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.

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