# Logic and arithmetic image operations

Institute for Environmental and Spatial Analysis...University of North Georgia

## 1   Bitwise logic operations

Mainly used for masking (selecting features to enhance)

### 1.1   NOT

$f(x,y)$NOT
01
10

### 1.2   AND

AND01
000
101
• $f$ AND 0xff?
• NOT ($f$ AND 0xff)?

OR01
001
111

XOR01
001
110

## 2   Exercise: Shadow enhancement using logic operations

1. Log the map: log_map
• Brighten everywhere, not just the shadow
• Identify the range of pixel values
• 255 for the shadow, 0 for the rest
• What range works best?

## 3   Arithmetic operations

### 3.1   Subtraction

Difference/motion detection

$g(x,y)=f(x,y)-h(x,y)$

### 3.2   Averaging

Let $f(x,y)$ be a noiseless image, but we have $g(x,y)=f(x,y)+\eta(x,y)$ where $\eta(x,y)$ is noise with a zero mean. For example, $\eta\sim\mathcal{N}(0,\sigma_\eta^2)$.

How can we enhance $g(x,y)$ and make it closer to $f(x,y)$?

### 3.3   Averaging: Need more noisy images?

Obtain more noisy images $g_i(x,y)$ for $1\leq i\leq K$.

### 3.4   Averaging: Take the average of these images

The average of these noisy images \begin{split} \bar{g}(x,y)&=\frac{1}{K}\sum_{i=1}^K g_i(x,y)\\ &=f(x,y)+\frac{1}{K}\sum_{i=1}^K\eta_i(x,y). \end{split}

### 3.5   Average: The expected value of the average

The expected value of $\bar{g}(x,y)$ $E\left[\bar{g}(x,y)\right]=f(x,y)$ because $f(x,y)$ is constant and $E\left[\frac{1}{K}\sum_{i=1}^K\eta_i(x,y)\right]=0$.

More important, $\sigma_{\bar{g}(x,y)}=\frac{1}{\sqrt{K}}\sigma_\eta(x,y),$ which means the more $g_i(x,y)$, the smaller error.

## 4   Exercise: Noise reduction using averaging

1. Add noise to the map: noisy_map_1
• Probability of noise: 0.5
• Max noise: 20
2. Repeat step 1 * 4 more times: noisy_map_2, ..., noisy_map_5
3. Subtract the map from noisy_map_1: diff_1
4. Average noisy_map_1 and noisy_map_2: average_map_2
5. Subtract the map from average_map_2: diff_2
6. Repeat steps 4-5 * 3 more times by adding one more noisy map each time
7. Compare the histograms of diff_1, ..., diff_5