## Revision 2e11cce90a6c9e960d8243c1465c3a872ba3b2f5 (click the page title to view the current version)

# Filters

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--- 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 ![Lenna with Salt and Pepper Noise](Images/lenna-snp.png) + 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. + Non-linear filters must 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. As the image, we can use [lenna-awgn.png](Images/lenna-awgn.png) with Gaussian noise or [lenna-snp.png](Images/lenna-snp.png) with Salt and Pepper noise. If we want to use convolution instead of 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 ```