Week 4: Image Filtering and Edge Detection

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## Image Filtering & Edge Detection

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## So far, we have learnt

1. Loading and accessing image pixels.
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1. Constructing a **histogram** for the image (probability distribution).
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1. Estimating a diagnostic parameters for a region (mean, std, variance).
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1. Color transformations.
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1. Fourier transform.
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1. Histogram Equalization (converting the probability distribution in hand to a uniform distribution).

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## Point operators (transfer functions)

#### Examples

1. Thresholding
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1. Image negation (negative image)
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1. Histogram equalization
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1. Linear scaling
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1. Nonlinear scaling
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1. RGB to Grayscale?

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## Histogram Processing
### Example: Histogram Equalization

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#### Transform the histogram to a uniform one.

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<img style="width:90%" src="../../images/pdf2uniform.png" />

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## Histogram Processing
### Example: Histogram Equalization

* Intensity image (3 bits): [0-7]
* Image size = 64x64 = 4096

### Step 1: compute the discrete PDF (histogram)
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<img style="width:40%" src="../../images/histexample.png" /> <img style="width:40%" src="../../images/pdfexample.png" />

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## Histogram Processing
### Example: Histogram Equalization

### Step 2: compute the discrete CDF (accumulative histogram)

| `i` | accumulative | accumulative x 7 | rounded |
|--|--|--|--|
| `s0` | 0.19 | 1.33 | 1 |
| `s1` | 0.44 | 3.08 | 3 |
| `s2` | 0.65 | 4.55 | 5 |
| `s3` | 0.81 | 5.67 | 6 |
| `s4` | 0.89 | 6.23 | 6 |
| `s5` | 0.95 | 6.65 | 7 |
| `s6` | 0.98 | 6.86 | 7 |
| `s7` | 1.00 | 7.00 | 7 |

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## Histogram Processing
### Example: Histogram Equalization

### Step 2: compute the discrete CDF (accumulative histogram)

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## Histogram Processing
### Example: Histogram Equalization

### Step 3: use the previous table to map the pixels values

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<img style="width:40%" src="../../images/equalizedexample.png" />

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## Histogram Processing
### Example: Histogram Matching

* Intensity image (3 bits): [0-7]
* Image size = 64x64 = 4096
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* Obtain an image with an **arbitrary distribution** instead of a uniform distribution
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* Target distribution

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## Histogram Processing
### Example: Histogram Matching

### Step 1: obtain the scaled histogram equalized values (previous example)

* `s0` = 1   `s1` = 3   `s2` = 5   `s3` = 6
* `s4` = 7   `s5` = 7   `s6` = 7   `s7` = 7

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## Histogram Processing
### Example: Histogram Matching

### Step 2: compute the discrete CDF of the target distribution

| `i` |  accumulative x 7 | rounded |
|--|--|--|
| `z0` | 0.00 | 0 |
| `z1` | 0.00 | 0 |
| `z2` | 0.00 | 0 |
| `z3` | 1.05 | 1 |
| `z4` | 2.45 | 2 |
| `z5` | 4.55 | 5 |
| `z6` | 5.95 | 6 |
| `z7` | 7.00 | 7 |

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## Histogram Processing
### Example: Histogram Matching

### Step 2: compute the discrete CDF of the target distribution

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## Histogram Processing
### Example: Histogram Matching

### Step 3: use the previous table to map the pixels values

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<img style="width:40%" src="../../images/matchedexample.png" />

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## Local operators (filters)
### How to filter an image

#### Convolution

$$\begin{equation}
g(x,y)= (\omega *f)(x,y)=\sum\_{s=-a}^a \sum\_{t=-b}^b w(s,t)f(x-s,y-t)  
\end{equation}
$$

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## 2D Convolution 
<img style="width:100%" src="../../images/2DConv.png" />

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## 2D Convolution Cont'd

Basic Steps are:

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1. Flip the kernel **horizontally and vertically**.
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2. Move over the array with kernel centered at interested point.
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3. Multiply kernel data with overlapped area.
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4. Sum or accumulate the output.

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* If you skip step (1), then we calculate a **Correlation**.

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### Original image

```python
def rgb2gray(rgb_image):
    return np.dot(rgb_image[...,:3], [0.299, 0.587, 0.114])

image = mpimg.imread("images/pegion.jpg")
image_gr = rgb2gray( image )

plt.figure("Original Image", figsize=figureSize)
plt.imshow(image_gr)
```

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## Image Denoising

### Original image

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## Image Denoising Cont'd

### Noisy image

```python`
weight = 0.9
noisy = image_gr + weight * image_gr.std() * np.random.random(image_gr.shape)
plt.figure("Noisy Image", figsize=figureSize)
plt.imshow(noisy)
plt.set_cmap("gray")
```

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## Image Denoising Cont'd

### Noisy image

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### Box = mean = average filter

```python
def box_filter( w ):
    return np.ones((w,w)) / (w*w)
```

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### Box = mean = average filter

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```python
box_filter(1)
```

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\[ 1 \]

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### Box = mean = average filter

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```python
box_filter(3)
```

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| 1/9 | 1/9 | 1/9 |
|-|-|-|
| 1/9 | 1/9 | 1/9 |
| 1/9 | 1/9 | 1/9 |

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### Box = mean = average filter

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```python
box_filter(5)
```

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| 1/25 | 1/25 | 1/25 | 1/25 | 1/25 |
|-|-|-|
| 1/25 | 1/25 | 1/25 | 1/25 | 1/25 |
| 1/25 | 1/25 | 1/25 | 1/25 | 1/25 |
| 1/25 | 1/25 | 1/25 | 1/25 | 1/25 |
| 1/25 | 1/25 | 1/25 | 1/25 | 1/25 |

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### Box = mean = average filter 
#### Using box filter of 1x1

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### Box = mean = average filter 
#### Using box filter of 3x3

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### Box = mean = average filter 
#### Using box filter of 5x5

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### Box = mean = average filter 
#### Using box filter of 7x7

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### Box = mean = average filter 
#### Using box filter of 9x9

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### Gaussian filter 
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#### More robust than box filter
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##### Higher weights to closer pixels

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Derived from 2D gaussian function $$\begin{equation}
h(u,v) = \frac{1}{2 \pi \sigma^2} e^{(- \frac{u^2 + v^2}{\sigma^2})}
\end{equation}
$$

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```python 
import numpy as np
from scipy import signal

def gaussian_kernel( kernlen , std ):
    """Returns a 2D Gaussian kernel array."""
    gkern1d = signal.gaussian(kernlen, std=std).reshape(kernlen, 1)
    gkern2d = np.outer(gkern1d, gkern1d)
    return gkern2d
```

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### Gaussian filter

```python
plt.imshow(gaussian_kernel(21,5), interpolation='none')
```
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$$ \text{size} = 21 \times 21$$
$$ \sigma = 5 $$

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<img style="width:60%" class="center" src="../../images/week3/gaussian_kernel_21x21_std5.png" />

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### Gaussian filter 
#### size = 7x7, std = 0.5

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### Gaussian filter 
#### size = 7x7, std = 1

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### Gaussian filter 
#### size = 7x7, std = 1.5

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### Gaussian filter 
#### size = 7x7, std = 2

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### Gaussian filter 
#### size = 7x7, std = 2.5

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### Median filter (nonlinear)

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<img style="width:70%" class="center" src="../../images/week3/median.gif" />

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* very efficient in removal of salt and pepper noise.