Original: Signal Processing for Implicit Neural Representations
Revised: Signal Processing for Implicit Neural Representations

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddress email
Revised: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Implicit Neural Representations (INR) encoding continuous multi-media data viamulti-layer perceptrons has shown undebatable promise in various computer visiontasks.
Revised: Implicit Neural Representations (INR) encoding continuous multi-media data viamulti-layer perceptrons has shown undebatable promise in various computer visiontasks.

Original: Despite many successful applications, editing and processing an INR remainsintractable as signals are represented by unintuitive parameters of a neural network.
Revised: Despite many successful applications, editing and processing an INR remainsintractable as signals are represented by unintuitive parameters of a neural network.

Original: Existing works manipulate such continuous representations via processing on theirdiscretized instance, which breaks down the compactness and continuous natureof INR.
Revised: Existing works manipulate such continuous representations via processing on theirdiscretized instance, which breaks down the compactness and continuous natureof INR.

Original: In this work, we present a pilot study on the question: how to directlymodify an INR without explicit decoding?
Revised: In this work, we present a pilot study on the question: how to directlymodify an INR without explicit decoding?

Original: We answer this question by proposingan implicit neural signal processing network, dubbed INSP-Net , via differentialoperators on INR.
Revised: We answer this question by proposingan implicit neural signal processing network, dubbed INSP-Net , via differentialoperators on INR.

Original: Our key insight is that spatial gradients of neural networks canbe computed analytically and are invariant to translation, while mathematicallywe show that any continuous convolution filter can be uniformly approximated bya linear combination of high-order differential operators.
Revised: Our key insight is that spatial gradients of neural networks canbe computed analytically and are invariant to translation, while mathematicallywe show that any continuous convolution filter can be uniformly approximated bya linear combination of high-order differential operators.

Original: With these two knobs,we instantiate the INR signal operator as a composition of computational graphscorresponding to the high-order derivatives, where the weighting parameters canbe either handcrafted or data-driven learned.
Revised: With these two knobs,we instantiate the INR signal operator as a composition of computational graphscorresponding to the high-order derivatives, where the weighting parameters canbe either handcrafted or data-driven learned.

Original: Based on our proposed INSP-Net, wefurther build the first Convolutional Neural Network (CNN) that implicitly runson INRs, named INSP-ConvNet .
Revised: Based on our proposed INSP-Net, wefurther build the first Convolutional Neural Network (CNN) that implicitly runson INRs, named INSP-ConvNet .

Original: Our experiments validate the expressiveness of
Revised: Our experiments validate the expressiveness of

Original: INSP-Net and INSP-ConvNet in fitting low-level image processing kernels (e.g.edge detection, blurring, deblurring, denoising, inpainting) as well as for high-leveltasks on implicit fields such as image classification.
Revised: INSP-Net and INSP-ConvNet in fitting low-level image processing kernels (e.g.edge detection, blurring, deblurring, denoising, inpainting) as well as for high-leveltasks on implicit fields such as image classification.

Original: We will release all codes.
Revised: We will release all codes.

Original:  Introduction
Revised:  Introduction

Original: The idea that our visual world can be represented continuously has attracted increasing popularityin the field of implicit neural representations (INR).
Revised: The idea that our visual world can be represented continuously has attracted increasing popularityin the field of implicit neural representations (INR).

Original: Also known as coordinate-based neural representations, INRs learn to encode a coordinate-to-value mapping for continuous multi-media data.
Revised: Also known as coordinate-based neural representations, INRs learn to encode a coordinate-to-value mapping for continuous multi-media data.

Original: Instead of storing the discrete signal values in a grid of pixels or voxels, INRs represent discretedata as samples of a continuous manifold.
Revised: Instead of storing the discrete signal values in a grid of pixels or voxels, INRs represent discretedata as samples of a continuous manifold.

Original: Using multi-layer perceptrons, INRs bring practicalbenefits to various computer vision applications, such as image and video compression [1, 2, 3],3D shape representation [4, 5, 6, 7, 8, 9, 10, 11], inverse problems [12, 2, 13, 14], and generativemodels [15, 16, 17, 18, 19, 20, 21, 22].
Revised: Using multi-layer perceptrons, INRs bring practicalbenefits to various computer vision applications, such as image and video compression [1, 2, 3],3D shape representation [4, 5, 6, 7, 8, 9, 10, 11], inverse problems [12, 2, 13, 14], and generativemodels [15, 16, 17, 18, 19, 20, 21, 22].

Original: Despite their recent success, INRs are not yet amenable to flexible editing and processing as thestandard images could do.
Revised: Despite their recent success, INRs are not yet amenable to flexible editing and processing as thestandard images could do.

Original: The encoded coordinate-to-value mapping is too complex to comprehendand the parameters stored in multi-layer perceptrons (MLPs) remains less explored.
Revised: The encoded coordinate-to-value mapping is too complex to comprehendand the parameters stored in multi-layer perceptrons (MLPs) remains less explored.

Original: One direction ofexisting approaches enables editing on INRs by training them with conditional input.
Revised: One direction ofexisting approaches enables editing on INRs by training them with conditional input.

Original: For example,[23, 24, 25, 20, 21, 26] utilize conditional codes to indicate different characteristics of the sceneincluding shape and color.
Revised: For example,[23, 24, 25, 20, 21, 26] utilize conditional codes to indicate different characteristics of the sceneincluding shape and color.

Original: Another main direction benefits from existing image editing techniquesand operates on discretized instances of continuous INRs such as pixels[27, 28] or voxels [29].
Revised: Another main direction benefits from existing image editing techniquesand operates on discretized instances of continuous INRs such as pixels or voxels.

Original: However, such solutions break down the continuous characteristic of INR and are not due to theprerequisite of decoding and discretizing before editing and processing.
Revised: However, suchsolutions break down the continuous characteristic of INR and are not due to the prerequisite ofdecoding and discretizing before editing and processing.

Original: [Figure]
Revised: [Figure]

Original: In this paper, we conduct the first pilot study on the question: how to generally modify an INR withoutexplicit decoding?
Revised: In this paper, we conduct the first pilot study on the question: how to generally modify an INR withoutexplicit decoding?

Original: The major challenge is that one cannot directly interpret what the parameters in an
Revised: The major challenge is that one cannot directly interpret what the parameters in an

Original: INR stand for, not to mention editing them correctly.
Revised: INR stand for, not to mention editing them correctly.

Original: Our key motivation is that spatial gradients canbe served as a favorable tool to tackle this problem as they can be computed analytically, and possessdesirable invariant properties.
Revised: Our key motivation is that spatial gradients canbe served as a favorable tool to tackle this problem as they can be computed analytically, and possessdesirable invariant properties.

Original: Theoretically, we prove that any continuous convolution filter can beuniformly approximated by a linear combination of high-order differential operators.
Revised: Theoretically, we prove that any continuous convolution filter can beuniformly approximated by a linear combination of high-order differential operators.

Original: Based on theabove two rationales, we propose an Implicit Neural Signal Processing Network, dubbed INSP-Net ,which processes INR utilizing high-order differential operators.
Revised: Based on theabove two rationales, we propose an Implicit Neural Signal Processing Network, dubbed INSP-Net ,which processes INR utilizing high-order differential operators.

Original: The proposed INSP-Net is composedof an inception fusion block connecting computational graphs corresponding to derivatives of INRs.
Revised: The proposed INSP-Net is composedof an inception fusion block connecting computational graphs corresponding to derivatives of INRs.

Original: The weights in the branchy part are loaded from the INR being processing, while the weights in thefusion block are parameters of therepresented operator, which can be either hand-crafted or learnedby the data-driven algorithm.
Revised: The weights in the branchy part are loaded from the INR being processed, while the weights inthe fusion block are parameters of the operator, which can be either hand-crafted or learned by thedata-driven algorithm.

Original: Even though we are not able to  surgery on neural network parameters,we can implicitly process them by retrofitting their architecture and reorganizing the spatial gradients.
Revised: Even though we are not able to perform surgery on neural network parameters,we can implicitly process them by retrofitting their architecture and reorganizing the spatial gradients.

Original: We further extend our framework to build the first Convolutional Neural Network (CNN) operatingdirectly on INRs, dubbed INSP-ConvNet.
Revised: We further extend our framework to build the first Convolutional Neural Network (CNN) operatingdirectly on INRs, dubbed INSP-ConvNet.

Original: Each layer of INSP-ConvNet is constructed by linearlycombining the derivative computational graphs of the former layers.
Revised: Each layer of INSP-ConvNet is constructed by linearlycombining the derivative computational graphs of the former layers.

Original: Nonlinear activation andnormalization are naturally supported as they are element-wise functions.
Revised: Nonlinear activation andnormalization are naturally supported as they are element-wise functions.

Original: Data augmentation can bealso implemented by augmenting the input coordinates of INRs.
Revised: Data augmentation can bealso implemented by augmenting the input coordinates of INRs.

Original: Under this pipeline (shown in Fig.
Revised: Under this pipeline (shown in Fig.

Original: 1),we demonstrate the expressiveness of our INSP-Net framework in fitting low-level image processingkernels including edge detection, blurring, deblurring, denoising, and image inpainting.
Revised: 1),we demonstrate the expressiveness of our INSP-Net framework in fitting low-level image processingkernels including edge detection, blurring, deblurring, denoising, and image inpainting.

Original: We alsosuccessfully apply our INSP-ConvNet to high-level tasks on implicit fields such as classification.
Revised: We alsosuccessfully apply our INSP-ConvNet to high-level tasks on implicit fields such as classification.

Original: Our main contributions can be summarized as follows:
Revised: Our main contributions can be summarized as follows:

Original: • We propose a novel signal processing framework, dubbed INSP-Net , that operates on INRsanalytically and continuously by closed-form high-order differential operators 1 .
Revised: • We propose a novel signal processing framework, dubbed INSP-Net , that operates on INRsanalytically and continuously by closed-form high-order differential operators 1 .

Original: Repeatedlycascading the computational paradigm of INSP-Net, we also build a convolutional network,called INSP-ConvNet, which directly runs on implicit fields for high-level tasks.
Revised: Repeatedlycascading the computational paradigm of INSP-Net, we also build a convolutional network,called INSP-ConvNet, which directly runs on implicit fields for high-level tasks.

Original: • We illustrate the advantage of adopting differential operators by revealing their inherent
Revised: • We illustrate the advantage of adopting differential operators by revealing their inherent

Original: group invariance.
Revised: group invariance.

Original: Furthermore, we rigorously prove that the convolution operator in the
Revised: Furthermore, we rigorously prove that the convolution operator in the

Original: continuous regime can be uniformly approximated by a linear combination of the gradients.
Revised: continuous regime can be uniformly approximated by a linear combination of the gradients.

Original: • Extensive experiments demonstrate the effectiveness of our approach in both low-level
Revised: • Extensive experiments demonstrate the effectiveness of our approach in both low-level

Original: processing (e
Revised: processing (e

Original: g. edge detection, blurring, deblurring, denoising, image inpainting) andhigh-level processing such as image classification.
Revised: g. edge detection, blurring, deblurring, denoising, image inpainting) andhigh-level processing such as image classification.

Original:  Preliminaries: Implicit Neural Representation
Revised:  Preliminaries: Implicit Neural Representation

Original: Implicit Neural Representation (INR) parameterizes continuous multi-media signals or vector fieldswith neural networks.
Revised: Implicit Neural Representation (INR) parameterizes continuous multi-media signals or vector fieldswith neural networks.

Original: Formally, we consider an INR as a continuous function Φ : R m → R that mapslow-dimension spatial/temporal coordinates to the value space 2 .
Revised: Formally, we consider an INR as a continuous function Φ : R m → R that mapslow-dimension spatial/temporal coordinates to the value space 2 .

Original: For example, to represent 2D image
Revised: For example, to represent 2D image

Original: [Figure]
Revised: [Figure]

Original: Concat
Revised: Concat

Original: [Figure]
Revised: [Figure]

Original: Inception Fusion Layer
Revised: Inception Fusion Layer

Original: MLP
Revised: MLP

Original: NL x
Revised: NL x

Original: NL NL
Revised: NL NL

Original: NL x
Revised: NL x

Original: NL
Revised: NL

Original: NL x x NL x
Revised: NL x x NL x

Original: NL NL
Revised: NL NL

Original: Original INR Network
Revised: Original INR Network

Original: First-order Derivative Network signals, the domain of Φ is ( x, y ) spatial coordinates, and the range of Φ are the pixel intensities.
Revised: First-order Derivative Network signals, the domain of Φ is ( x, y ) spatial coordinates, and the range of Φ are the pixel intensities.

