with degree K.
((r)
Using induction we can get the similar results for the PIN output (this is simlar to Roddenberyy et al.
2023 (r(r)) = WIZL-1 + by is polynomial of sb W x(r) + be) with degree K KL-1 where the number of layers.
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A.1.2 WHY DOES PIN OUTPERFORM SAACE-FREQENCY COMPACT INRs?
Activations such as Gauss and Gabor are also peace-frqquence compact, but with rapid decay in both domains. However, as evidenced by PIN's results, PIN consistently outperforms these baselines. The primary reasons for this are its high-energy concentration and the implementation of PSWFs, which can be explained as follows. As PSWFs are derived through an optimization procedure that ensures optimal energy concentration throughout the given space, they must be constructed using numerical methods. Unlike the Gauss and Gabor functions, at any given band, the PSWF has the maximum energy concentration, tapering off with oscillations but not zeroing out all values outside of the support. This optimal decay of PSWFs allows them to retain both geghffreyenccy and low- frequency components, passing essential elements through the activation function without losing critical information due to rapid decay.
For Gauss and Gabor functions, to obtain better results, the parameters w or a are often large (e.g., 20, 10 in the WIRE paper). These parameters make these functions behave like Dirac delta, zeroing out most values throughout the considered space. We hypothesize that this property makes the parameters of Gauss and Gabor functions inefficiently learnable. This can be explained mathematically and shown experimentally as follows, using the Taylor expansion. For simplicity, consider the expansion of a Gaussian:
(sx²)²
(sz?)3
exp( szz = 1- sx
2!
3!
where s is a trainable parameter. For large s, this behaves like a Dirac delta, filtering most data during the forward pass, which may be undesirable for given signal. Therefore, backpropagation instructs all trainable parameters, including the activation function parameter, to adjust their values to minimize the overall loss. At certain epoch, backpropagation may seek to expand or compress the activation function to pass more features or limit them, respectively. As these functions are implemented numerically using a truncated Taylor series of order 2n, the parameter s appears non- linearly in 2(n 1) coefficients. Therefore, adjusting s might exhibit a certain level of reluctance, as it must be simultaneously modified 2n 1 places. Consequently, the value of s unlikely to change significantly, even though backpropagation attempts to adjust it. This means the shape of the activation curve does not change much, preventing optimal filtering of features to be passed to the final layer. This leads to either the loss of valuable information or the passage of redundant informa- tion during each forward pass. However, in PIN, we use a discretized solution from the differential equation with cubic spline approximation, denoted as d(x), and ((sx) is given by,
(<x) = ao + 01 (sx) + az (sx)) ag())))
Therefore, in PIN, 8 appears only in three coefficients, making it easier to adjust s in the loss land- scape when necessary, leading to quicker convergence. To substantiate this, we conducted experi- ments with trainable parameters in PIN and WIRE. For this experiment, we perturbed the parameters as described in section A.3.I applying variance of 25. The variation of parameters are shown in figure [8] and it illustrates the adjustments of each trainable parameter across epochs for both PIN, and WIRE. In figure the subscript number in each parameter shows which layer it belongs. For instance, Ti represents the amplitude parameter of PSWF activation in the first hidden layer.
The first three plots of figure show that PIN's parameters quickly adjust in way that they locate a local minimum, resulting in a PSNR of 35.56 dB. In contrast, the last two plots of figure g de- piet WIRE's slower parameter adjustments due to its reluctance, resulting in PSNR of 20.40 dB. Without special treatment, these parameters are unlikely to move effectively.
A.2 FURTHER EXPERIMENTAL DETAILS
A.2.1 EXPERIMENTAL SETUP
PIN, based on PSWFs, concentrates maximum energy in a given frequency band, and outperforms the existing INRs in representation capabilities, accuracy, and generalization across all visual sig- nals. Rigorous experiments show that PIN precisely learns the underlying continuous mapping of INRs, enabling exceptional performance in inverse vision problems. For numerical experiments,
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Denator MUX AUL
Figure 8: PIN efficiently locates local minima: PIN's parameters evolve within a few epochs to local minimum, resulting in PSNR of 35.56 dB. In contrast, WIRE's parameters do not move significantly or move very slowly, leading to PSNR of 20.40 dB. The subscript in each parameter denotes the layer number.
we used the Py PyTorch framework and the Adam optimizer (learning rate 0.001) with a Multilayer Perceptron (MLP) comprising layers of 300 neurons in each hidden layer. Image performance was assessed using PSNR, SSIM, and IoU for occupancy fields.
