image denoising

Image denoising is the task of recovering the clean image hidden inside a noisy observation. It is fundamentally an ill-posed inverse problem: infinitely many clean images could have produced the same noisy pixels, so you cannot solve it by logic alone, you must assume something about what real images look like. The art of denoising is encoding good priors, that images are mostly smooth, that edges are sparse, that patches repeat across the image, while still letting through genuine detail.

The model is observed = clean + noise, and the central tension is the bias-variance trade-off: smooth too aggressively and you blur away real detail (bias), smooth too little and noise survives (variance). Classical methods span a spectrum. Local linear filters (Gaussian, mean) are fast but blur edges; the median handles impulses; the bilateral and guided filters preserve edges. Total variation (TV) denoising minimizes the total amount of intensity change subject to staying close to the data, favouring piecewise-flat results. The two most powerful classical methods exploit self-similarity: non-local means averages distant patches that look alike, and BM3D groups similar patches into 3D stacks and filters them collaboratively, BM3D remained the benchmark to beat for years.

Since the mid-2010s, learned denoisers have taken the lead. A convolutional network such as DnCNN is trained on pairs of noisy and clean images and often learns to predict the noise itself (residual learning). Self-supervised schemes like Noise2Noise (train on two noisy copies) and Noise2Void (train on a single noisy image) remove the need for clean ground truth. Most recently, diffusion models act as powerful learned image priors that can be plugged into denoising and restoration. Quality is measured by PSNR and SSIM for fidelity, and increasingly by perceptual metrics like LPIPS, because high-PSNR results can look unnaturally smooth.