Original: Thetypical use of INR is to solve a feasibility problem where Φ is sought to satisfy a set of N constraints{C (Φ , a m | Ω m ) }
Revised: Thetypical use of INR is to solve a feasibility problem where Φ is sought to satisfy a set of N constraints{C (Φ , a m | Ω m ) }

Original: Nm =1 , where C is a functional that relates function Φ to some observable quantitiesa m evaluating over a measurable domain Ω m ⊆
Revised: Nm =1 , where C is a functional that relates function Φ to some observable quantitiesa m evaluating over a measurable domain Ω m ⊆

Original: R m .
Revised: R m .

Original: This problem can be cast into an optimizationproblem that minimizes deviations from each of the constraints:
Revised: This problem can be cast into an optimizationproblem that minimizes deviations from each of the constraints:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: For instance, we can let C =
Revised: For instance, we can let C =

Original: Φ( x m ) − a m with Ω m = { x m } , then our objective is reduced to apoint-to-point supervision which memorizes a signal into Φ [30].
Revised: Φ( x m ) − a m with Ω m = { x m } , then our objective is reduced to apoint-to-point supervision which memorizes a signal into Φ [27].

Original: When functional C is a combinationof differential operators taking values in a point set, i.e., C ( a ( x ) , Φ( x ) , ∇ Φ( x ) , · · · ) , ∀ x ∈ Ω m , Eq.1 is objective to solving a bunch of differential equations [27, 7, 31].
Revised: When functional C is a combinationof differential operators taking values in a point set, i.e., C ( a ( x ) , Φ( x ) , ∇ Φ( x ) , · · · ) , ∀ x ∈ Ω m , Eq.1 is objective to solving a bunch of differential equations [28, 7, 29].

Original: Note that in this paper, withoutparticular specification, the gradients are all computed with respect to the input coordinate x .
Revised: Note that in this paper, withoutparticular specification, the gradients are all computed with respect to the input coordinate x .

Original: C canalso form an integral equation system over some intervals Ω m [12].
Revised: C canalso form an integral equation system over some intervals Ω m [12].

Original: In practice of computer vision, wereconstruct a signal by capturing sparse observations D
Revised: In practice of computer vision, wereconstruct a signal by capturing sparse observations D

Original: = { (Ω m , a m ) }
Revised: = { (Ω m , a m ) }

Original: Nm =1 from unknown functionΦ , and dynamically sampling a mini-batch from D to minimize Eq.
Revised: Nm =1 from unknown functionΦ , and dynamically sampling a mini-batch from D to minimize Eq.

Original: 1 to obtain a feasible Φ .
Revised: 1 to obtain a feasible Φ .

Original: A handy parameterization of function Φ is a fully-connected neural network, which enables solving
Revised: A handy parameterization of function Φ is a fully-connected neural network, which enables solving

Original: Eq. 1 via gradient descent through a differentiable C .
Revised: Eq. 1 via gradient descent through a differentiable C .

Original: Common INR networks consist of pure
Revised: Common INR networks consist of pure

Original: Multi-Layer Perceptrons (MLP) with periodic activation functions.
Revised: Multi-Layer Perceptrons (MLP) with periodic activation functions.

Original: Fourier Feature Mapping (FFM)[30] places a sinusoidal transformation before the MLP, while Sinusoidal Representation Network(SIREN)
Revised: Fourier Feature Mapping (FFM)[27] places a sinusoidal transformation before the MLP, while Sinusoidal Representation Network(SIREN)

Original: [27] replaces every piece-wise linear activation with a sinusoidal function.
Revised: [28] replaces every piece-wise linear activation with a sinusoidal function.

Original: Below we give aunified formulation of INR networks:
Revised: Below we give aunified formulation of INR networks:

Original: [Equation]
Revised: [Equation]

Original: where W i ∈ R
Revised: where W i ∈ R

Original: d i − 1 ×
Revised: d i − 1 ×

Original: d
Revised: d

Original: i , b i ∈ R
Revised: i , b i ∈ R

Original: d
Revised: d

Original: i are the weight matrix and bias of the i -th layer, respectively, n is thenumber of layers, and σ i ( · ) is an element-wise nonlinear activation function.
Revised: i are the weight matrix and bias of the i -th layer, respectively, n is thenumber of layers, and σ i ( · ) is an element-wise nonlinear activation function.

Original: For FFM architecture,σ i = sin( · ) when i = 1 denotes the positional encoding layer [12, 32] and otherwise σ i = ReLU( · ) .
Revised: For FFM architecture,σ i = sin( · ) when i = 1 denotes the positional encoding layer [12, 30] and otherwise σ i = ReLU( · ) .

Original: For SIREN, σ i = sin( · ) for every layer i = 1 , · · · , n − 1 .
Revised: For SIREN, σ i = sin( · ) for every layer i = 1 , · · · , n − 1 .

Original:  Implicit Representation Processing via Differential Operators
Revised:  Implicit Representation Processing via Differential Operators

Original: Digital Signal Processing (DSP) techniques have been widely applied in computer vision tasks, suchas image restoration [33], signal enhancement [34] and geometric processing [35].
Revised: Digital Signal Processing (DSP) techniques have been widely applied in computer vision tasks, suchas image restoration [31], signal enhancement [32] and geometric processing [33].

Original: Even modern deeplearning models are consisting of the most basic signal processing operators.
Revised: Even modern deeplearning models are consisting of the most basic signal processing operators.

Original: Suppose we already acquire an Implicit Neural Representation (INR)
Revised: Suppose we already acquire an Implicit Neural Representation (INR)

Original: Φ : R m → R , now we are interested in whether we canrun a signal processing program on the implicitly represented signals.
Revised: Φ : R m → R , now we are interested in whether we canrun a signal processing program on the implicitly represented signals.

Original: One straightforward solutionis to rasterize the implicit field with a 2D/3D lattice and run a typical kernel on the pixel/voxel grids.
Revised: One straightforward solutionis to rasterize the implicit field with a 2D/3D lattice and run a typical kernel on the pixel/voxel grids.

Original: However, this decoding strategy produces a finite resolution and discretizes signals, which is memory inefficient and unfriendly to modeling fine details.
Revised: However, this decoding strategy produces a finite resolution and discretizes signals, which is memory inefficient and unfriendly to modeling fine details.

Original: In this section, we introduce a computationparadigm that can process an INR analytically with spatial/temporal derivatives.
Revised: In this section, we introduce a computationparadigm that can process an INR analytically with spatial/temporal derivatives.

Original: We show that our pro- posed method serves as a universal operator that can represent any continuous convolutional kernels.
Revised: We show that our pro- posed method serves as a universal operator that can represent any continuous convolutional kernels.

Original: 3.1 Computational Paradigm
Revised: 3.1 Computational Paradigm

Original: It has not escaped our notice that spatial/temporal gradients on INRs ∇ k Φ can be computed analytically due to the differentiable characteristics of neural networks.
Revised: It has not escaped our notice that spatial/temporal gradients on INRs ∇ k Φ can be computed analytically due to the differentiable characteristics of neural networks.

Original: Inspired by this, we propose an
Revised: Inspired by this, we propose an

Original: Implicit Neural Signal Processing (INSP) framework that composes a class of closed-form operatorsfor INRs using functional combinations of high-order derivatives.
Revised: Implicit Neural Signal Processing (INSP) framework that composes a class of closed-form operatorsfor INRs using functional combinations of high-order derivatives.

Original: We denote our proposed signal processing operator by A built upon high-order derivatives.
Revised: We denote our proposed signal processing operator by A built upon high-order derivatives.

Original: Given anacquired INR Φ , we denote the resultant INR processed by operator A as Ψ =
Revised: Given anacquired INR Φ , we denote the resultant INR processed by operator A as Ψ =

Original: A Φ : R m → R .
Revised: A Φ : R m → R .

Original: Toevaluate point x ∈ R m of processed INR, we propose the following computational paradigm:
Revised: Toevaluate point x ∈ R m of processed INR, we propose the following computational paradigm:

Original: [Equation]
Revised: [Equation]

Original: where Π : R M → R can be arbitrary continuous functions, which can be either handcrafted or learnedfrom data.
Revised: where Π : R M → R can be arbitrary continuous functions, which can be either handcrafted or learnedfrom data.

Original: To learn an operator A from data, we represent Π by Multi-Layer Perceptrons (MLP) withparameters θ .
Revised: To learn an operator A from data, we represent Π by Multi-Layer Perceptrons (MLP) withparameters θ .

Original: Here we can slightly abuse the notation of ∇ k to be a flattened vector of high-orderderivatives without multiplicity since differential operators defined over continuous functions forma commutative ring.
Revised: Here we can slightly abuse the notation of ∇ k to be a flattened vector of high-orderderivatives without multiplicity since differential operators defined over continuous functions forma commutative ring.

Original: The input dimension of Π depends on the highest order of used derivatives.
Revised: The input dimension of Π depends on the highest order of used derivatives.

Original: Suppose we compute derivatives up to K -th order, then M = (cid:80) Kk =0 (cid:0) k
Revised: Suppose we compute derivatives up to K -th order, then M = (cid:80) Kk =0 (cid:0) k

Original: + m − 1 k (cid:1) =
Revised: + m − 1 k (cid:1) =

Original: ( K +1) (cid:0) K
Revised: ( K +1) (cid:0) K

Original: + m K +

Revised: + m K +

Original: (cid:1)
Revised: (cid:1)

Original: /m ,where (cid:0)
Revised: /m ,where (cid:0)

Original: k
Revised: k

Original: + m − 1 k (cid:1) is the number of distinctive k -th order differential operators 3 .
Revised: + m − 1 k (cid:1) is the number of distinctive k -th order differential operators 3 .

Original: As we will showin Sec. 3.2, Π can construct both shift-invariant and rotation-invariant operators, which introducesfavorable inductive bias to images and 3D geometry processing.
Revised: As we will showin Sec. 3.2, Π can construct both shift-invariant and rotation-invariant operators, which introducesfavorable inductive bias to images and 3D geometry processing.

Original: More importantly, we prove that Eq.3 is also a universal approximator of arbitrary convolutional operators.
Revised: More importantly, we prove that Eq.3 is also a universal approximator of arbitrary convolutional operators.

Original: We note that Ψ( x ) as a whole can also be regarded as a neural network.
Revised: We note that Ψ( x ) as a whole can also be regarded as a neural network.

Original: Recall the architecture ofΦ( x ) in Eq.
Revised: Recall the architecture ofΦ( x ) in Eq.

Original: 2, its k -th order derivative is another computational graph parameterized by W i andb i that maps x to ∇ k Φ( x ) .
Revised: 2, its k -th order derivative is another computational graph parameterized by W i andb i that maps x to ∇ k Φ( x ) .

Original: For example, the first-order gradient will have the following form:
Revised: For example, the first-order gradient will have the following form:

Original: [Equation]
Revised: [Equation]

Original: where ˆ ϕ i ( y ) =
Revised: where ˆ ϕ i ( y ) =

Original: W ⊤ i σ ′ i − 1 ( W i − 1 y + b k − 1 ) , and σ ′ i ( · ) is the first-order derivative of σ i ( · ) .
Revised: W ⊤ i σ ′ i − 1 ( W i − 1 y + b k − 1 ) , and σ ′ i ( · ) is the first-order derivative of σ i ( · ) .

Original: Since ˆ ϕ ishares the weights with ϕ i , ∇ Φ is represented by a closed-form computational network re-using theweights from Φ , which we refer to as the first-order derivative network .
Revised: Since ˆ ϕ ishares the weights with ϕ i , ∇ Φ is represented by a closed-form computational network re-using theweights from Φ , which we refer to as the first-order derivative network .

Original: The higher-order derivativesshould induce the derivative network of similar forms.
Revised: The higher-order derivativesshould induce the derivative network of similar forms.