A.2.2 POSITIONAL. ENCODING
For the positional encoding, we leverage the method used in (Chen et al. 2023a). Here, we describe the implementation in detail.
Let W and H be the width and height of the image. The coordinate riy, embedded to:
i
((rs)) = 2H 1' 2W - sin(P: cos(p;; "in s
(3)
where Pij = iH + +j the lexicographically ordered location of coordinate riy: d et, where fx = 2(k-)))o 22 k = 1, I., I. is the dimension of embedding, / is the length of the sequence.
A.2.3 NUMERICAL IMPLEMENTATION OF PSWFS
Here, we explain the implementation details of the PSWF activation function. There many methods sove the integral operator problem:
sin a(x t) π - 1)
Un (c,
wn Ac))
The classical one would be based on the expansion of Legendre polynomial:
2 (c, = Σ (t),
(4)
k=
where 8 are coefficients of the corresponding Legendre polynomials, which can be obtained by recursions:
(A: + 2)(k 1)
2h:(h: + 1)
2 B
k(k+ +
62 -Xn
(2k 3) (2k + (2k 1)
(2k: + 32k: -1)
AE.
A
(5)
k(K
+
2 36 = 0,
(2k 1) (2k (2k + 1)
Autz,
recursions equation 5 can be written as:
(A Xa 1) (8x) O,
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detailed implementation can be found in Osipov et al. (2013). Here = 0, and 1 to construct our PSWF activation function template vb.
To implement the PSWF then shift and mirror the function according to the y axis to make it suitable for the activation function. The instantiated PSWF function shown in figure [ Once this discritized mirror reflection is obtained, the next step is to adapt it as a neural network activation function. During forward propagation, the transformed coordinates, computed through linear layers, must pass through the activation function. To obtain this, the activation function needs to be continuous, rather than a set of sampled values. When converting these discrete values into a continuous representation, we employed cubic spline approximation. This method provides third-order approximation between successive points while ensuring continuity and differentiability. Even though a simple regression could transform the discretized points into a continuous curve, it often fails to retain the exact data points and the differentiability properties between discrete points, both of which are essential for backpropagation. Therefore, cubic spline approximation, which is third-order and computationally efficient, serves as an optimal choice and fulfills the intended purpose.
activation function, use the prolate spheroidal radial function,
A.3 ABLATION STUDIES
A.3.1 ROBUSTNESS OF PIN'S S PARAMETERS
PIN has three trainable parameters: T. w, and b. and they are initialized as T 1, L, and b = 0. To demonstrate the robustness of parameters T, w. and b. we conducted an additional four experiments. In these experiments, the parameters were not initialized as T = w = and 6 but drawn from Normal distributions, denoted as N. Specifically, T and w are now sampled from (11, (2) and from (0, o²), with the first number in the parentheses representing the mean and the second the variance of the distribution. Here we varied o² and observed the final PNSR. The obtained results are shown in table [ and each result shown in the table represents the mean of four trials, and the number next to the + sign indicates the standard deviation of the obtained four trials. As can be seen from the results, even under a perturbation of variance 25, PIN is able to locate a local minimum in the loss landscape which provides good performance. However, when the variance is 100 for parameters drawn from Normal distribution, PIN only showed a convergence to 1 trial among the 4 trials conducted. Therefore, it can be stated that when PIN's parameters disturbed by a distribution with a variance of 25, PIN is capable of adjusting its parameters and locating local minimum. However, when the variance is 100 it is unlikely that PIN converges to a local minimum. To showcase how these parameters are robust compared to baselines, we also experimented with WIRE's s parameters under the same perturbations. For WIRE, the optimal values for w and are 20 and 10, respectively. Then, we drew from N(00, o²) and a from N(10, o²). The results of these experiments are shown in the table.
In general, PIN's parameters are trainable for all experiments. However, we further analyzed whether the parameters T. w, and can be used with fixed values. To check this, we trained two different configurations. Configl: T = 1. w 1, and b = 0 for all epochs. Config2: T,, and w are drawn from (1, 25). and is drawn from (0, 25), and the realized values are kept constant. follows. Configl: PSNR=35.70 76 dB, and Config2: PSNR=31.66 dB. The training
The results curves for these configurations are shown in figure 9 Therefore, can be concluded that even when have a highly perturbed T, w, and b PIN showcases good performance and gets very closer to WIRE. However, in this case due to its inability to move PIN's parameters in the loss landscape such that a way that it finds better local minimum, it ends up with giving 31,66 dB final PSNR.
