Question 5 of 6: Application — Restoration System Design

Question 5: Application — Restoration System Design (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A degraded image with four simultaneous defects: blur, poor contrast, low brightness, and high-frequency multiplicative noise.

Find. An ordered image-restoration pipeline that corrects all four defects without one stage undoing or worsening another.

Approach. Handle the noise first (in a domain where it becomes additive and therefore linearly filterable), fix illumination/contrast next, and leave the noise-sensitive sharpening step for last.

1. Log transform. Multiplicative noise $g(x,y) = f(x,y)\cdot n(x,y)$ cannot be removed by an ordinary linear (additive-noise) filter without also blurring the signal. Taking logarithms converts the multiplicative degradation into an additive one: $$\ln g(x,y) = \ln f(x,y) + \ln n(x,y)$$ As a side benefit, the log transform compresses the image's dynamic range, which directly helps the dark, low-contrast complaint by boosting the (relatively) darker pixel values.
2. Noise reduction in the log domain. With the noise now additive, a linear smoothing/Wiener filter suppresses it in the usual way, exploiting the noise's high-frequency character (a low-pass or frequency-domain Wiener filter attenuates exactly the band the noise occupies while leaving the lower-frequency image structure intact).
3. Exponential transform. Undo step 1: $\hat{f}(x,y) = \exp(\widehat{\ln f}(x,y))$, returning to the (now denoised) intensity domain.
4. Contrast and brightness enhancement. Adaptive histogram equalization (CLAHE) or a gamma correction $s = c\, r^{\gamma}$ (with $\gamma < 1$ to brighten and expand contrast in the dark range) is applied now, on an already-denoised image, so that stretching the intensity range does not simultaneously stretch and amplify residual noise texture the way it would if applied to the raw, noisy input.
5. Deblurring / sharpening — done last. Only once the image is denoised and correctly exposed is unsharp masking (or, if the blur's point-spread function is known, a Wiener deconvolution) applied to restore edge sharpness. Sharpening is a high-pass operation and is the step most sensitive to noise amplification, so it is deliberately sequenced after denoising and contrast correction rather than before or in between — sharpening a still-noisy or badly-exposed image would amplify noise and clipping artifacts instead of genuine edges.