Original: Therefore, the processed INR Ψ will have an
Revised: Therefore, the processed INR Ψ will have an

Original: Inception-like architecture, namely, a multi-branch structure connecting the original INR network andweight-sharing derivative subnetworks followed by a fusion layer Π .
Revised: Inception-like architecture, namely, a multi-branch structure connecting the original INR network andweight-sharing derivative subnetworks followed by a fusion layer Π .

Original: We call the entire model ( Ψ =
Revised: We call the entire model ( Ψ =

Original: A Φ or Eq. 3) an Implicit Neural Signal Processing Network or an INSP-Net .
Revised: A Φ or Eq. 3) an Implicit Neural Signal Processing Network or an INSP-Net .

Original: Note that the only parameters of INSP-Net θ are located at the last fusion layer, and can be trained in an end-to-end manner.
Revised: Note that the only parameters of INSP-Net θ are located at the last fusion layer, and can be trained in an end-to-end manner.

Original: We illustrate an INSP-Net in Fig.
Revised: We illustrate an INSP-Net in Fig.

Original: 2 where the color indicates the weight-sharing scheme.
Revised: 2 where the color indicates the weight-sharing scheme.

Original: A similarweight-sharing scheme is also adopted in AutoInt [36].
Revised: A similarweight-sharing scheme is also adopted in AutoInt [34].

Original: In practice, we employ auto-differentiationin PyTorch [37] to automatically create such derivatives networks and reassemble them parallellyto constitute the architecture of an INSP-Net.
Revised: In practice, we employ auto-differentiationin PyTorch [35] to automatically create such derivatives networks and reassemble them parallellyto constitute the architecture of an INSP-Net.

Original: When inputting an INR, we load the weights of the
Revised: When inputting an INR, we load the weights of the

Original: INR to our model following the weight-sharing scheme, and then we obtain an INSP-Net, whichimplicitly and continuously represents the processed INR Ψ( x ) .
Revised: INR to our model following the weight-sharing scheme, and then we obtain an INSP-Net, whichimplicitly and continuously represents the processed INR Ψ( x ) .

Original: To effectively express high-orderderivatives, we choose SIREN as the base model [27].
Revised: To effectively express high-orderderivatives, we choose SIREN as the base model [28].

Original: 3.2 Theoretical Analysis
Revised: 3.2 Theoretical Analysis

Original: In this section, we provide a theoretical justification for the design of our INSP-Net.
Revised: In this section, we provide a theoretical justification for the design of our INSP-Net.

Original: We will focuson discussing the latent invariance property and the expressive power of INSP-Net.
Revised: We will focuson discussing the latent invariance property and the expressive power of INSP-Net.

Original: Translation and Rotation Invariance.
Revised: Translation and Rotation Invariance.

Original: Group invariance has testified as a favorable inductivebias for image [38], video [39], and geometry processing [40].
Revised: Group invariance has testified as a favorable inductivebias for image [36], video [37], and geometry processing [38].

Original: It has also been well-known thatgroup invariance is an intrinsic property of Partial Differential Equations (PDEs) [41, 42].
Revised: It has also been well-known thatgroup invariance is an intrinsic property of Partial Differential Equations (PDEs) [39, 40].

Original: Sinceour INSP-Net is built using differential operators, we are motivated to reveal its hidden invarianceproperty to demonstrate its advantage in processing visual signals.
Revised: Sinceour INSP-Net is built using differential operators, we are motivated to reveal its hidden invarianceproperty to demonstrate its advantage in processing visual signals.

Original: In this section, we only consider two transformation groups: translation group T ( m ) and the specialorthogonal group SO ( m ) (a.k.a.
Revised: In this section, we only consider two transformation groups: translation group T ( m ) and the specialorthogonal group SO ( m ) (a.k.a.

Original: rotation group).
Revised: rotation group).

Original: Elements T v ∈ T ( m ) in translation group shifts the function Φ by some offset v ∈ R m .
Revised: Elements T v ∈ T ( m ) in translation group shifts the function Φ by some offset v ∈ R m .

Original: The shifted function can be denoted as Φ ◦ T v ( x ) =
Revised: The shifted function can be denoted as Φ ◦ T v ( x ) =

Original: Φ( x + v ) .
Revised: Φ( x + v ) .

Original: Similarly, elements in rotation group perform a coordinate transformation on function Φ by a rotationmatrix R ∈ SO ( m ) .
Revised: Similarly, elements in rotation group perform a coordinate transformation on function Φ by a rotationmatrix R ∈ SO ( m ) .

Original: The transformed function can be written as Φ ◦ R ( x ) =
Revised: The transformed function can be written as Φ ◦ R ( x ) =

Original: Φ( Rx ) .
Revised: Φ( Rx ) .

Original: Groupinvariance means deforming the input space of a function first and then processing it via an operatoris equivalent to directly applying the transformation to the processed function.
Revised: Groupinvariance means deforming the input space of a function first and then processing it via an operatoris equivalent to directly applying the transformation to the processed function.

Original: For a more rigorousargument, A is said to be translation-invariant if ∀ T v ∈ T ( m ) , Ψ( x + v ) =
Revised: For a more rigorousargument, A is said to be translation-invariant if ∀ T v ∈ T ( m ) , Ψ( x + v ) =

Original: A [Φ ◦ T v ]( x ) .
Revised: A [Φ ◦ T v ]( x ) .

Original: Likewise,
Revised: Likewise,

Original: A is rotation-invariant if ∀ R ∈
Revised: A is rotation-invariant if ∀ R ∈

Original: SO
Revised: SO

Original: ( m ) we have Ψ( Rx ) =
Revised: ( m ) we have Ψ( Rx ) =

Original: A [Φ ◦ R ]( x ) .
Revised: A [Φ ◦ R ]( x ) .

Original: Below we provide Theorem1 to characterize the invariance property for our model.
Revised: Below we provide Theorem1 to characterize the invariance property for our model.

Original: Theorem 1.
Revised: Theorem 1.

Original: Given function Π : R M → R , the composed operator A (Eq. 3) can satisfy:
Revised: Given function Π : R M → R , the composed operator A (Eq. 3) can satisfy:

Original: shift invariance for every Π .2.
Revised: shift invariance for every Π .2.

Original: rotation invariance if Π has the form: Π( y ) =
Revised: rotation invariance if Π has the form: Π( y ) =

Original: f ( ∥ y ∥ 2 ) for some f : R → R .
Revised: f ( ∥ y ∥ 2 ) for some f : R → R .

Original: We prove Theorem 1 in Appendix A. Our Theorem 1 implies that operator
Revised: We prove Theorem 1 in Appendix A. Our Theorem 1 implies that operator

Original: A is inherently shiftinvariant.
Revised: A is inherently shiftinvariant.

Original: This is due to the shift-invariant intrinsics of differential operators as we show in the proof.
Revised: This is due to the shift-invariant intrinsics of differential operators as we show in the proof.

Original: Rotation invariance is not guaranteed in general.
Revised: Rotation invariance is not guaranteed in general.

Original: However, if one carefully designs Π , it can alsobe achieved via our framework.
Revised: However, if one carefully designs Π , it can alsobe achieved via our framework.

Original: Moreover, we also suggest a feasible solution to constructing arotation-invariant operator A in Theorem 1.
Revised: Moreover, we also suggest a feasible solution to constructing arotation-invariant operator A in Theorem 1.

Original: In our construction, Π first isotropically pools over thesquares of all directional derivatives, and then maps the summarized information through anotherfunction f , which can be either handcrafted or a trainable neural network.
Revised: In our construction, Π first isotropically pools over thesquares of all directional derivatives, and then maps the summarized information through anotherfunction f , which can be either handcrafted or a trainable neural network.

Original: We refer interested readers[41] for more group invariance in differential forms.
Revised: We refer interested readers[39] for more group invariance in differential forms.

Original: Universal Approximation.
Revised: Universal Approximation.

Original: Convolution, formally known as the linear shift-invariant operator,has served as one of the most prevalent signal processing tools in the vision domain.
Revised: Convolution, formally known as the linear shift-invariant operator,has served as one of the most prevalent signal processing tools in the vision domain.

Original: Given two(real-valued) signals f and g , we denote their convolution as g ⋆ f = f
Revised: Given two(real-valued) signals f and g , we denote their convolution as g ⋆ f = f

Original: ⋆ g .
Revised: ⋆ g .

Original: In this section, weexamine the expressiveness of our INSP-Net (Eq. 3) by showing it can represent any convolutionalfilter.
Revised: In this section, weexamine the expressiveness of our INSP-Net (Eq. 3) by showing it can represent any convolutionalfilter.

Original: To draw this conclusion, we first present an informal version of our main results as follows:
Revised: To draw this conclusion, we first present an informal version of our main results as follows:

Original: Theorem 2.
Revised: Theorem 2.

Original: (Informal statement)
Revised: (Informal statement)

Original: For every real-valued function g : R → R , there exists a polynomialp ( x )
Revised: For every real-valued function g : R → R , there exists a polynomialp ( x )

Original: = a 0 + a 1 x + · · · where a 0 , a 1 , · · · ∈ R , such that p ( ∇ ) f can uniformly approximate g ⋆ f byarbitrary precision for all real-valued signals f .
Revised: = a 0 + a 1 x + · · · where a 0 , a 1 , · · · ∈ R , such that p ( ∇ ) f can uniformly approximate g ⋆ f byarbitrary precision for all real-valued signals f .

Original: The formal statement and proof can be found in Appendix B. Theorem 2 involves the notion ofpolynomials in differential operators.
Revised: The formal statement and proof can be found in Appendix B. Theorem 2 involves the notion ofpolynomials in differential operators.

Original: We define p ( ∇ ) =
Revised: We define p ( ∇ ) =

Original: a 0 + a 1 ∇ + a 2 ∇ 2 + · · · , where the mapthat takes polynomial p ( x ) to p ( ∇ ) is a ring homomorphism from polynomial ring R [ x ] to thering of endomorphism of a function space.
Revised: a 0 + a 1 ∇ + a 2 ∇ 2 + · · · , where the mapthat takes polynomial p ( x ) to p ( ∇ ) is a ring homomorphism from polynomial ring R [ x ] to thering of endomorphism of a function space.

Original: Theorem 2 can be regarded as an application of the
Revised: Theorem 2 can be regarded as an application of the

Original: Stone-Weierstrass approximation theorem on the Fourier domain.
Revised: Stone-Weierstrass approximation theorem on the Fourier domain.

Original: However, we note that functionsobtained by the Fourier transform are generally complex functions.
Revised: However, we note that functionsobtained by the Fourier transform are generally complex functions.

Original: The prominence of our results isthat we can constrain the range of the polynomial coefficients into the real domain, which makes itimplementable via a common deep learning infrastructure.
Revised: The prominence of our results isthat we can constrain the range of the polynomial coefficients into the real domain, which makes itimplementable via a common deep learning infrastructure.

Original: The implication of Theorem 2 is that themapping between convolution and derivative is as simple as a linear transformation.
Revised: The implication of Theorem 2 is that themapping between convolution and derivative is as simple as a linear transformation.

Original: Recent works[43, 44, 45] show similar results that derivatives can be computed via a linear combination of discreteconvolution.
Revised: Recent works[41, 42, 43] show similar results that derivatives can be computed via a linear combination of discreteconvolution.

Original: Theorem 2 can be served as their converse argument but in the continuous regime.
Revised: Theorem 2 can be served as their converse argument but in the continuous regime.

Original: Inour proof, k -th order derivatives correspond to k -th order monomial function in spectral space.
Revised: Inour proof, k -th order derivatives correspond to k -th order monomial function in spectral space.

Original: Thisindicates that differential operators form a basis of the linear shift-invariant operator space.
Revised: Thisindicates that differential operators form a basis of the linear shift-invariant operator space.

Original: In otherwords, any convolution can be written as a linear combination of differential operators, i.e., p ( ∇ ) .
Revised: In otherwords, any convolution can be written as a linear combination of differential operators, i.e., p ( ∇ ) .

Original: Since a linear function is not difficult to be approximated by a neural network Π θ , we can derive thenext straightforward result Corollary 3.
Revised: Since a linear function is not difficult to be approximated by a neural network Π θ , we can derive thenext straightforward result Corollary 3.

Original: Corollary 3.
Revised: Corollary 3.

Original: For every real-valued function g , there exists a neural network Π θ such that Ψ
Revised: For every real-valued function g , there exists a neural network Π θ such that Ψ

Original: =
Revised: =

Original: A Φ(Eq.
Revised: A Φ(Eq.