Table 1: Robustness of PIN's parameters 0² o²=
INR
0==1
r² 25
0² = 100
PIN
35.8610.88
35.70=0.71
35.96=1.775
Diverges
WIRE
31.69-00.6
30.30=1.399
20.25
Diverges
A.3.2 LEARNING CURVES
Learning curves are a valuable tool that provide insights into how effectively and quickly an INR learns over time. For this task, we selected the 15th image from the Kodak dataset, as featured in the
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Variation of PSNR of PIN Parameters are Kept Consnat
35
PSNR(dB)
20
Disturbed From the Default Configuration Default Configuration
250
750
1000
1250
1500
1750
2000
Epochs
Figure 9: Variation of PSNR without the explicit control of parameters: As can be seen, without explicit control of parameters, PIN needs more epochs to obtain convergence. This demonstrates the importance of explicit parameter control for faster convergence.
main paper. The learning curves for various INRs, including PIN, are depicted in figure 10 which illustrates the variations in PSNR (dB) across the number of epochs. Notably, from around the 500th epoch, PIN consistently maintains a gap of nearly 3 dB compared to SIREN. This substantial difference indicates that PIN encodes explicit signals into its weights and biases much faster rate than any other existing INR.
Epochs
30
PSMMU66
10
SPEN
CAUSS
Epochs
Figure 10: PSNR variation with the number of epochs: PIN Learns Faster. The learning curves clearly show that PIN quickly reaches PSNR of nearly 30 dB within just 125 epochs, and ap- proaches 35 dB by the th0th epoch. Consequently, it can be concluded that PSWFs encode explicit representations into an INR both efficiently and rapidly.
A.3.3 WEIGHT INITIALIZATION
Previous studies (Saragadam et al. 2023 Ramasinghe & Lucey 2022). have shown that space- frequency compact activation functions like PSWFS generally insensitive weight initialization
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methods. Consistent with this, all experiments in paper obtained using Pytorch's default weight initialization mechanism. However, to experimentally validate this claim, we initialized PIN with different Normal, Uniform, and SIREN-like weight distributions. The resulting PSNRs are: SIREN-like at 35.67, Uniform at 35.39, Normal at 36.06, Pytorch default at 36.00; this shows that weight initialization does not significantly impact the performance.
A.4 ADDITIONAL RESULTS
1.4.1 IMAGE REPRESENTATION ON KODAK
As we have evaluated the capabilities of PIN using the Kodak dataset, some of the decoded repre- sentations from PIN, along with other models, are displayed in figure [T] The decoded images are organized into rows corresponding to their positions in the dataset: the first, second, and third rows represent the 1st 5th and 20th images of the Kodak dataset, respectively. In addition to evaluating PIN against WIRE, SIREN, GAUSS, and ReLU+PE, it is also compared against INCODE (Kaze- rouni et al. 2024) and FR-INR (Shi et al. 2024). The following table summarizes the obtained results across the Kodak dataset.
Table 2: Comparison of methods based on PSNR (dB) the Kodak dataset.
Method PIN
PSNR (dB)
40.17
INCODE
34.07
FR-INR
37.91
Ground Truth
PIN
WIRE
SIREN
GAUSS
Figure 11: Additional image representation results:The decoded representations clearly demon- strate that PIN is the only INR that consistently aligns its outputs with the ground truth. is evi- dent that some baseline models produce overly smoothed images, and others struggle with accurate color reproduction. Additionally, most baseline models introduce extraneous noise into regions that should be smooth, due to their limited ability to handle signals across broad frequency spectrum. However, PIN, which is derived from PSWFs, optimally processes each area, thereby producing the most visually coherent decoded images and effectively minimizing distortions.
A.4.2 IMAGE REPRESENTATION ON DIV2K
In addition to the evaluation on the entire Kodak dataset (Franzen. 1999), 30 randomly selected images from the DIV2K dataset (Agustsson Timofte (017)) were used to demonstrate the effec- tiveness of the proposed method. These images represented using different INRs. For this purpose, an MLP with 5 layers and 300 hidden neurons was used. The following table summarizes the obtained results.
