Convolutional Neural Networks (CNNs) achieve strong denoising performance by exploiting spatial context from neighboring pixels. Deep Image Prior (DIP) leverages this property to restore images from a single noisy input without requiring large datasets. However, the over-parameterized architecture of DIP often leads to noise fitting during optimization. In this paper, we propose Pool-DIP, a convolution-free architecture that incorporates pooling-based contrast modeling to capture spatial context efficiently. Pool-DIP improves denoising performance while significantly reducing the number of parameters and computational complexity compared to convolution-based DIP models. Experimental results show that Pool-DIP achieves competitive performance across multiple datasets, including a real-world benchmark. Spectral analysis further reveals that Pool-DIP stabilizes the evolution of high-frequency components during optimization and suppresses erroneous high-frequency signals. The proposed architecture also generalizes well to other image restoration tasks such as super-resolution and inpainting.