Original: 3) can uniformly approximate g ⋆ Φ by arbitrary precision for every real-valued signals Φ .
Revised: 3) can uniformly approximate g ⋆ Φ by arbitrary precision for every real-valued signals Φ .

Original: As we discussed in Theorem 1, A are constantly shift-invariant.
Revised: As we discussed in Theorem 1, A are constantly shift-invariant.

Original: This means when approximating aconvolutional kernel, the trajectory of A is restricted into the shift-invariant space.
Revised: This means when approximating aconvolutional kernel, the trajectory of A is restricted into the shift-invariant space.

Original: Moreover, weemphasize that INSP-Net is far more expressive than convolutional kernels since Π θ can also fit anynonlinear continuous functions due to the universal approximation theorem [46, 47].
Revised: Moreover, weemphasize that INSP-Net is far more expressive than convolutional kernels since Π θ can also fit anynonlinear continuous functions due to the universal approximation theorem [44, 45].

Original: 3.3 Building CNNs for Implicit Neural Representations
Revised: 3.3 Building CNNs for Implicit Neural Representations

Original: Convolutional Neural Networks (CNN) are capable of extracting informative semantics by only pilingup basic signal processing operators.
Revised: Convolutional Neural Networks (CNN) are capable of extracting informative semantics by only pilingup basic signal processing operators.

Original: This motivates us to build CNNs based on INSP-Net that can directly run on INRs for high-level downstream tasks.
Revised: This motivates us to build CNNs based on INSP-Net that can directly run on INRs for high-level downstream tasks.

Original: In fact, to simulate exact convolution,
Revised: In fact, to simulate exact convolution,

Original: our
Revised: our

Original: Theorem 2 suggests simplify Π θ to a linear mapping.
Revised: Theorem 2 suggests simplify Π θ to a linear mapping.

Original: Then our former computational paradigm Eq.3 is changed to:
Revised: Then our former computational paradigm Eq.3 is changed to:

Original: [Equation]
Revised: [Equation]

Original: where θ k ∈ R ( k + m − 1 k ) are parameters of the operator p ( ∇ ) .
Revised: where θ k ∈ R ( k + m − 1 k ) are parameters of the operator p ( ∇ ) .

Original: We name this special case of Eq. 3 as
Revised: We name this special case of Eq. 3 as

Original: INSP-Conv.
Revised: INSP-Conv.

Original: One plausible implementation of INSP-Conv is to employ a one-layer MLP to representΠ θ .
Revised: One plausible implementation of INSP-Conv is to employ a one-layer MLP to representΠ θ .

Original: When A = p ( ∇ ) , INSP-Conv preserves both linearity and shift-invariance when evolving duringthe training.
Revised: When A = p ( ∇ ) , INSP-Conv preserves both linearity and shift-invariance when evolving duringthe training.

Original: We propose to repeatedly apply INSP-Conv with non-linearity to INRs that mimicsa CNN-like architecture.
Revised: We propose to repeatedly apply INSP-Conv with non-linearity to INRs that mimicsa CNN-like architecture.

Original: We name this class of CNNs composed by multi-layer INSP-Conv (Eq.5) as INSP-ConvNet.
Revised: We name this class of CNNs composed by multi-layer INSP-Conv (Eq.5) as INSP-ConvNet.

Original: Previous works [48, 49] extracting semantic features from INR either lacklocal information by point-wisely mapping INR’s intermediate representation to a semantic space orexplicitly rasterize INR into regular grids.
Revised: Previous works [46, 47] extracting semantic features from INR either lacklocal information by point-wisely mapping INR’s intermediate representation to a semantic space orexplicitly rasterize INR into regular grids.

Original: To the best of our knowledge, it is the first time that onecan run a CNN directly on an implicit representation thanks to closed-formness of INSP-Net.
Revised: To the best of our knowledge, it is the first time that onecan run a CNN directly on an implicit representation thanks to closed-formness of INSP-Net.

Original: Theoverall architecture of INSP-ConvNet can be formulated as:
Revised: Theoverall architecture of INSP-ConvNet can be formulated as:

Original: [Equation]
Revised: [Equation]

Original: 6) where σ is an element-wise non-linear activation, L is the number of INSP-Net layers, and Φ is aninput INR.
Revised: 6) where σ is an element-wise non-linear activation, L is the number of INSP-Net layers, and Φ is aninput INR.

Original: We use the symbol · to denote operator functioning, and ◦ to denote function composition.
Revised: We use the symbol · to denote operator functioning, and ◦ to denote function composition.

Original: Due to page limit, we defer detailed introduction to INSP-ConvNet to Appendix C.
Revised: Due to page limit, we defer detailed introduction to INSP-ConvNet to Appendix C.

Original:  Related Works
Revised:  Related Works

Original: 4.1 Implicit Neural Representation
Revised: 4.1 Implicit Neural Representation

Original: Implicit Neural Representation (INR) represents signals by continuous functions parameterized bymulti-layer perceptrons (MLPs) [27, 30], which is different from traditional discrete representations(e.g., pixel, mesh).
Revised: Implicit Neural Representation (INR) represents signals by continuous functions parameterized bymulti-layer perceptrons (MLPs) [28, 27], which is different from traditional discrete representations(e.g., pixel, mesh).

Original: Compared with other representations, the continuous implicit representations arecapable of representing signals theoretically at infinite resolution and have become prevailing to beapplied upon image fitting [27], image compression [1, 50] and video compressing [3].
Revised: Compared with other representations, the continuous implicit representations arecapable of representing signals theoretically at infinite resolution and have become prevailing to beapplied upon image fitting [28], image compression [1, 48] and video compressing [3].

Original: In addition,
Revised: In addition,

Original: INR has been applied to more efficient and effective shape representation [4, 5, 6, 7, 8, 9, 10, 11],texture mapping [51, 52], inverse problems [12, 2, 13, 14] and generative models [15, 16, 17, 18,19, 20, 21, 22].
Revised: INR has been applied to more efficient and effective shape representation [4, 5, 6, 7, 8, 9, 10, 11],texture mapping [49, 50], inverse problems [12, 2, 13, 14] and generative models [15, 16, 17, 18,19, 20, 21, 22].

Original: There are also efforts speeding up the fitting of INRs [53] and improving therepresentation efficiency [54].
Revised: There are also efforts speeding up the fitting of INRs [51] and improving therepresentation efficiency [52].

Original: Nowadays, editing and manipulating multi-media objects gainsincreasing interest and demand [55].
Revised: Nowadays, editing and manipulating multi-media objects gainsincreasing interest and demand [53].

Original: Thus, signal processing on implicit neural representation isessentially an important task worth investigating.
Revised: Thus, signal processing on implicit neural representation isessentially an important task worth investigating.

Original: 4.2 Editable Implicit Fields
Revised: 4.2 Editable Implicit Fields

Original: Editing implicit fields has recently attracted much research interest.
Revised: Editing implicit fields has recently attracted much research interest.

Original: Several methods have beenproposed to allow editing the reconstructed 3D scenes by rearranging the objects or manipulatingthe shape and appearance.
Revised: Several methods have beenproposed to allow editing the reconstructed 3D scenes by rearranging the objects or manipulating theshape and appearance.

Original: One line of work alters the structure and color of objects by conditioninglatent codes for different characteristics of the scene [25, 20, 21, 26].
Revised: One line of work alters the structure and color of objects by conditioning latentcodes for different characteristics of the scene [25, 20, 21, 26].

Original: Another direction involvesdiscretizing the continuous fields.
Revised: Another direction involves discretizingthe continuous fields.

Original: By converting the implicit fields into pixels[27, 28] or voxels [29],traditional image and voxel editing techniques  can be applied effortlessly.
Revised: By converting the implicit fields into pixels or voxels, traditional image andvoxel editing techniques [54, 55] can be applied effortlessly.

Original: Recently, NFGP [56]proposes to use neural fields for geometry processing.
Revised: Recently, NFGP [56] proposes touse neural fields for geometry processing.

Original: These approaches, however, are not capableof directly performing signal processing on continuous INRs.
Revised: These approaches, however, are not capable of directlyperforming signal processing on continuous INRs.

Original: Our INSP-Net makes smart use ofclosed-form differential operators and thus does not require explicit decoding.
Revised: Our INSP-Net makes smart use of closed-formdifferential operators and thus does not require explicit decoding.

Original: 4.3 PDE based Image Processing
Revised: 4.3 PDE based Image Processing

Original: Partial differential equations (PDEs) have been successfully applied to many tasks in image processing and computer vision, such as image enhancement [57, 58, 59], segmentation [60, 42], imageregistration [61], saliency detection [62] and optical flow computation [63].
Revised: Partial differential equations (PDEs) have been successfully applied to many tasks in image processing and computer vision, such as image enhancement [57, 58, 59], segmentation [60, 40], imageregistration [61], saliency detection [62] and optical flow computation [63].

Original: Early traditional PDEsare written directly based on mathematical and physical understanding of the PDEs (e.g., anisotropicdiffusion [57], shock filter [58] and curve evolution based equations [64, 65, 66]).
Revised: Early traditional PDEsare written directly based on mathematical and physical understanding of the PDEs (e.g., anisotropicdiffusion [57], shock filter [58] and curve evolution based equations [64, 65, 66]).

Original: Variational designmethods [59, 67, 66] start from an energy function describing the desired properties of output images and compute the Euler-Lagrange equation to derive the evolution equations.
Revised: Variational designmethods [59, 67, 66] start from an energy function describing the desired properties of output images and compute the Euler-Lagrange equation to derive the evolution equations.

Original: Learning-basedattempts [42, 62] build PDEs from image pairs based on the assumption (without proof) that PDEs could be written as linear combinations of fundamental differential invariants.
Revised: Learning-basedattempts [40, 62] build PDEs from image pairs based on the assumption (without proof) that PDEs could be written as linear combinations of fundamental differential invariants.

Original: Although it might befeasible to let INRs solve this bunch of signal processing PDEs, one needs to per-case re-fit an INRwith an additional temporal axis, which is severely sampling inefficient.
Revised: Although it might befeasible to let INRs solve this bunch of signal processing PDEs, one needs to per-case re-fit an INRwith an additional temporal axis, which is severely sampling inefficient.

Original: The multi-layer structureappearing in INSP-Net can be viewed as an unfolding algorithm [68, 69] that discretizes the evolutionof time-variant PDEs [70].
Revised: The multi-layer structureappearing in INSP-Net can be viewed as an unfolding algorithm [68, 69] that discretizes the evolutionof time-variant PDEs [70].

Original:  Experiments
Revised:  Experiments

Original: In this section, we evaluate the proposed INSP framework on several challenging tasks, using differentcombinations of Π .
Revised: In this section, we evaluate the proposed INSP framework on several challenging tasks, using differentcombinations of Π .

Original: First, we build low-level image processing filters using either hand-crafted orlearnable Π .
Revised: First, we build low-level image processing filters using either hand-crafted orlearnable Π .

Original: Then, we construct convolutional neural networks with our INSP-ConvNet frameworkand validate its performance on image classification.
Revised: Then, we construct convolutional neural networks with our INSP-ConvNet frameworkand validate its performance on image classification.

Original: More results and implementation details areprovided in the Appendix.
Revised: More results and implementation details areprovided in the Appendix.

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Original: For low-level image processing, we operate on natural images from Set5 dataset [71], Set14dataset [72], and DIV-2k dataset [73].
Revised: For low-level image processing, we operate on natural images from Set5 dataset [71], Set14dataset [72], and DIV-2k dataset [73].

Original: Originally designed for super-resolution, the images arediverse in style and content.
Revised: Originally designed for super-resolution, the images arediverse in style and content.

Original: Note that the unprocessed images presented are the images decodedfrom unprocessed INRs.
Revised: Note that the unprocessed images presented are the images decodedfrom unprocessed INRs.

Original: 5.1 Handcrafted Π for Edge Detection
Revised: 5.1 Handcrafted Π for Edge Detection

Original: We demonstrate the proposed INSP framework is capable of expressing low-level image processingfilters such as for edge detection.
Revised: We demonstrate the proposed INSP framework is capable of expressing low-level image processingfilters such as for edge detection.