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Table 3: PSNR Variation DIV2K Dataset
Method
PSNR (dB) on DIV2K
PIN
41.46
WIRE
30.12
SIREN
38.77
GAUSS
28.13
ReLU+PE
26.74
A.4.3 NERF
Additional views on the novel view synthesis task from various positions and viewing angles are depicted in figure [2 is evident that the novel views obtained demonstrate PIN's excellence the task, closely resembling the ground truth. Moreover, intricate details are well-preserved with PIN compared the other INRs.
Ground Truth
WIRE
GAUSS
Additional novel views from differrnt positions and angles: As can be seen, PIN clearly outperforms other INRs like WIRE, SIREN, and GAUSS, particularly in capturing intricate details of the drum set. The rods supporting the drums are more precisely reconstructed, showing sharper geometry and finer features that are blurred or lost in the other methods. PIN also excels in rendering lighting effects, capturing reflections on the cymbals and shadows across the red drums with greater accuracy. As the synthesized views are closer to the ground truth, PIN stands out as superior choice for novel view synthesis tasks.
Figure 12:
A.5 ADDITIONAL EXPERIMENTS
A.5.1 HIGH FREQUENCY ENCODING CAPABILITIES
The ability of INRs encode high frequencies can be effectively evaluated using images with sharp color transitions. By examining an image featuring abrupt changes in color color-cornne cor---poorrponnngggg to high-frequency elements in the frequency domain- it possible to gauge the proficiency of INRs in capturing high- high-frequency information. For this study, we opted to utilize the image showcased on the left side of the illustration provided in figure 13 Once each INR is trained on this image, the learned
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implicit representation is decoded, and the absolute enror between the decoded representation and ground truth is obtained. As shown in figure [3 the PIN model yields the lowest residual plot, demonstrating its remarkable signal encoding capabilities. In contrast, INRs such WIRE and SIREN exhibit inefficiencies not only in encoding sharp transitions but also in handling constant color areas. The exceptional performance of the PIN model may be attributed to the PSWF's optimal management of energy concentration in both spatial and frequency domains
Ground Truth
PIN
WIRE
SIREN
Figure 13: High frequency encoding capabilities of PIN: This figure presents the error plots ob- tained after training each INR on the image shown on the left. evident that the PIN produces the smallest residual plot, excelling in encoding high frequencies exceptionally well compared to other INRs, thanks to the optimal energy concentration balance provided by PSWFs. Conversely, WIRE and SIREN exhibit errors not only in encoding frggefreyeenyy content but also in constant color areas as well.
A.5.2 IMAGE DENOSING
In order to find out the possibility of image denoising, we have taken an image which has 512#512 of spatial resolution, and followed the procedure curated in Saragadam et al. (2023), Photon noise was simulated in our experiment by assigning a Poisson random variable, distributed independently each pixel, with a maximum mean photon count of 30 and readout count of which results noise image with PSNR and SSIM of 15,49, 0.173 respectively, and this can be considered as an extremely noisy situation with the size of the image. Then, INRs were tested on the ability image denoising as can be seen from figure 14 PIN is the only architecture which was able to recover the PSNR to 27.35 dB, which signifies an improvement of nearly dB. This substantial increase in PSNR reflects significant reduction in noise and restoration of image quality. Additionally, PIN achieved the highest SSIM value of 0.78, illustrating its effectiveness not only reducing noise also in maintaining the structural integrity and perceptual quality of the image. On the other hand, the ReLU+PE combination emerged as the second most effective activation for denoising extremely noisy images, demonstrating the second most robust performance.
A.5.3 PIN AS RELAIBLE EDGE DETECTOR
INRs represent signals implicitly, and a crucial aspect of good INR is its capability to perform tasks associated with explicit representations. In image processing, explicit representations allow us to extract edges using filters like Prewitt, Sobel, or Gaussian, vital for tasks such object boundary identification, text detection, or facial feature recognition, which heavily depend on accurate edge
Mrn 15.49 dB
Mils
26.54 dB
24.42 dB
26.87 dB
0.780
0.663
0.728
0.535
0.768
GT
Noisy Image
PIN
WIRE
SIREN
GAUSS
ReLU+PE
Figure 14: Image denoising capabilities of PIN: In this task, INRs were assigned to denoise images initially characterized by PSNR of 15. 49dB and SSIM of 0733. The recovered images demonstrate the denoising capabilities of INRs. PIN's superior metrics show its proficiency in restoring images with minimal distortion while preserving maximum structural information.
