1. Digital Image Processing
Image Enhancement
Spatial Filtering
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Duong Anh Duc - Digital Image Processing

2. Image Enhancement: Spatial Filtering
Image enhancement in the spatial domain can be represented as:
g(m,n) = T(f)(m,n)
The transformation T maybe linear or nonlinear. We will mainly study linear operators T but will see one important nonlinear operation.
Transformation
Enhanced
Image
Given Image
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3. Duong Anh Duc - Digital Image Processing
How to specify T
If the operator T is linear and shift invariant (LSI), characterized by the point-spread sequence (PSS) h(m,n) , then (recall convolution)
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Duong Anh Duc - Digital Image Processing

4. Duong Anh Duc - Digital Image Processing
How to specify T
In practice, to reduce computations, h(m,n) is of “finite extent:
h(k,l) = 0 for (k,l)  
where D is a small set (called neighborhood). D is also called as the support of h.
In the frequency domain, this can be represented as:
G(u,v) = He(u,v) Fe(u,v)
where He(u,v) and Fe(u,v) are obtained after appropriate zeropadding.
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Duong Anh Duc - Digital Image Processing

5. Duong Anh Duc - Digital Image Processing
How to specify T
Many LSI operations can be interpreted in the frequency domain as a “filtering operation.” It has the effect of filtering frequency components (passing certain frequency components and stopping others).
The term filtering is generally associated with such operations.
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Duong Anh Duc - Digital Image Processing

6. Duong Anh Duc - Digital Image Processing
How to specify T
Examples of some common filters (1-D case):
Lowpass filter Highpass filter
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7. Duong Anh Duc - Digital Image Processing
If h(m, n) is a 3 by 3 mask given by
then w1 w2 w3 h= w4 w5 w6 w7 w8 w9
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Duong Anh Duc - Digital Image Processing

8. Duong Anh Duc - Digital Image Processing
The output g(m, n) is computed by sliding the mask over each pixel of the image f(m, n). This filtering procedure is sometimes referred to as moving average filter.
Special care is required for the pixels at the border of image f(m, n). This depends on the so-called boundary condition. Common choices are:
The mask is truncated at the border (free boundary)
The image is extended by appending extra rows/columns at the boundaries. The extension is done by repeating the first/last row/column or by setting them to some constant (fixed boundary).
The boundaries “wrap around” (periodic boundary).
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Duong Anh Duc - Digital Image Processing

9. Duong Anh Duc - Digital Image Processing
In any case, the final output g(m, n) is restricted to the support of the original image f(m, n).
The mask operation can be implemented in MATLAB using the filter2 command, which is based on the conv2 command.
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Duong Anh Duc - Digital Image Processing

10. Duong Anh Duc - Digital Image Processing
Smoothing Filters
Image smoothing refers to any image-to-image transformation designed to “smooth” or flatten the image by reducing the rapid pixel-to-pixel variation in grayvalues.
Smoothing filters are used for:
Blurring: This is usually a preprocessing step for removing small (unwanted) details before extracting the relevant (large) object, bridging gaps in lines/curves,
Noise reduction: Mitigate the effect of noise by linear or nonlinear operations.
10/13/2018
Duong Anh Duc - Digital Image Processing

11. Image smoothing by averaging (lowpass spatial filtering)
Smoothing is accomplished by applying an averaging mask.
An averaging mask is a mask with positive weights, which sum to 1. It computes a weighted average of the pixel values in a neighborhood. This operation is sometimes called neighborhood averaging.
Some 3 x 3 averaging masks:
This operation is equivalent to lowpass filtering.
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12. Example of Image Blurring
Original Image
Avg. Mask
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13. Example of Image Blurring
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14. Example of Image Blurring
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15. Example of noise reduction
Noise-free Image
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16. Example of noise reduction
Zero-mean Gaussian noise, Variance = 0.01
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Duong Anh Duc - Digital Image Processing

17. Example of noise reduction
Zero-mean Gaussian noise, Variance = 0.05
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18. Duong Anh Duc - Digital Image Processing
Median Filtering
The averaging filter is best suited for noise whose distribution is Gaussian:
The averaging filter typically blurs edges and sharp details.
The median filter usually does a better job of preserving edges.
Median filter is particularly suited if the noise pattern exhibits strong (positive and negative) spikes. Example: salt and pepper noise.
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Duong Anh Duc - Digital Image Processing