Original: Since the edges correspond to gradients in the images, using
Revised: Since the edges correspond to gradients in the images, using

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: gradients of INRs to obtain edges is straightforward.
Revised: gradients of INRs to obtain edges is straightforward.

Original: θ 1 is set to 1 while other coefficients are set to 0.
Revised: θ 1 is set to 1 while other coefficients are set to 0.

Original: In this way, we implicitly obtain a new INR which can be decoded to obtain the edges.
Revised: In this way, we implicitly obtain a new INR which can be decoded to obtain the edges.

Original: We providevisual comparisons against Sobel filter [74], Canny detector [75] and Prewitt operator [76] in Fig.
Revised: We providevisual comparisons against Sobel filter [74], Canny detector [75] and Prewitt operator [76] in Fig.

Original: 5.2 Learnable Π for Image Blurring, Denoising, Deblurring, and Inpainting
Revised: 5.2 Learnable Π for Image Blurring, Denoising, Deblurring, and Inpainting

Original: Since our method operates directly on INRs, we firstly fit the images with INRs and then feedthe INRs into our framework.
Revised: Since our method operates directly on INRs, we firstly fit the images with INRs and then feedthe INRs into our framework.

Original: The final output is another INR which can be decoded into desiredimages.
Revised: The final output is another INR which can be decoded into desiredimages.

Original: The training set of our method consists of 90 examples of INRs, where each INR is built on
Revised: The training set of our method consists of 90 examples of INRs, where each INR is built on

Original: SIREN [27] architectures.
Revised: SIREN [28] architectures.

Original: Image Denoising For classical image denoising filters, we compare against the median filter andmean filter.
Revised: Image Denoising For classical image denoising filters, we compare against the median filter andmean filter.

Original: We use DnCNN [77] as a baseline learning-based method.
Revised: We use DnCNN [77] as a baseline learning-based method.

Original: DnCNN is a pioneer work ofusing residual CNN for denoising.
Revised: DnCNN is a pioneer work ofusing residual CNN for denoising.

Original: The input noisy images are synthesized using additive gaussiannoise.
Revised: The input noisy images are synthesized using additive gaussiannoise.

Original: Visual results are provided in Fig.
Revised: Visual results are provided in Fig.

Original: Image Blurring Image blurring is a low-pass filtering operation.
Revised: Image Blurring Image blurring is a low-pass filtering operation.

Original: We provide a visual comparisonagainst classical filters including 3 × 3 box filter and 3 × 3 gaussian filter.
Revised: We provide a visual comparisonagainst classical filters including 3 × 3 box filter and 3 × 3 gaussian filter.

Original: The target images used fortraining our INSP-Net are the results of the Gaussian filter.
Revised: The target images used fortraining our INSP-Net are the results of the Gaussian filter.

Original: Visual results are provided in Fig.
Revised: Visual results are provided in Fig.

Original: Image Deblurring We compare the proposed method with both traditional algorithms (e.g., wienerfilter [78]) and learning-based algorithms(e.g.
Revised: Image Deblurring We compare the proposed method with both traditional algorithms (e.g., wienerfilter [78]) and learning-based algorithms(e.g.

Original: , MPRNet [79]).
Revised: , MPRNet [79]).

Original: We synthesize blurry images using
Revised: We synthesize blurry images using

Original: Gaussian filters.
Revised: Gaussian filters.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 5, Wiener Filter produce severe artifacts and MPRNet successfullyreconstructs clear textures.
Revised: 5, Wiener Filter produce severe artifacts and MPRNet successfullyreconstructs clear textures.

Original: INSP-Net is capable of generating competitive results against MPRNetand outperforms the Wiener Filter.
Revised: INSP-Net is capable of generating competitive results against MPRNetand outperforms the Wiener Filter.

Original: Image Inpainting We conduct two kinds of experiments in image inpainting, to inpaint 30%random masked pixels or to remove the texts (“INSP-Net”).
Revised: Image Inpainting We conduct two kinds of experiments in image inpainting, to inpaint 30%random masked pixels or to remove the texts (“INSP-Net”).

Original: Comparison methods include meanfilter, median filter, and LaMa [80].
Revised: Comparison methods include meanfilter, median filter, and LaMa [80].

Original: LaMa is a learning-based method using Fourier convolution for inpainting.
Revised: LaMa is a learning-based method using Fourier convolution for inpainting.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 7, mean filter and median filter partially restore the masked pixels,but severely hurt the visual quality of the rest parts.
Revised: 7, mean filter and median filter partially restore the masked pixels,but severely hurt the visual quality of the rest parts.

Original: Also, they can not handle the text region.
Revised: Also, they can not handle the text region.

Original: LaMa successfully removes the text and inpaint the masked pixels.
Revised: LaMa successfully removes the text and inpaint the masked pixels.

Original: Our proposed method largelyoutperforms the filter-based algorithms and performs as well as the LaMa.
Revised: Our proposed method largelyoutperforms the filter-based algorithms and performs as well as the LaMa.

Original: 5.3 Learnable Π for Geometry Processing
Revised: 5.3 Learnable Π for Geometry Processing

Original: [Figure]
Revised: [Figure]

Original: We demonstrate that the proposed INSP framework is not only capable of processing images, butalso capable of processing geometry.
Revised: We demonstrate that the proposed INSP framework is not only capable of processing images, butalso capable of processing geometry.

Original: We first fit a SIREN using the unprocessed mesh following [25].
Revised: We first fit a SIREN using the unprocessed mesh following [25].

Original: Then we use our INSP-Net to implicitly process it into a new INR.
Revised: Then we use our INSP-Net to implicitly process it into a new INR.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 8, our method isable to smooth the surface of the mesh.
Revised: 8, our method isable to smooth the surface of the mesh.

Original: 5.4 Convolutional Neural Networks for Image Classification
Revised: 5.4 Convolutional Neural Networks for Image Classification

Original: We demonstrate that the proposed INSP framework is not only capable to express low-level imageprocessing filters, but also supports high-level tasks such as image classification.
Revised: We demonstrate that the proposed INSP framework is not only capable to express low-level imageprocessing filters, but also supports high-level tasks such as image classification.

Original: To achieve thisgoal, we construct a 2-layer INSP-ConvNet.
Revised: To achieve thisgoal, we construct a 2-layer INSP-ConvNet.

Original: The INSP-ConvNet consists of 2 INSP-Net layers.
Revised: The INSP-ConvNet consists of 2 INSP-Net layers.

Original: Each of them decomposes the INR via the differential operator and combines them with learnable Π .
Revised: Each of them decomposes the INR via the differential operator and combines them with learnable Π .

Original: Since each INSP-Net layer combines the derivative computational graphs of the former layers andapproximates a convolution filter, we build another 2-layer depthwise ConvNets running on pixels asthe baseline, for a fair comparison.
Revised: Since each INSP-Net layer combines the derivative computational graphs of the former layers andapproximates a convolution filter, we build another 2-layer depthwise ConvNets running on pixels asthe baseline, for a fair comparison.

Original: We evaluate the proposed INSP-ConvNet on MNIST ( 28 × 28 resolution) and CIFAR-10 ( 32 × 32resolution) datasets, respectively.
Revised: We evaluate the proposed INSP-ConvNet on MNIST ( 28 × 28 resolution) and CIFAR-10 ( 32 × 32resolution) datasets, respectively.

Original: For each dataset, we will firstly fit each image into an implicitrepresentation using SIREN [27].
Revised: For each dataset, we will firstly fit each image into an implicitrepresentation using SIREN [28].

Original: Both experiments take 1000 epochs to optimize, using AdamWoptimizer [81] and a learning rate of 10 − 4 .
Revised: Both experiments take 1000 epochs to optimize, using AdamWoptimizer [81] and a learning rate of 10 − 4 .

Original: Results are shown in Tab. 1.
Revised: Results are shown in Tab. 1.

Original: Accuracy
Revised: Accuracy

Original: Depthwise CNN
Revised: Depthwise CNN

Original: INSP-ConvNet
Revised: INSP-ConvNet

Original: MNIST 87.
Revised: MNIST 87.

Original: [Table]
Revised: [Table]

Original:  Conclusion
Revised:  Conclusion

Original: Contribution.
Revised: Contribution.

Original: We present INSP-Net framework, an implicit neural signal processing network thatis capable of directly modifying an INR without explicit decoding.
Revised: We present INSP-Net framework, an implicit neural signal processing network thatis capable of directly modifying an INR without explicit decoding.

Original: By incorporating differentialoperators on INR, we can instantiate the INR signal operator as a composition of computationalgraphs approximating any continuous convolution filter.
Revised: By incorporating differentialoperators on INR, we can instantiate the INR signal operator as a composition of computationalgraphs approximating any continuous convolution filter.

Original: Furthermore, we make the first effort tobuild a convolutional neural network that implicitly runs on INRs.
Revised: Furthermore, we make the first effort tobuild a convolutional neural network that implicitly runs on INRs.

Original: While all other methods runon discrete grids, our experiment demonstrates our INSP-Net can achieve competitive results withentirely implicit operations.
Revised: While all other methods runon discrete grids, our experiment demonstrates our INSP-Net can achieve competitive results withentirely implicit operations.

Original: Limitation.
Revised: Limitation.

Original: (Theory)
Revised: (Theory)

Original: Our theory only guarantees the expressiveness of convolution by allowinginfinite sequence approximation.
Revised: Our theory only guarantees the expressiveness of convolution by allowinginfinite sequence approximation.

Original: Construction of more expressive operators and more effectiveparameterization for convolution remain widely open questions.
Revised: Construction of more expressive operators and more effectiveparameterization for convolution remain widely open questions.

Original: (Practice) INSP-Net requires thecomputation of high-order derivatives which is neither memory efficient nor numerical stable.
Revised: (Practice) INSP-Net requires thecomputation of high-order derivatives which is neither memory efficient nor numerical stable.

Original: Thishinders the scalability of our INSP-ConvNet that even requires recursive computation of derivatives.
Revised: Thishinders the scalability of our INSP-ConvNet that even requires recursive computation of derivatives.

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] Please see the conclusion section.
Revised: [Yes] Please see the conclusion section.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [No]
Revised: [No]

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Original: them?
Revised: them?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] The detailed proof is
Revised: [Yes] The detailed proof is

Original: provided in the Appendix.
Revised: provided in the Appendix.

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)?
Revised: mental results (either in the supplemental material or as a URL)?

Original: [No]
Revised: [No]

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters,
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters,

Original: how they
Revised: how they

Original: were chosen)?
Revised: were chosen)?

Original: [Yes] More details are provided in the Appendix.
Revised: [Yes] More details are provided in the Appendix.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi-
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experi-

Original: ments multiple times)?
Revised: ments multiple times)?

Original: [No]
Revised: [No]

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., type
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type

Original: of GPUs, internal cluster, or cloud provider)?
Revised: of GPUs, internal cluster, or cloud provider)?

Original: [Yes] More details are provided in the
Revised: [Yes] More details are provided in the

Original: Appendix.
Revised: Appendix.

Original: All experiments are run on a A6000 GPU.
Revised: All experiments are run on a A6000 GPU.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] We cite them carefully.
Revised: [Yes] We cite them carefully.

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [No] The data are in public domain.
Revised: [No] The data are in public domain.

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [No]
Revised: [No]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?
Revised: using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: information or offensive content?
Revised: information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]

Original: A Proof of Theorem 1
Revised: A Proof of Theorem 1

Original: To begin with, we give the formal definitions of translation and rotation group, along with the notionof shift invariance and rotation invariance.
Revised: To begin with, we give the formal definitions of translation and rotation group, along with the notionof shift invariance and rotation invariance.

Original: Definition 1.
Revised: Definition 1.

Original: (Translation Group & Shift Invariance) Translation group T ( m ) is a transformationgroup isomorphic to m -dimension Euclidean space, where each group element T v transforms avector x ∈ R m by T v ( x ) =
Revised: (Translation Group & Shift Invariance) Translation group T ( m ) is a transformationgroup isomorphic to m -dimension Euclidean space, where each group element T v transforms avector x ∈ R m by T v ( x ) =

Original: x + v .
Revised: x + v .

Original: An operator A is said to be shift-invariant if A (Φ ◦ T v )( x ) =
Revised: An operator A is said to be shift-invariant if A (Φ ◦ T v )( x ) =

Original: A Φ( T v ( x ))
Revised: A Φ( T v ( x ))

Original: =
Revised: =

Original: A Φ( x + v ) .
Revised: A Φ( x + v ) .

