C$^{2}$-INR: Customized Convolutional Implicit Neural Representation

Jinglei Shi; Xinran Chang; Jiaqi Cui; Yingjie Xia; Zhaolin Xiao; Chongyi Li

cs.CVSep 19, 2026

C $^{2}$ -INR: Customized Convolutional Implicit Neural Representation

Authors: Jinglei Shi, Xinran Chang, Jiaqi Cui, Yingjie Xia, Zhaolin Xiao, Chongyi Li

Organizations: VCIP & TMCC & DISSec, College of Computer Science, Nankai University · College of Computer Science and Engineering, Xi’an University of Technology

Abstract

Implicit Neural Representation (INR) leverages neural networks to represent discrete signals such as images as continuous ones, where the network weights serve as a compact form of the signal itself. Most existing INR methods adopt Multi-Layer Perceptrons (MLPs) as their backbone. Since these models render each pixel independently, they inherently fail to exploit the spatial correlations that exist between neighboring pixels. In contrast,convolutional INRs can process pixels in parallel while inherently accounting for inter-pixel dependencies, making them a more natural fit for representing images. Nevertheless, convolutional INRs remain relatively underexplored, and the majority of them rely on fixed architectural settings, leaving little room for image-specific adaptation. In this paper, we investigate network customization for convolutional INRs. We replace conventional filters with irregular directional kernels, whose allocation is guided by the directional energy in the image spectrum, i.e., directions exhibiting stronger energy are assigned a larger number of kernels, enabling content-tailored convolution settings. These kernels are further reformulated via an orthogonal basis to achieve a superior sparse representation. Moreover, we introduce an annealed Gumbel-Softmax-based mechanism for kernel-level activation function selection, which gives the most suitable activation function for each convolution kernel. Extensive experiments demonstrate that our method, namely C $^{2}$ -INR, achieves superior performance against state-of-the-art approaches under comparable parameter budgets across a wide range of image processing tasks, including representation, inpainting, and super-resolution.

Figures & tables

Figure 1: Illustration of example images and their directional spectral magnitude distributions. The top image exhibits predominantly vertical variations, with its large-magnitude spectral components concentrated around $\pi/2$ , while the bottom image contains mostly horizontal variations, with its energy peaking around 0. This implies that different images should have different optimal INR architectures.

Figure 2: Overview of the proposed C 2 -INR. The upper part illustrates the overall network backbone, which consists of five stages, each comprising SGDCB, BN, UP, and AGS-AFS modules, followed by a final $1\times 1$ convolutional decoding layer. The structure of SGDCB (bottom left) is determined prior to training: it focuses on analyzing the frequency energy in each direction and uses this information to allocate directional kernel numbers. These kernels are then reformulated via orthogonal basis decomposition with learnable parameters. Meanwhile, AGS-AFS (bottom right) operates during training, where it estimates scores for each candidate activation function using the Gumbel-Softmax function and adopts an annealing mechanism to gradually select the best-fitting activation function for each kernel.

	#Param $\sim$ 108k		#Param $\sim$ 215k		#Param $\sim$ 437k
Methods	PSNR $\uparrow$	SSIM $\uparrow$	PSNR $\uparrow$	SSIM $\uparrow$	PSNR $\uparrow$	SSIM $\uparrow$
DD	32.58	0.8796	33.99	0.9081	35.48	0.9305
INCODE	25.97	0.6423	30.95	0.8159	35.59	0.9025
SL 2 A-INR	32.27	0.8348	36.23	0.9104	42.51	0.9698
MIRE	31.77	0.8750	33.11	0.9018	35.52	0.9305
MSA	31.90	0.8523	34.62	0.9085	39.03	0.9626

Table 1: Quantitative results for the image representation task using C 2 -INR (Ours) and other SOTA methods. The best and second-best results are highlighted in bold and underlined .