19. Duong Anh Duc - Digital Image Processing
Median Filtering
Median filter is a nonlinear filter, that also uses a mask. Each pixel is replaced by the median of the pixel values in a neighborhood of the given pixel.
Suppose A ={a1, a2, …, ak} are the pixel values in a neighborhood of a given pixel with a1  a2  … ak. Then
Note: Median of a set of values is the “center value,” after sorting.
For example: If A ={0,1,2,4,6,6,10,12,15} then median(A) = 6.
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20. Example of noise reduction
Gaussian noise: s = 0.2 Salt & Pepper noise: prob. = 0.2
MSE =
MSE = 0.062
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21. Example of noise reduction
Output of 3x3 Averaging filter
MSE =
MSE =
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22. Example of noise reduction
Output of 3x3 Median filter
MSE =
MSE =
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23. Duong Anh Duc - Digital Image Processing
Image Sharpening
This involves highlighting fine details or enhancing details that have been blurred.
Before
After Sharpening
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24. Basic highpass spatial filtering
This can be accomplished by a linear shift-invariant operator, implemented by means of a mask, with positive and negative coefficients.
This is called a sharpening mask, since it tends to enhance abrupt graylevel changes in the image.
The mask should have a positive coefficient at the center and negative coefficients at the periphery. The coefficients should sum to zero.
10/13/2018
Duong Anh Duc - Digital Image Processing

25. Basic highpass spatial filtering
Example:
This is equivalent to highpass filtering.
A highpass filtered image g can be thought of as the difference between the original image f and a lowpass filtered version of f :
g(m,n) = f(m,n) – lowpass(f(m,n))
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Duong Anh Duc - Digital Image Processing

26. Duong Anh Duc - Digital Image Processing
Example
Original Image
Highpass filtering
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27. High-boost filtering
This is a filter whose output g is produced by subtracting a lowpass (blurred) version of f from an amplified version of f
g(m,n) = Af(m,n) – lowpass(f(m,n))
This is also referred to as unsharp masking.
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Duong Anh Duc - Digital Image Processing

28. Duong Anh Duc - Digital Image Processing
High-boost filtering
Observe that
g(m,n) = Af(m,n) – lowpass(f(m,n))
= (A-1)f(m,n) + f(m,n) – lowpass(f(m,n))
= (A-1)f(m,n) + hipass(f(m,n))
For 1 > A , part of the original image is added back to the highpass filtered version of f.
The result is the original image with the edges enhanced relative to the original image.
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29. Duong Anh Duc - Digital Image Processing
Example
Original Image
Highpass filtering
High-boost filtering
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30. Duong Anh Duc - Digital Image Processing
Derivative filter
Averaging tends to blur details in an image. Averaging involves summation or integration.
Naturally, differentiation or “differencing” would tend to enhance abrupt changes, i.e., sharpen edges.
Most common differentiation operator is the gradient:
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31. Duong Anh Duc - Digital Image Processing
Derivative filter
The magnitude of the gradient is:
Discrete approximations to the magnitude of the gradient is normally used.
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32. Duong Anh Duc - Digital Image Processing
Derivative filter
Consider the following image region:
We may use the approximation: z1 z2 z3 z4 z5 z6 z7 z8 z9
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33. Duong Anh Duc - Digital Image Processing
Derivative filter
This can implemented using the masks:
As follows:
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34. Duong Anh Duc - Digital Image Processing
Derivative filter
Alternatively, we may use the approximation:
This can implemented using the masks:
As follows:
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35. Duong Anh Duc - Digital Image Processing
Derivative filter
The resulting maks are called Roberts cross-gradient operators.
The Roberts operators and the Prewitt/Sobel operators (described later) are used for edge detection and are sometimes called edge detectors.
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36. Example: Roberts cross-gradient operator
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37. Example: Roberts cross-gradient operator
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38. Duong Anh Duc - Digital Image Processing
Prewitt operators
Better approximations to the gradient can be obtained by:
This can be implemented using the masks:
as follows:
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39. Duong Anh Duc - Digital Image Processing
Sobel operators
Another approximation is given by the masks:
The resulting masks are called Sobel operators.
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40. Duong Anh Duc - Digital Image Processing
Example
Prewitt
Sobel
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Duong Anh Duc - Digital Image Processing