Original: Definition 2.
Revised: Definition 2.

Original: (Rotation Group & Rotation Invariance)
Revised: (Rotation Group & Rotation Invariance)

Original: Rotation group SO ( m ) is a transformationgroup also known as the special orthogonal group, where each group element R ∈
Revised: Rotation group SO ( m ) is a transformationgroup also known as the special orthogonal group, where each group element R ∈

Original: R m × m satisfying
Revised: R m × m satisfying

Original: R ⊤ R
Revised: R ⊤ R

Original: = I transforms a vector x ∈ R m by R ( x ) .
Revised: = I transforms a vector x ∈ R m by R ( x ) .

Original: An operator A is said to be rotation-invariant if
Revised: An operator A is said to be rotation-invariant if

Original: A (Φ ◦ R )( x ) =
Revised: A (Φ ◦ R )( x ) =

Original: A Φ( Rx ) .
Revised: A Φ( Rx ) .

Original: Proof.
Revised: Proof.

Original: (Shift Invariance)
Revised: (Shift Invariance)

Original: To show the shift invariance of our model Eq.
Revised: To show the shift invariance of our model Eq.

Original: 3, it is equivalent to showany differential operators are shift-invariant.
Revised: 3, it is equivalent to showany differential operators are shift-invariant.

Original: For the first-order derivatives (gradients), we considerarbitrary shift operator T v ∈ T , by chain rule we will have:
Revised: For the first-order derivatives (gradients), we considerarbitrary shift operator T v ∈ T , by chain rule we will have:

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where the Jacobian matrix of T v ( x ) is an identity matrix.
Revised: where the Jacobian matrix of T v ( x ) is an identity matrix.

Original: Eq.
Revised: Eq.

Original: 7 implies that gradient operator isshift-invariant.
Revised: 7 implies that gradient operator isshift-invariant.

Original: By induction, any high-order differential operators must also be shift-invariant:
Revised: By induction, any high-order differential operators must also be shift-invariant:

Original: [Equation]
Revised: [Equation]

Original: Therefore, we can conclude Π(Φ , ∇ Φ , ∇ 2 Φ , · · · ) is shift-invariant for any Π combining derivativesin any form.(Rotation Invariance)
Revised: Therefore, we can conclude Π(Φ , ∇ Φ , ∇ 2 Φ , · · · ) is shift-invariant for any Π combining derivativesin any form.(Rotation Invariance)

Original: Similarly, we prove the rotation invariance via induction.
Revised: Similarly, we prove the rotation invariance via induction.

Original: First we consider thefollowing case: given arbitrary function Φ : R m → R , and for every rotation matrix R ∈ SO ( m ) ,we compute the gradient:
Revised: First we consider thefollowing case: given arbitrary function Φ : R m → R , and for every rotation matrix R ∈ SO ( m ) ,we compute the gradient:

Original: [Equation]
Revised: [Equation]

Original: Then due to the orthogonality of R , the norm of all the gradients can be written as:
Revised: Then due to the orthogonality of R , the norm of all the gradients can be written as:

Original: [Equation]
Revised: [Equation]

Original: The induction can be proceed by assuming ∥∇ k
Revised: The induction can be proceed by assuming ∥∇ k

Original: [Φ ◦ R ]( x ) ∥ 22 = ∥∇ k Φ( Rx ) ∥ 22 , where with slightabuse of notation, we flatten the k -th order Jacobian matrix to a (cid:0) k
Revised: [Φ ◦ R ]( x ) ∥ 22 = ∥∇ k Φ( Rx ) ∥ 22 , where with slightabuse of notation, we flatten the k -th order Jacobian matrix to a (cid:0) k

Original: + m − 1 k (cid:1) -dimension vector withoutmultiplicity.
Revised: + m − 1 k (cid:1) -dimension vector withoutmultiplicity.

Original: Then for the ( k + 1) -th order Jacobian matrix (represented as an m by (cid:0) k
Revised: Then for the ( k + 1) -th order Jacobian matrix (represented as an m by (cid:0) k

Original: + m − 1 k (cid:1) matrix),we have
Revised: + m − 1 k (cid:1) matrix),we have

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Then by the induction hypothesis, we can obtain:
Revised: Then by the induction hypothesis, we can obtain:

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where
Revised: where

Original: Eq.
Revised: Eq.

Original: 12 is due to the linearity of differential operators.
Revised: 12 is due to the linearity of differential operators.

Original: As we show in Eq.
Revised: As we show in Eq.

Original: 10, k = 0satisfies the induction hypothesis, then we can draw the conclusion: for arbitrary differentiablefunction Φ and for every integer k > 0 , operator ∥∇ k Φ( x ) ∥ 22 is rotation-invariant.
Revised: 10, k = 0satisfies the induction hypothesis, then we can draw the conclusion: for arbitrary differentiablefunction Φ and for every integer k > 0 , operator ∥∇ k Φ( x ) ∥ 22 is rotation-invariant.

Original: Hence, Π =
Revised: Hence, Π =

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: k =0 ∥∇ k Φ( x ) ∥ 22 (cid:17) is also rotation-invariant asfor arbitrary f mapping the L 2 norm of derivatives.
Revised: k =0 ∥∇ k Φ( x ) ∥ 22 (cid:17) is also rotation-invariant asfor arbitrary f mapping the L 2 norm of derivatives.

Original: B Proof of Theorem 2
Revised: B Proof of Theorem 2

Original: To give a formal statement of Theorem 2, we first list the following definitions:
Revised: To give a formal statement of Theorem 2, we first list the following definitions:

Original: Definition 3.
Revised: Definition 3.

Original: (Differential Operator)
Revised: (Differential Operator)

Original: For a sake of simplicity and consistency, we denote D = ddx [ · ]as the first-order differential operator, D n = d n dx n [ · ] as the n -th order differential operator.
Revised: For a sake of simplicity and consistency, we denote D = ddx [ · ]as the first-order differential operator, D n = d n dx n [ · ] as the n -th order differential operator.

Original: Definition 4.
Revised: Definition 4.

Original: (Fourier Transform) Given real-valued function f : R → C that satisfies Dirichletcondition 4 , then Fourier transform F is defined as:
Revised: (Fourier Transform) Given real-valued function f : R → C that satisfies Dirichletcondition 4 , then Fourier transform F is defined as:

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Inverse Fourier transform F − 1 exists and has the form of:
Revised: Inverse Fourier transform F − 1 exists and has the form of:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Definition 5.
Revised: Definition 5.

Original: (Convolution)
Revised: (Convolution)

Original: Given two real-valued functions f : R → R and g : R → R , convolutionbetween f
Revised: Given two real-valued functions f : R → R and g : R → R , convolutionbetween f

Original: and g is defined as:
Revised: and g is defined as:

Original: [Equation]
Revised: [Equation]

Original: Then we denote function f
Revised: Then we denote function f

Original: ⋆ g =
Revised: ⋆ g =

Original: T g
Revised: T g

Original: [ f ] , where T g represents a functional operator with respect to g .
Revised: [ f ] , where T g represents a functional operator with respect to g .

Original: Definition 6.
Revised: Definition 6.

Original: (Band-limited Signal Space)
Revised: (Band-limited Signal Space)

Original: Define the signal space F as a Hilbert space of realvalued functions such that for ∀ f ∈ F , (1) f ∈ C ∞ ( R ) is continuous and smooth over real, (2) 639 (cid:82) ∞
Revised: Define the signal space F as a Hilbert space of realvalued functions such that for ∀ f ∈ F , (1) f ∈ C ∞ ( R ) is continuous and smooth over real, (2) 639 (cid:82) ∞

Original: −∞ | f ( x ) | dx < ∞ is L 1 integrable, (3) F [ f ]( ω )
Revised: −∞ | f ( x ) | dx < ∞ is L 1 integrable, (3) F [ f ]( ω )

Original: = 

Revised: =

Original: if | ω | > a has a limited width of spectrum,and (cid:82) ∞
Revised: if | ω | > a has a limited width of spectrum,and (cid:82) ∞

Original: −∞ |F [ f ]( ω ) | dω ≤ C has limited spectral power, for some a > 0 , C > 0 .
Revised: −∞ |F [ f ]( ω ) | dω ≤ C has limited spectral power, for some a > 0 , C > 0 .

Original: Definition 7.
Revised: Definition 7.

Original: (Convolution Space)
Revised: (Convolution Space)

Original: Define a convolutional operator space T of all T g such that∀T g ∈ T , (1) g : R → R is real-valued function, (2) F [ g ] ∈ C 1 ( R ) has a continuous spectrum.
Revised: Define a convolutional operator space T of all T g such that∀T g ∈ T , (1) g : R → R is real-valued function, (2) F [ g ] ∈ C 1 ( R ) has a continuous spectrum.

Original: Before we prove Theorem 2, we enumerate the following results as our key mathematical tools:
Revised: Before we prove Theorem 2, we enumerate the following results as our key mathematical tools:

Original: Lemma 4.
Revised: Lemma 4.

Original: (Convolution Theorem)
Revised: (Convolution Theorem)

Original: For every T g ∈ T , it always holds that F ◦ T g [ f ]( ω ) =
Revised: For every T g ∈ T , it always holds that F ◦ T g [ f ]( ω ) =

Original: F [ f ]( ω ) ·
Revised: F [ f ]( ω ) ·

Original: F [ g ]( ω ) .
Revised: F [ g ]( ω ) .

Original: Lemma 5.
Revised: Lemma 5.

Original: (Weierstrass Approximation Theorem) Suppose f is a continuous real-valued function defined on a compact space X .
Revised: (Weierstrass Approximation Theorem) Suppose f is a continuous real-valued function defined on a compact space X .

Original: For every ϵ > 0 , there exists a polynomial p ( x ) such thatsup x ∈X | f ( x ) − p ( x ) | < ϵ .
Revised: For every ϵ > 0 , there exists a polynomial p ( x ) such thatsup x ∈X | f ( x ) − p ( x ) | < ϵ .

Original: Lemma 6.
Revised: Lemma 6.

Original: Suppose f is a continuous real-valued function satisfying Dirichlet condition.
Revised: Suppose f is a continuous real-valued function satisfying Dirichlet condition.

Original: Then
Revised: Then

Original: F [ f ]( ω ) =
Revised: F [ f ]( ω ) =

Original: F [ f ]( − ω ) ∗ , i.e., the spectrum of real-valued function is conjugate symmetric.
Revised: F [ f ]( − ω ) ∗ , i.e., the spectrum of real-valued function is conjugate symmetric.

Original: Below is the formal statement of our Theorem 2 and its detailed proof.
Revised: Below is the formal statement of our Theorem 2 and its detailed proof.

Original: Theorem 7.
Revised: Theorem 7.

Original: For every T g ∈ T and arbitrary small ϵ > 0 , there exists a polynomial p ( x )
Revised: For every T g ∈ T and arbitrary small ϵ > 0 , there exists a polynomial p ( x )

Original: =
Revised: =

Original: a 0 + a 1 x + · · · where a 0 , a 1 , · · · ∈ R , such that sup x ∈ R |T g
Revised: a 0 + a 1 x + · · · where a 0 , a 1 , · · · ∈ R , such that sup x ∈ R |T g

Original: [ f ]( x ) − p ( D )[ f ]( x ) | < ϵ for all f ∈ F .
Revised: [ f ]( x ) − p ( D )[ f ]( x ) | < ϵ for all f ∈ F .

Original: Proof.
Revised: Proof.

Original: For every f ∈ F and T g ∈ T , by Lemma 4, one can rewrite:
Revised: For every f ∈ F and T g ∈ T , by Lemma 4, one can rewrite:

Original: [Equation]
Revised: [Equation]

Original: (18) where we use ˆ f : R → C and ˆ g : R → C to denote the Fourier transform of f and g , respectively.
Revised: (18) where we use ˆ f : R → C and ˆ g : R → C to denote the Fourier transform of f and g , respectively.

Original: By
Revised: By

Original: Lemma 6, we can construct an invertible mapping ϕ by letting:
Revised: Lemma 6, we can construct an invertible mapping ϕ by letting:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: which is also known as the Hartley transform.
Revised: which is also known as the Hartley transform.