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Published as a conference paper at ICLR 2025
PIN
WIRE
SIREN
GAUSS
ReLU+PE
Figure 15: PIN edge detector: The results above showcase the edge detection capabilities of INRs. Among the various models evaluated, PIN emerges as a standout performer, demonstrating its ability produce the cleanest and most well-defined edge maps with minimal false edge detections.
detection. While theoretically, we could derive edge maps from an INR via its decoded representa- tions, INRs offer the advantage of differentiable learned representations. Therefore, a superior INR should not only represent the signal accurately but also provide access to its gradient information. edge detector, it should encode pixel-level correspondences in
For INR to effectively act its weights and biases without regularization. The potential of identifying edge maps through INRs becomes evident when presented with images containing visually identifiable edges, such as the butterfly image depicted on the top left of figure Initially, the image is represented using INRs, and thereafter once the representation is encoded into the INR, the gradient operator is applied be- tween the learned representation and the training coordinates. figure 15 depicts the butterfly's RGB image alongside its corresponding edge maps obtained through INRs. PIN stands out by providing the cleanest and most well-defined edge map, identifying necessary edges with minimal false edge identifications. Among other INRs, GAUSS, despite producing smooth edge maps, sacrifices intri- cate details, which can diminish the true edge signal and result in less reliable representation. Even though WIRE, and ReLU+PE showcase the boundaries, as an edge detector they fail significatly due to noise enhancement effects. Therefore, this demonstrate that even though these INRs capable of doing the representation, when it comes to down stream tasks like accessing gradients through the encoded representation, their weights and biases do not adequetly maintain the necessary pixel level correspondances. On the other hand, SIREN also functions well as an edge detector, benefiting from its specific weight initialization, specific frequency parameter, and inherent edge detection capabil- ities. Even without such inherent features, PIN accurately identifies edges, emphasizing its ability retain intricate lixel-Ievel correspondences without any regularization schemes while providing accurate representations. Thus, PIN stands out as a reliable edge detector compared to existing state-of-the-ard INRs.
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Reviews:
[{"review_id": "jVrE2odAXj", "reviewer": "Reviewer_8yGC", "summary": "The paper proposes Prolate Spheroidal Wave Function-based Implicit Neural Representations (PIN), an effective representation inspired by Prolate Spheroidal Wave Functions (PSWFs). The proposed PIN outperforms other INR baselines in various reconstruction tasks.", "questions": "- I am interested in the runtime of the proposed PIN. Will the new formulation hurt the speed of INRs? Is there a trade-off between quality and runtime?\n- How is the experiment conducted for the Image Inpainting task? Are the missing pixels marked as black and fed into the INR? Or are those coordinates masked out and not used during training? Are the pixel masks provided to the INR?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The proposed PIN outperforms other INR baselines in various reconstruction tasks, including image reconstruction, image inpainting, and occupancy field and NeRF.\n- Detailed ablation studies are conducted.", "weaknesses": "- The metrics provided in the paper are only evaluated on a few individual examples. I appreciate the evaluation on the Kodak Lossless\nTrue Color Image Dataset. However, it only contains 24 images. It is ideal to perform larger-scale evaluations, e.g. calculating the mean PSNR for a thousand images. For example, one may consider using the DIV2K dataset [1]. \n\n[1] Agustsson E, Timofte R. Ntire 2017 challenge on single image super-resolution: Dataset and study[C]//Proceedings of the IEEE conference on computer vision and pattern recognition workshops. 2017: 126-135.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "XuGJichbLI", "reviewer": "Reviewer_Yext", "summary": "The paper proposes a novel Implicit Neural Representation (INR) utilizing Prolate Spheroidal Wave Functions (PSWFs) to improve performance and generalization in computer vision tasks. By leveraging the optimal space-frequency domain concentration of PSWFs, the proposed method addresses the noise artifacts over smoother areas and poor generalization of existing INRs, demonstrating superior results in image inpainting, novel view synthesis, edge detection, and image denoising.", "questions": "See Weakness.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper clearly explains the limitations of current INRs and proposes a novel activation function (PSWFs) for INR to overcome these limitations.\n2. The localization and expressivity properties of PIN have been theortically proven.