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# Local image enhancement and image-smoothing spatial filters

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

## 1   Local enhancement

What if we just want to enhance an image locally?

For example, only dark or light areas?

### 1.1   $n$th moment of $r$ about its mean

$\mu_n(r)=\sum_{i=0}^{L-1}(r_i-m)^n p(r_i)$ where

• $r$: Image
• $L$: Gray levels
• $m_r$: Mean gray level or $m_r=\sum_{i=0}^{L-1}r_ip(r_i)$
• $p(r_i)$: Probability of gray level $i$ or $p(r_i)=\frac{n_i}{n}$
• $n_i$: Number of pixels with gray level $i$
• $n$: Total number of pixels

### 1.2   Global statistics of $r$

$\mu_0(r)=1$ $\mu_1(r)=0$ $\sigma_r^2=\mu_2(r)=\sum_{i=0}^{L-1}(r_i-m)^2p(r_i)$

### 1.3   Local statistics of $r$

$m_{S_{xy}}=\sum_{(s,t)\in S_{xy}}r_{s,t}p(r_{s,t})$ $\sigma_{S_{xy}}^2=\sum_{(s,t)\in S_{xy}}(r_{s,t}-m_{S_{xy}})^2p(r_{s,t})$ where

• $S_{xy}$: Neighborhood of pixel at $(x,y)$
• $(s,t)$: Pixels in $S_{xy}$
• $r_{s,t}$: Gray level of pixel $(s,t)$

### 1.4   Conditions for the enhancement of local dark areas

Locally darker than the entire image $m_{S_{xy}}\leq k_0m_r\qquad 0<k_0<1$

Locally lower contrast than the entire image, but not constant areas $k_1\sigma_r\leq\sigma_{S_{xy}}\leq k_2\sigma_r\qquad 0<k_1<k_2<1$

Make those pixels that meet the above criteria brighter $g(x,y)=E\cdot f(x,y)\qquad E>1$

### 1.5   Transformation for local dark areas

$g(x,y)= \begin{cases} E\cdot f(x,y)& \text{if }m_{S_{xy}}\leq k_0m_r\text{ and }k_1\sigma_r\leq\sigma_{S_{xy}}\leq k_2\sigma_r\\ f(x,y)& \text{otherwise} \end{cases}$ where

• $0<k_0<1$
• $0<k_1<k_2<1$
• $E>1$

## 2   Exercise: Shadow enhancement using local enhancement

1. Local statistics
• Neighborhood width: 3
• Neighborhood height: 3
• Local mean: local_3x3_mean.tif
• Local standard deviation: local_3x3_std.tif
2. Local enhance
• Raster layer: map.tif
• Local mean: local_3x3_mean.tif
• Local standard deviation: local_3x3_std.tif
• Gray-level multiplier: 4
• Mean parameter: 0.4
• Lower standard deviation parameter: 0.02
• Upper standard deviation parameter: 0.4
• Output TIFF: local_3x3_enhanced.tif

Try 5x5!

## 3   Linear spatial filtering

A filter or mask is a $m\times n$ subimage with coefficients $w(s,t)$.

An $m\times n$ neighborhood determines the value of the center pixel.

$m$ and $n$ are odd integers.

### 3.1   Convolution

$g(x,y)=\sum_{s=-a}^a\sum_{t=-b}^b w(s,t)f(x+s,y+t)$ where

• $a=\frac{m-1}{2}$
• $b=\frac{n-1}{2}$
• $w(s,t)$: Weight
• $f(x,y)$: Image

## 4   Smoothing spatial filters

Smoothing/blurring requires the “convolution” or “integration” of neighborhood pixel values and filter coefficients.

### 4.1   Smoothing linear filters

Weighted average of

### 4.2   Order-statistics filters

Non-linear percentile filters

• Median for noise reduction
• Max filter for detecting the brightest points
• Min filter for detecting the darkest points

## 5   Exercise: Smoothing

• Weighted average
• Raster layer: map.tif
• Neighborhood width: 3
• Neighborhood height: 3
• Weights: 1 1 1 1 1 1 1 1 1
• Output: average_3x3.tif
• Weighted average
• Raster layer: map.tif
• Neighborhood width: 3
• Neighborhood height: 3
• Weights: 1 2 1 2 4 2 1 2 1
• Output: weighted_3x3.tif

## 6   Exercise: Noise reduction

1. Local statistics
• Raster layer: noisy_1.tif
• Neighborhood width: 3
• Neighborhood height: 3
• Local median: local_3x3_median.tif

## 7   Homework: Local image enhancement

Write the transformation equation that enhances local bright areas just like the one for local dark areas in the last slide.