	$\times$ 2 SR			$\times$ 4 SR
Methods	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
DD	26.45	0.7488	0.3181	24.25	0.6650	0.4241
INCODE	26.06	0.6764	0.2413	20.90	0.4047	0.5324
SL 2 A-INR	25.68	0.7215	0.2540	22.15	0.5478	0.5415
MIRE	24.72	0.6308	0.3108	19.57	0.3240	0.6117
MSA	25.50	0.7274	0.2541	22.42	0.5558	0.4416

Table 2: Quantitative results for the image

\times 2

and

\times 4

SR tasks using C 2 -INR (Ours) and other SOTA methods. The best and second-best results are highlighted in bold and underlined .

	30% Pixels Masked			50% Pixels Masked
Methods	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$
DD	35.83	0.9437	0.0369	33.63	0.9276	0.0423
INCODE	34.35	0.8340	0.0453	32.15	0.7981	0.0720
SL 2 A-INR	34.85	0.8477	0.0467	32.44	0.8039	0.2813
MIRE	35.53	0.9398	0.0359	31.49	0.8850	0.0903
MSA	34.16	0.9354	0.0293	31.02	0.8870	0.0797

Table 3: Quantitative results for the image inpainting tasks using C 2 -INR (Ours) and other SOTA methods, with 30% and 50% pixels being randomly masked. The best and second-best results are highlighted in bold and underlined .

Panel of Figure 3 — Figure 3: Qualitative comparisons on the Kodak ( Kodak 2018 ) and Set14 ( Zeyde et al. 2010 ) datasets. The first two rows present results for the image representation task, while the 3rd and 4th rows show results for the image $\times 2$ SR task. Under identical parameter budget, our C 2 -INR consistently produces cleaner images, with noticeable reductions in distortions and artifacts, e.g., the horizontal window mullions and aircraft fuselage paintings in the figures.

Panel of Figure 4 — Figure 4: Qualitative comparison on the Set5 ( Bevilacqua et al. 2012 ) dataset for the task of image inpainting (masking 30% pixels). Our C 2 -INR more effectively leverages inter-pixel correlations to fill missing pixels, preserving the sharpness of the hat’s woven texture and achieving superior visual quality.

Variants	DKA	KRB	AFS	PLF	PSNR $\uparrow$	SSIM $\uparrow$
C 2 -INR*	✗	✗	✗	✗	39.60	0.9634
C 2 -INR †	✓	✗	✗	✗	41.39	0.9736
C 2 -INR ‡	✓	✓	✗	✗	42.60	0.9789
C 2 -INR ⋆	✓	✓	✓	✗	44.03	0.9831
C 2 -INR	✓	✓	✓	✓	44.51	0.9846

Table 4: Ablation study on different modules. The baseline (

*

) denotes the vanilla C 2 -INR without any of our enhancements. The variants

\dagger

\ddagger

, and

\star

correspond to progressive integrations of DKA, KRB, and AFS, respectively, with C 2 -INR representing the full model.

Panel of Figure 5 — Figure 5: Image and error map visualizations for a horizontally dominated example (108k params), comparing horizontal-only, vertical-only, and four-direction kernel settings, suggesting that kernel allocation should adapt to image content.

Kernel Types	Proportions	PSNR $\uparrow$	SSIM $\uparrow$
$3\times 3$	Equal Prop.	42.76	0.9788
$0^{\circ}$ + $90^{\circ}$	Equal Prop.	43.79	0.9835
$45^{\circ}$ + $135^{\circ}$	Equal Prop.	44.03	0.9840
$0^{\circ}$ + $90^{\circ}$ + $45^{\circ}$ + $135^{\circ}$	Equal Prop.	44.30	0.9843
$0^{\circ}$ + $90^{\circ}$ + $45^{\circ}$ + $135^{\circ}$	w.r.t. Grad.	44.20	0.9841
$0^{\circ}$ + $90^{\circ}$ + $45^{\circ}$ + $135^{\circ}$	w.r.t. FT	44.51	0.9846

Table 5: Ablation on different kernel allocation strategies.

Figure 6: Representation quality evolution curves for each activation function selection strategies, implying that our joint AFS and PLF stategies gives the best efficiency.

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Abstract

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