Revision 47b5cb064f06e97ccc753a7419c8f8bfa4f24eb4 (click the page title to view the current version)
Changes from 47b5cb064f06e97ccc753a7419c8f8bfa4f24eb4 to 90d2802d33ffff9c011064134f8ae99b387113a0
---
title: Filters
categories: lecture
---
# What are filters
Today we will investigate the use of blurring filters on images.
This is just one of many applications of filters in image processing,
and in the next session we will use the same basic techniques for
feature detection.
Blurring filters do exactly what it says on the tin. They blur the
image. This has the effect of smoothing out noise, so it is a useful
technique for noise removal.
One of the groups using machine vision in the final year project
last year had problems with the camera failing to return a value in
some pixel. The result was black pixels, so-called pepper noise.
A blurring filter solves this problem very effectively.
If you have read signal processing or image processing previously,
all of this may be familiar to you already.
## Conceptual Application
+ Noisy image. Some pixels are just off.
+ Recalculate pixels using a neighbourhood
+ Use a matrix of coefficients for a weighted average
+ You could implement this with a fairly simple for-loop,
but it would be slow.
## Convolution
$$(f*g)[x] = \sum_{k=-\infty}^{\infty} f[k]g[x-k]$$
+ Note the convolution of a sampled signal and a continuous one
+ [Demo](https://youtu.be/HW4IamyQnzw)
**Note** OpenCV uses correlation instead of convolution.
Conceptually this is different, but in practice, the only effect
is to mirror the filter.
$$(f\star g)[x] = \sum_{k=-\infty}^{\infty} f[k]g(k+x)$$
## Non-linear filtering
+ Convolution is a linear operation
+ Simple and fast
+ Non-linear filters may be better for denoising.
+ However, we need linear filters when we move on to edge detection.
+ Non-linear filters for be for another course.
## Blurring in Python
```python
def normaliseImage(frame):
grey = cv.cvtColor(frame, cv.COLOR_BGR2GRAY).astype(np.float64)
grey /= 255
return grey
```
```python
frame = cv.imread("lenna.png")
grey = normaliseImage(frame)
f7 = np.ones((7,7)) / 49
blurred = cv.filter2D(grey,-1,f7)
```
The -1 means the same depth is used for output as for input.
If we want to use convolution instead of auto-correlation, we have
(requiring extra imports)
```python
snp1 = sig.convolve2d(snp,f7)
```
## Gaussian filter
Above, we used a uniform filter (`f7`).
```python
shape = (5,5)
sigma = 0.5
m,n = [(ss-1.)/2. for ss in shape]
y,x = np.ogrid[-m:m+1,-n:n+1]
h = np.exp( -(x*x + y*y) / (2.*sigma*sigma) )
h /= h.sum()
```
This gives a different matrix which we can use as a blurring filter.
```
Out[39]:
array([[6.96247819e-08, 2.80886418e-05, 2.07548550e-04, 2.80886418e-05, 6.96247819e-08],
[2.80886418e-05, 1.13317669e-02, 8.37310610e-02, 1.13317669e-02, 2.80886418e-05],
[2.07548550e-04, 8.37310610e-02, 6.18693507e-01, 8.37310610e-02, 2.07548550e-04],
[2.80886418e-05, 1.13317669e-02, 8.37310610e-02, 1.13317669e-02, 2.80886418e-05],
[6.96247819e-08, 2.80886418e-05, 2.07548550e-04, 2.80886418e-05, 6.96247819e-08]])
```
Note, large numbers in the middle, smaller towards the edges. Since
we divided by the sum in the last line, the entries now add to one.
## Generating Noisy Images for Testing
```python
def addGaussNoise(image,var=0.02):
row,col = image.shape
mean = 0
sigma = var**0.5
gauss = np.random.normal(mean,sigma,(row,col))
gauss = gauss.reshape(row,col)
noisy = image + gauss
return noisy
```
```python
def addSaltPepperNoise(image,amount=0.01):
row,col = image.shape
s_vs_p = 0.5
out = np.copy(image)
# Salt mode
num_salt = np.ceil(amount * image.size * s_vs_p)
coords = [np.random.randint(0, i - 1, int(num_salt))
for i in image.shape]
out[tuple(coords)] = 1
# Pepper mode
num_pepper = np.ceil(amount* image.size * (1. - s_vs_p))
coords = [np.random.randint(0, i - 1, int(num_pepper))
for i in image.shape]
out[tuple(coords)] = 0
return out
```