Original: Then ˜ f
Revised: Then ˜ f

Original: := ϕ
Revised: := ϕ

Original: [ ˆ f ] and ˜ g :
Revised: [ ˆ f ] and ˜ g :

Original: = ϕ
Revised: = ϕ

Original: [ˆ g ] turn out to be real-valuedfunctions.
Revised: [ˆ g ] turn out to be real-valuedfunctions.

Original: Since ˆ f is only supported in [ − a, a ] , we only consider ˜ g within the interval [ − a, a ] .
Revised: Since ˆ f is only supported in [ − a, a ] , we only consider ˜ g within the interval [ − a, a ] .

Original: By
Revised: By

Original: Lemma 5, there exists a polynomial ˜ p ( x ) =
Revised: Lemma 5, there exists a polynomial ˜ p ( x ) =

Original: ˜ a 0 + ˜ a 1 x + · · · + ˜ a d x d where d = deg(˜ p ) is the degreeof ˜ p and ˜ a 0 , ˜ a 1 , · · · , ˜ a d ∈ R such that sup ω ∈ [ − a,a ] | ˜ g ( ω ) − ˜ p ( ω ) | < ϵ/ 2 C for every ϵ > 0 .
Revised: ˜ a 0 + ˜ a 1 x + · · · + ˜ a d x d where d = deg(˜ p ) is the degreeof ˜ p and ˜ a 0 , ˜ a 1 , · · · , ˜ a d ∈ R such that sup ω ∈ [ − a,a ] | ˜ g ( ω ) − ˜ p ( ω ) | < ϵ/ 2 C for every ϵ > 0 .

Original: Applying ϕ − 1 to ˜ p , we will obtain a new polynomial ˆ p :=
Revised: Applying ϕ − 1 to ˜ p , we will obtain a new polynomial ˆ p :=

Original: ϕ − 1 [˜ p ] over C such that:
Revised: ϕ − 1 [˜ p ] over C such that:

Original: [Equation]
Revised: [Equation]

Original: Then we show the difference between ˆ g and ˆ p :
Revised: Then we show the difference between ˆ g and ˆ p :

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: C .
Revised: C .

Original: [Equation]
Revised: [Equation]

Original: Hereby, we have shown that we can construct a complex polynomial ˆ p ( x ) =
Revised: Hereby, we have shown that we can construct a complex polynomial ˆ p ( x ) =

Original: ˆ a 0 +ˆ a 1 x + · · · +ˆ a d fromthe real polynomial ˜ p ( x ) =
Revised: ˆ a 0 +ˆ a 1 x + · · · +ˆ a d fromthe real polynomial ˜ p ( x ) =

Original: ˜ a 

Revised: ˜ a

Original: +˜ a 1 x + · · · +˜ a d by letting ˆ a 2 k = ˜
Revised: +˜ a 1 x + · · · +˜ a d by letting ˆ a 2 k = ˜

Original: a 2 k , k = 0 , 1 , · · · , ⌊ d 2 ⌋ and ˆ a 2 k +

Revised: a 2 k , k = 0 , 1 , · · · , ⌊ d 2 ⌋ and ˆ a 2 k +

Original: =
Revised: =

Original: i ˜ a 2 k +1 , k = 0 , 1 , · · · , ⌊ d − 12 ⌋ ,
Revised: i ˜ a 2 k +1 , k = 0 , 1 , · · · , ⌊ d − 12 ⌋ ,

Original: such that polynomial
Revised: such that polynomial

Original: ˆ p satisfies that sup ω ∈ [ − a,a ] | ˆ g ( ω ) − ˆ p ( ω ) | < ϵ/C .
Revised: ˆ p satisfies that sup ω ∈ [ − a,a ] | ˆ g ( ω ) − ˆ p ( ω ) | < ϵ/C .

Original: In the meanwhile, f ∈ F is L 1 integrable over R implies lim x →∞ f ( x ) = 0 .
Revised: In the meanwhile, f ∈ F is L 1 integrable over R implies lim x →∞ f ( x ) = 0 .

Original: Then by the fundamentaltheorem of calculus, we have the following equation:
Revised: Then by the fundamentaltheorem of calculus, we have the following equation:

Original: [Equation]
Revised: [Equation]

Original: which implies F ◦ D
Revised: which implies F ◦ D

Original: [Equation]
Revised: [Equation]

Original: ]( ω ) .
Revised: ]( ω ) .

Original: Define a sequence q n ( ω ) =
Revised: Define a sequence q n ( ω ) =

Original: (2 πiω ) n .
Revised: (2 πiω ) n .

Original: By induction,the n -th order derivative of f satisfies
Revised: By induction,the n -th order derivative of f satisfies

Original: [Equation]
Revised: [Equation]

Original: We note that { q n , n = 0 , 2 , · · · , } form a basis of the polynomial ring, where q 2 k only has real part,q 2 k +1 only has imaginary part, for k = 0 , 1 , · · · .
Revised: We note that { q n , n = 0 , 2 , · · · , } form a basis of the polynomial ring, where q 2 k only has real part,q 2 k +1 only has imaginary part, for k = 0 , 1 , · · · .

Original: Thus, we can find a series of a 0 , a 1 , · · · , a d ∈ Rsuch that ˆ p ( x ) = a 0 + a 1 q 1 ( ω ) + · · · + a d q d ( ω ) by letting a n = ˆ a n / (2 πi ) n ∈ R , n = 0 , 1 , · · · , d .
Revised: Thus, we can find a series of a 0 , a 1 , · · · , a d ∈ Rsuch that ˆ p ( x ) = a 0 + a 1 q 1 ( ω ) + · · · + a d q d ( ω ) by letting a n = ˆ a n / (2 πi ) n ∈ R , n = 0 , 1 , · · · , d .

Original: So far, we obtain a 0 , a 1 , · · · , a d ∈ R such that sup ω ∈ [ − a,a ] | ˆ g ( ω ) − (cid:80) dn =

Revised: So far, we obtain a 0 , a 1 , · · · , a d ∈ R such that sup ω ∈ [ − a,a ] | ˆ g ( ω ) − (cid:80) dn =

Original: a n q n ( ω ) | < ϵ/C .
Revised: a n q n ( ω ) | < ϵ/C .

Original: By linearity of Fourier transform and Eq. 30, element-wisely applying (cid:80) d n =0 a n q n to ˆ f will lead toa transform on the real domain:
Revised: By linearity of Fourier transform and Eq. 30, element-wisely applying (cid:80) d n =0 a n q n to ˆ f will lead toa transform on the real domain:

Original: [Equation]
Revised: [Equation]

Original: ˆ y ( l ) ← (cid:104) y
Revised: ˆ y ( l ) ← (cid:104) y

Original: ( l − 1)
Revised: ( l − 1)

Original: ∂y ( l − 1) ∂ x ⊤ ∂ 2 y ( l
Revised: ∂y ( l − 1) ∂ x ⊤ ∂ 2 y ( l

Original: [Equation]
Revised: [Equation]

Original: 1) ∂ x 2 ⊤ · · · ∂ K y
Revised: 1) ∂ x 2 ⊤ · · · ∂ K y

Original: l
Revised: l

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: x K ⊤ (cid:105) θ ( l ) ▷ Convolutional layer l ) ← ReLU(InstanceNorm1D( where p ( x ) =
Revised: x K ⊤ (cid:105) θ ( l ) ▷ Convolutional layer l ) ← ReLU(InstanceNorm1D( where p ( x ) =

Original: a 0 + a 1 x + · · · + a d x d over ring of differentiation operators.
Revised: a 0 + a 1 x + · · · + a d x d over ring of differentiation operators.

Original: Now we bound thedifference between T g [ f ] and p ( D )[ f ] for every f ∈ F and x ∈ R :|T
Revised: Now we bound thedifference between T g [ f ] and p ( D )[ f ] for every f ∈ F and x ∈ R :|T

Original: g
Revised: g

Original: [ f ]( x ) − p (
Revised: [ f ]( x ) − p (

Original: [Equation]
Revised: [Equation]

Original: (cid:12)(cid:12)(cid:12)(cid:12) (cid:12) (cid:90) a − a exp(2 πiωx )
Revised: (cid:12)(cid:12)(cid:12)(cid:12) (cid:12) (cid:90) a − a exp(2 πiωx )

Original: ˆ f ( ω ) (cid:32)
Revised: ˆ f ( ω ) (cid:32)

Original: ˆ g ( ω ) − d (cid:88) n = a n q n ( ω ) (cid:33) dω (cid:12)(cid:12)(cid:12)(cid:12) (cid:12) (32) ≤ (cid:90) a − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ f ( ω ) (cid:32)
Revised: ˆ g ( ω ) − d (cid:88) n = a n q n ( ω ) (cid:33) dω (cid:12)(cid:12)(cid:12)(cid:12) (cid:12) (32) ≤ (cid:90) a − a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ f ( ω ) (cid:32)

Original: ˆ g ( ω ) − d (cid:88) n = a n q n ( ω
Revised: ˆ g ( ω ) − d (cid:88) n = a n q n ( ω

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where Eq.
Revised: where Eq.

Original: 33 holds due to Jensen’s inequality, Eq.
Revised: 33 holds due to Jensen’s inequality, Eq.

Original: 34 follows from Hölder’s inequality, and Eq. 35 isobtained by substituting the upper bound of difference | ˆ g ( ω ) − (cid:80) dn =

Revised: 34 follows from Hölder’s inequality, and Eq. 35 isobtained by substituting the upper bound of difference | ˆ g ( ω ) − (cid:80) dn =

Original: a n q n ( ω ) | and the L 1 norm ofˆ f ( ω ) .
Revised: a n q n ( ω ) | and the L 1 norm ofˆ f ( ω ) .

Original: C Implementation Details of INSP-ConvNet
Revised: C Implementation Details of INSP-ConvNet

Original: We have formulated exact convolution form and INSP-Conv in Sec. 3.3.
Revised: We have formulated exact convolution form and INSP-Conv in Sec. 3.3.

Original: We provide a pseudocodeto illustrate the forward pass of INSP-ConvNet in Algorithm 1.
Revised: We provide a pseudocodeto illustrate the forward pass of INSP-ConvNet in Algorithm 1.

Original: Below we elaborate each maincomponent:
Revised: Below we elaborate each maincomponent:

Original: Convolutional Layer.
Revised: Convolutional Layer.

Original: Each A ( l ) represents an implicit convolution layer.
Revised: Each A ( l ) represents an implicit convolution layer.

Original: We follow the closedform solution in Eq.
Revised: We follow the closedform solution in Eq.

Original: 5 to parameterize A ( l ) with θ ( l ) .
Revised: 5 to parameterize A ( l ) with θ ( l ) .

Original: We point out that ConvNet[Φ] also correspondsto a computational graph, which can continuously map coordinates to the output features.
Revised: We point out that ConvNet[Φ] also correspondsto a computational graph, which can continuously map coordinates to the output features.

Original: To constructthis computational graph, we recursively call for gradient networks of the previous layer until the firstlayer.
Revised: To constructthis computational graph, we recursively call for gradient networks of the previous layer until the firstlayer.

Original: For example, A ( l ) will request the gradient network of A ( l − 1) · σ ◦· · ·◦A (1) · Φ , and then A ( l − 1)will request the gradient network of the rest part.
Revised: For example, A ( l ) will request the gradient network of A ( l − 1) · σ ◦· · ·◦A (1) · Φ , and then A ( l − 1)will request the gradient network of the rest part.

Original: This procedure will proceed until the first layer,which directly returns the derivative network of Φ .
Revised: This procedure will proceed until the first layer,which directly returns the derivative network of Φ .

Original: Kernels in CNNs typically perform multi-channelconvolution.
Revised: Kernels in CNNs typically perform multi-channelconvolution.

Original: However, this is not memory friendly to gradient computing in our framework.
Revised: However, this is not memory friendly to gradient computing in our framework.

Original: To thisend, we run channel-wise convolution first and then employ a linear layer to mix channels [82].
Revised: To thisend, we run channel-wise convolution first and then employ a linear layer to mix channels [82].

Original: Nonlinear Activation and Normalization.
Revised: Nonlinear Activation and Normalization.

Original: Nonlinear activation and normalization are naturallyelement-wise functions.
Revised: Nonlinear activation and normalization are naturallyelement-wise functions.

