Hands-on: MLE and MAP Denoising
Use the same noisy photograph. Change the weight of a smoothness prior and compare the restoration.
Each prior has its own weight range. Equal λ values do not represent equal smoothing strength.
Objective: ‖x − y‖² + λR(x)
Prior belief: neighboring pixels usually have similar values.
- Set λ = 0. Is any noise removed?
- Increase λ for each prior. Compare noise reduction and lost detail, especially at edges.
- Change σ. Would you choose the same λ?
- Hide the reference. How would you judge the result without knowing the original?
Experiment details
The images are actual optimization results, not illustrative retouching. Noise is independent Gaussian noise; the same random sample is scaled for the three noise levels. Results are precomputed. Quadratic uses FFT to solve the objective directly; TV uses 700 iterations of a primal-dual optimization algorithm. Both use periodic boundary conditions. TV is isotropic, applied separately to each RGB channel: R(x) = Σ √((Δhorizontal x)² + (Δvertical x)²). Pixel values are clipped only for display. λ absorbs the relative scale of the likelihood and prior. The prior is a designer’s smoothness assumption; it was not learned from this photo. Larger λ does not automatically give a better image.
Photo: existing CSED105 course materials. The clean photograph is used only to simulate noise and evaluate error.