\n3. The results validate the effectiveness of the proposed INR across various tasks. The authors not only show the good representation ability of PIN for various signals, such as image, Occupancy Filed, and NeRF, but also show that PIN has a very good performance on image Image Inpainting.", "weaknesses": "1. The activation function seems to be rather computationally heavy, making it necessary to report its speed in applications. \n2. As I have reviewed the previous submission of this paper, I notice that the Fig.4 is replaced with a scene with better results. I wonder how these new results are obtained and why these results are not attached in previous submission? Are they obtained by tuning parameters for each scene specifically?\n3. Since the first NeurIPS submission of this paper, several new INRs have been proposed, including FINER (and its extension, FINER++) and H-SIREN. However, these new INRs are not cited or compared within the current manuscript.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "iqXrssMmA3", "reviewer": "Reviewer_1n58", "summary": "This paper introduces the use of prolate spheroidal wave functions (PSWF) for implicit neural representation (INR). By employing PSWF as the activation function in INR, the method excels not only in representing images and 3D shapes but also significantly outperforms existing approaches in various vision tasks that rely on INR generalization, including image inpainting, novel view synthesis, edge detection, and image denoising.", "questions": "I noticed that the authors use initial INR methods as their baselines. However, there are several approaches aimed at enhancing the expressivity and generalizability of INR, including improvements in training strategies [1] and input signals [2]. Could PSWF be applied to these methods to further enhance the representation performance of INR?\n\n\n[1] Improved Implicit Neural Representation with Fourier Reparameterized Training, CVPR 2024\n\n[2] Disorder-invariant implicit neural representation, CVPR 2023.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. Extensive experiments across various vision tasks demonstrate the effectiveness of PSWF.\n\n2. A comprehensive theoretical analysis highlights the advantages of using PSWF.", "weaknesses": "1. When comparing different INR methods, do you ensure that the same parameters are used? Could you provide the specific parameters for each INR method?\n\n2. I am curious about the decoding complexity of the PSWF-based INR. Could you provide the decoding speed or time for the different INR methods?\n\n3. Besides the vision tasks mentioned in the paper, can PSWF also improve performance in image super-resolution?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "PQKlRsjwnN", "reviewer": "Reviewer_BT7N", "summary": "This paper argues that the Gaussian and Gabor wavelet activation functions cannot achieve the optimal space-frequency trade-off, and thus may not effectively capture distant relevance. Motivated by this, the paper proposes using the Legendre polynomial numerical estimation of the Prolate Spheroidal Wave Function (PSWF) as the activation function, which has been proven in previous work to offer the optimal space-frequency trade-off. The authors demonstrate improvements through experiments on various tasks, including image regression, neural radiance fields, and so on.", "questions": "1. Could you clarify the approximation order used for the Prolate Spheroidal Wave Function (PSWF)? Additionally, does this approximation affect the theoretical properties of the PSWF? The paper mentions that PSWF possesses infinite support in space, but it remains unclear whether the finite-order approximation may potentially diminish this property.\n2. I observed that a bandwidth parameter \\(c\\) governs the frequency. Could you provide guidance on how to select this parameter effectively? Does it influence model performance, and if so, what strategies should users follow to determine an optimal value, especially for tasks in continuous domains like Neural Radiance Fields (NeRF), where there might be no frequency bounds, unlike the image regression scenario?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The motivation presented in the paper is compelling and provides a solid foundation for the proposed approach.  \n2. This paper offers a insightful critique of previous works, identifying limitations in existing activation functions and highlighting the need for improvement.  \n3. The experiments encompass a diverse range of tasks, allowing for a more comprehensive and thorough comparison. This diversity enhances the validity of the findings and demonstrates the effectiveness of the proposed method across different domains.", "weaknesses": "1. The exact implementation of the numerical estimation of the Prolate Spheroidal Wave Function lacks clarity in the main text. For instance, the approximation order is not explicitly stated, and the impact of the approximation order on the computational complexity (in terms of space/time consumption) and the model's performance is not clearly explained.\n2. The experiments in tasks, such as Occupancy Field Representation and Neural Radiance Field, appear to be conducted on a subset of the entire dataset, which may undermine the persuasiveness of the results.\n3. The presentation could be improved by refining certain details, such as employing the vector graphics and employing the \\autoref{} command for figure citations to ensure consistency and clarity.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)