Original: They are point-wisely applied to the output of an INSP-Net and participatethe computational graph construction process.
Revised: They are point-wisely applied to the output of an INSP-Net and participatethe computational graph construction process.

Original: This corresponds to the line 6 of Algorithm 1.
Revised: This corresponds to the line 6 of Algorithm 1.

Original: Training Recipe.
Revised: Training Recipe.

Original: Given a dataset D
Revised: Given a dataset D

Original: = { (Φ i , y i ) } with a set of pre-trained INRs Φ i and theircorresponding labels y i , our goal is to learn a ConvNet[ · ] that can process each example.
Revised: = { (Φ i , y i ) } with a set of pre-trained INRs Φ i and theircorresponding labels y i , our goal is to learn a ConvNet[ · ] that can process each example.

Original: In contrastto standard ConvNets that are designed for grid-based images, the computational graph of INSPConvNet contains parameters of both the input INR Φ i and learnable kernels A ( l ) .
Revised: In contrastto standard ConvNets that are designed for grid-based images, the computational graph of INSPConvNet contains parameters of both the input INR Φ i and learnable kernels A ( l ) .

Original: During thetraining stage, we randomly sample a mini-batch (Φ i , y i ) from D to optimize INSP-ConvNet.
Revised: During thetraining stage, we randomly sample a mini-batch (Φ i , y i ) from D to optimize INSP-ConvNet.

Original: Thecorresponding loss will be evaluated according to the network output, and then back-propagate the calculated gradients to the learnable parameters in A ( l ) , using the stochastic gradient descentoptimization.
Revised: Thecorresponding loss will be evaluated according to the network output, and then back-propagate the calculated gradients to the learnable parameters in A ( l ) , using the stochastic gradient descentoptimization.

Original: Along the whole process, the parameters of Φ i is fixed and only the parameters in A ( l )is optimized.
Revised: Along the whole process, the parameters of Φ i is fixed and only the parameters in A ( l )is optimized.

Original: Standard data augmentations are included by default, including rotation, zoom in/out,etc.
Revised: Standard data augmentations are included by default, including rotation, zoom in/out,etc.

Original: In practice, we implement these augmentations by using affine transformation on the coordinatesof INRs.
Revised: In practice, we implement these augmentations by using affine transformation on the coordinatesof INRs.

Original: D Connection with PDE based Signal Processing
Revised: D Connection with PDE based Signal Processing

Original: Partial Differential Equation (PDE) has been successfully applied to image processing domain aswe discussed in Sec. 4.3.
Revised: Partial Differential Equation (PDE) has been successfully applied to image processing domain aswe discussed in Sec. 4.3.

Original: In this section, we focus on their connection with our INSP-Net.
Revised: In this section, we focus on their connection with our INSP-Net.

Original: Wesummarize the methods of this line of works [57, 42, 62] in the following formulation:
Revised: Wesummarize the methods of this line of works [57, 40, 62] in the following formulation:

Original: [Equation]
Revised: [Equation]

Original: where M t ( · ) is a time-variant function that remaps the direct output and high-order derivatives offunction Ψ .
Revised: where M t ( · ) is a time-variant function that remaps the direct output and high-order derivatives offunction Ψ .

Original: For heat diffusion, M t boils down to be an stationary isotropic combination of secondorder derivatives.
Revised: For heat diffusion, M t boils down to be an stationary isotropic combination of secondorder derivatives.

Original: In [57], M t is chosen to be a gradient magnitude aware diffusion operator runningon divergence operators.
Revised: In [57], M t is chosen to be a gradient magnitude aware diffusion operator runningon divergence operators.

Original: [42, 62] degenerate M t to a time-dependent linear mapping of pre-definedinvariants of the maximal order two.
Revised: [40, 62] degenerate M t to a time-dependent linear mapping of pre-definedinvariants of the maximal order two.

Original: We note that Eq.
Revised: We note that Eq.

Original: 36 can be naturally solved with INRs, as
Revised: 36 can be naturally solved with INRs, as

Original: INRs are amenable to solving complicated differential equation shown by [27].
Revised: INRs are amenable to solving complicated differential equation shown by [28].

Original: One straightforwardsolution is to parameterize M t by another time-dependent coordinate network [30] and enforce theboundary condition Ψ( x , 0) =
Revised: One straightforwardsolution is to parameterize M t by another time-dependent coordinate network [27] and enforce theboundary condition Ψ( x , 0) =

Original: Φ( x ) and minimize the difference between the two hands of the Eq.36.
Revised: Φ( x ) and minimize the difference between the two hands of the Eq.36.

Original: However, foreseeable problem falls in sampling inefficiency over the time axis.
Revised: However, foreseeable problem falls in sampling inefficiency over the time axis.

Original: Suppose wediscretize the time axis into small intervals 

Revised: Suppose wediscretize the time axis into small intervals

Original: = t 0 < t 1 < · · · < t N , then Eq.
Revised: = t 0 < t 1 < · · · < t N , then Eq.

Original: 36 has a closed-formsolution given M t :
Revised: 36 has a closed-formsolution given M t :

Original: [Equation]
Revised: [Equation]

Original: One can see Eq.
Revised: One can see Eq.

Original: 38 can be regarded as a special case of our model Eq.
Revised: 38 can be regarded as a special case of our model Eq.

Original: 3, where we absorb M t n , timeinterval t n +1 − t n , and the residual term Ψ( x , t n ) into one Π .
Revised: 3, where we absorb M t n , timeinterval t n +1 − t n , and the residual term Ψ( x , t n ) into one Π .

Original: Considering our multi-layer model
Revised: Considering our multi-layer model

Original: INSP-ConvNet (see Sec. 3.3), we can analogize t n to the layer number, and then solving Eq.
Revised: INSP-ConvNet (see Sec. 3.3), we can analogize t n to the layer number, and then solving Eq.

Original: 36 attime t N is approximately equal to forward passing an N -layer INSP-ConvNet.
Revised: 36 attime t N is approximately equal to forward passing an N -layer INSP-ConvNet.

Original: E More Experiment Details
Revised: E More Experiment Details

Original: We implement our INSP framework using PyTorch.
Revised: We implement our INSP framework using PyTorch.

Original: The gradients are obtained directly using theautograd package from PyTorch.
Revised: The gradients are obtained directly using theautograd package from PyTorch.

Original: All learnable parameters are trained with AdamW optimizer anda learning rate of 1e-4.
Revised: All learnable parameters are trained with AdamW optimizer anda learning rate of 1e-4.

Original: For low-level image processing kernels, images are obtained from Set5dataset [71], Set14 dataset [72], and DIV-2k dataset [73].
Revised: For low-level image processing kernels, images are obtained from Set5dataset [71], Set14 dataset [72], and DIV-2k dataset [73].

Original: These datasets are original collected forsuper-resolution task, so the images are diverse in style and content.
Revised: These datasets are original collected forsuper-resolution task, so the images are diverse in style and content.

Original: In our experiments, we construct
Revised: In our experiments, we construct

Original: SIREN [27] on each image.
Revised: SIREN [28] on each image.

Original: For efficiency, we resized the images to 256 × 256 .
Revised: For efficiency, we resized the images to 256 × 256 .

Original: We use 90 images toconstruct the INRs used for training, and use the other images for evaluation.
Revised: We use 90 images toconstruct the INRs used for training, and use the other images for evaluation.

Original: For image classification, we construct a 2-layer INSP-ConvNet framework.
Revised: For image classification, we construct a 2-layer INSP-ConvNet framework.

Original: Each INSP layer constructs the derivative computational graphs of the former layers and combines them with learnableΠ .
Revised: Each INSP layer constructs the derivative computational graphs of the former layers and combines them with learnableΠ .

Original: The INSP-layer is capable of approximating a convolution filter.
Revised: The INSP-layer is capable of approximating a convolution filter.

Original: For a fair comparison, we buildanother 2-layer depthwise convolutional network running on image pixels as the baseline.
Revised: For a fair comparison, we buildanother 2-layer depthwise convolutional network running on image pixels as the baseline.

Original: Both our
Revised: Both our

Original: INSP-ConvNet and the ConvNet running on pixels are trained with the same hyper-parameters.
Revised: INSP-ConvNet and the ConvNet running on pixels are trained with the same hyper-parameters.

Original: Bothexperiments take 1000 training epochs, with a learning rate of 1 e − 4 using AdamW optimizer.
Revised: Bothexperiments take 1000 training epochs, with a learning rate of 1 e − 4 using AdamW optimizer.

Original: F More Experimental Results
Revised: F More Experimental Results

Original: In this section, we provide more experimental results.
Revised: In this section, we provide more experimental results.

Original: Fig.
Revised: Fig.

Original: 9 provides comparisons on edge detectiontask.
Revised: 9 provides comparisons on edge detectiontask.

Original: Fig.
Revised: Fig.

Original: 10 shows image denoising results.
Revised: 10 shows image denoising results.

Original: Fig.
Revised: Fig.

Original: 11 demonstrates image deblurring results.
Revised: 11 demonstrates image deblurring results.

Original: Fig.
Revised: Fig.

Original: 12shows image blurring results.
Revised: 12shows image blurring results.

Original: Fig.
Revised: Fig.

Original: 13 shows image inpainting results.
Revised: 13 shows image inpainting results.

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: where the synthetic noise is rgb gaussian noise.
Revised: where the synthetic noise is rgb gaussian noise.

Original: The noise is similar to the ones seen during the training of MPRNet and MAXIM, so they obtain better performance with the help of a much wider training set.
Revised: The noise is similar to the ones seen during the training of MPRNet and MAXIM, so they obtain better performance with the help of a much wider training set.

Original: We also provide quantitative comparisons on the test set in Tab. 2 and Tab. 3.
Revised: We also provide quantitative comparisons on the test set in Tab. 2 and Tab. 3.

Original: The test set consistsof 100 INRs fitted from 100 images in DIV-2k dataset [73].
Revised: The test set consistsof 100 INRs fitted from 100 images in DIV-2k dataset [73].

Original: In Tab. 2, the synthetic noise used isa gray-scale Gaussian noise.
Revised: In Tab. 2, the synthetic noise used isa gray-scale Gaussian noise.

Original: We additionally clip the value of the noise so that only the positivepart remains.
Revised: We additionally clip the value of the noise so that only the positivepart remains.

Original: Under such situations, MPRNet [79] and MAXIM [83] do not generalize well becausethe synthetic noise differs from the SIDD dataset they were trained on.
Revised: Under such situations, MPRNet [79] and MAXIM [83] do not generalize well becausethe synthetic noise differs from the SIDD dataset they were trained on.

Original: However, in Tab. 3, theirperformance is better when the synthetic noise becomes three-channel Gaussian noise.
Revised: However, in Tab. 3, theirperformance is better when the synthetic noise becomes three-channel Gaussian noise.

Original: The syntheticnoise is similar to those seen during the training process of MPRNet [79] and MAXIM [83], so theybenefit from their much wider training set.
Revised: The syntheticnoise is similar to those seen during the training process of MPRNet [79] and MAXIM [83], so theybenefit from their much wider training set.

Original: We additionally validate the ability of our INSP framework by processing audio signals.
Revised: We additionally validate the ability of our INSP framework by processing audio signals.

Original: We addsynthetic Gaussian noise onto the audio and use it to fit a SIREN.
Revised: We addsynthetic Gaussian noise onto the audio and use it to fit a SIREN.

Original: The noisy audio decoded from the
Revised: The noisy audio decoded from the

Original: INR is shown in Fig.
Revised: INR is shown in Fig.

Original: 14(b).
Revised: 14(b).

Original: Then we use our INSP-Net to implicitly process it to a new INR that canbe further decoded into denoised audio.
Revised: Then we use our INSP-Net to implicitly process it to a new INR that canbe further decoded into denoised audio.

Original: It’s decoded result is shown in Fig.
Revised: It’s decoded result is shown in Fig.

Original: 14(c).
Revised: 14(c).

Original: We also providevisualization of the denoising effect in Fig.
Revised: We also providevisualization of the denoising effect in Fig.

Original: 14(f).
Revised: 14(f).

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Original: Mean Filter
Revised: Mean Filter

Original: LaMa
Revised: LaMa

Original: INSP-Net
Revised: INSP-Net

Original: Target Image
Revised: Target Image

Original: [Figure]
Revised: [Figure]