Neighbourhood Operations
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1 Neighbourhood Operations Neighbourhood operations simply operate on a larger neighbourhood o piels than point operations Origin Neighbourhoods are mostly a rectangle around a central piel Any size rectangle and any shape ilter are possible Neighbourhood (, y) y Image (, y)
2 Neighbourhood Operations For each piel in the origin image, the outcome is written on the same location at the target image. Origin Target Neighbourhood (, y) y Image (, y)
3 Simple Neighbourhood Operations Simple neighbourhood operations eample: Min: Set the piel value to the minimum in the neighbourhood Ma: Set the piel value to the maimum in the neighbourhood
4 The Spatial Filtering Process Origin a b c j k l d e * m n o g h i p q r Simple 3*3 Neighbourhood y Image (, y) e 3*3 Filter Original Image Piels Filter (w) e processed = n*e + j*a + k*b + l*c + m*d + o* + p*g + q*h + r*i The above is repeated or every piel in the original image to generate the iltered image
5 Spatial Filtering: Equation Form a a s b b t t y s t s w y g ), ( ), ( ), ( Filtering can be given in equation orm as shown above Notations are based on the image shown to the let
6 Smoothing Spatial Filters One o the simplest spatial iltering operations we can perorm is a smoothing operation Simply average all o the piels in a neighbourhood around a central value Especially useul in removing noise rom images Also useul or highlighting gross detail / 9 / 9 / 9 / 9 / 9 / 9 / 9 / 9 / 9 Simple averaging ilter
7 Smoothing Spatial Filtering Origin / 9 / 9 / * / 9 / 9 / 9 / 9 / 9 / 9 Simple 3*3 Neighbourhood 04 / 00 9 / 9 08 / 9 99 / 06 9 / 3*3 Smoothing 9 98 / 9 / 9 / 9 / 9 Filter y Image (, y) Original Image Piels Filter e = / 9 *06 + / 9 *04 + / 9 *00 + / 9 *08 + / 9 *99 + / 9 *98 + / 9 *95 + / 9 *90 + / 9 *85 = The above is repeated or every piel in the original image to generate the smoothed image
8 Image Smoothing Eample The image at the top let is an original image o size 500*500 piels The subsequent images show the image ater iltering with an averaging ilter o increasing sizes 3, 5, 9, 5 and 35 Notice how detail begins to disappear
9 Image Smoothing Eample
10 Image Smoothing Eample
11 Image Smoothing Eample
12 Image Smoothing Eample
13 Image Smoothing Eample
14 Image Smoothing Eample
15 Weighted Smoothing Filters More eective smoothing ilters can be generated by allowing dierent piels in the neighbourhood dierent weights in the averaging unction Piels closer to the central piel are more important Oten reerred to as a weighted averaging / 6 / 6 / 6 / 6 4 / 6 / 6 / 6 / 6 / 6 Weighted averaging ilter
16 Another Smoothing Eample By smoothing the original image we get rid o lots o the iner detail which leaves only the gross eatures or thresholding Original Image Smoothed Image Thresholded Image
17 Averaging Filter Vs. Median Filter Eample Original Image With Noise Image Ater Averaging Filter Image Ater Median Filter Filtering is oten used to remove noise rom images Sometimes a median ilter works better than an averaging ilter
18 Averaging Filter Vs. Median Filter Eample Original
19 Averaging Filter Vs. Median Filter Eample Averaging Filter
20 Averaging Filter Vs. Median Filter Eample Median Filter
21 Strange Things Happen At The Edges! At the edges o an image we are missing piels to orm a neighbourhood Origin e e e e e e y Image (, y) e
22 Strange Things Happen At The Edges! (cont ) There are a ew approaches to dealing with missing edge piels: Omit missing piels Only works with some ilters Can add etra code and slow down processing Pad the image Typically with either all white or all black piels Replicate border piels Truncate the image
23 Correlation & Convolution The iltering we have been talking about so ar is reerred to as correlation with the ilter itsel reerred to as the correlation kernel Convolution is a similar operation, with just one subtle dierence a b c d e e g h Original Image Piels * r s t u v w y z Filter e processed = v*e + z*a + y*b + *c + w*d + u*e + t* + s*g + r*h For symmetric ilters it makes no dierence
24 Sharpening Spatial Filters Previously we have looked at smoothing ilters which remove ine detail Sharpening spatial ilters seek to highlight ine detail Remove blurring rom images Highlight edges Sharpening ilters are based on spatial dierentiation
25 Spatial Dierentiation Dierentiation measures the rate o change o a unction Let s consider a simple dimensional eample
26 Spatial Dierentiation A B
27 st Derivative The ormula or the st derivative o a unction is as ollows: ( ) ( ) It s just the dierence between subsequent values and measures the rate o change o the unction
28 st Derivative (cont ) Image Strip () st 0 Derivative ()
29 nd Derivative The ormula or the nd derivative o a unction is as ollows: ( ) ( ( ) Simply takes into account the values both beore and ater the current value )
30 nd Derivative (cont ) Image Strip () nd Derivative 0 ()
31 st and nd Derivative Image Strip () st Derivative () nd Derivative 0 ()
32 Using Second Derivatives For Image Enhancement The nd derivative is more useul or image enhancement than the st derivative Stronger response to ine detail Simpler implementation We will come back to the st order derivative later on The irst sharpening ilter we will look at is the Laplacian Isotropic One o the simplest sharpening ilters We will look at a digital implementation
33 The Laplacian The Laplacian is deined as ollows: where the partial st order derivative in the direction is deined as ollows: and in the y direction as ollows: y ), ( ), ( ), ( y y y ), ( ), ( ), ( y y y y
34 The Laplacian (cont ) So, the Laplacian can be given as ollows: [ (, y) (, y) (, y ) (, y )] 4 (, y) We can easily build a ilter based on this
35 The Laplacian (cont ) Applying the Laplacian to an image we get a new image that highlights edges and other discontinuities Original Image Laplacian Filtered Image Laplacian Filtered Image Scaled or Display
36 But That Is Not Very Enhanced! The result o a Laplacian iltering is not an enhanced image We have to do more work in order to get our inal image Subtract the Laplacian result rom the original image to generate our inal sharpened enhanced image g(, y) (, y) Laplacian Filtered Image Scaled or Display
37 Laplacian Image Enhancement - = Original Image Laplacian Filtered Image Sharpened Image In the inal sharpened image edges and ine detail are much more obvious
38 Laplacian Image Enhancement
39 Simpliied Image Enhancement The entire enhancement can be combined into a single iltering operation ), ( ), ( [ ), ( y y y ), ( ), ( y y )], ( 4 y y y g ), ( ), ( ), ( ), ( ), ( 5 y y y ), ( ), ( y y
40 Simpliied Image Enhancement (cont ) This gives us a new ilter which does the whole job or us in one step
41 Simpliied Image Enhancement (cont )
42 Variants On The Simple Laplacian There are lots o slightly dierent versions o the Laplacian that can be used: Simple Laplacian -8 Variant o Laplacian
43 Unsharp Mask & Highboost Filtering Using sequence o linear spatial ilters in order to get Sharpening eect. -Blur - Subtract rom original image - add resulting mask to original image
44 Highboost Filtering
45 st Derivative Filtering Implementing st derivative ilters is diicult in practice For a unction (, y) the gradient o at coordinates (, y) is given as the column vector: G G y y
46 st Derivative Filtering (cont ) The magnitude o this vector is given by: For practical reasons this can be simpliied as: ( ) mag G y G y G G y
47 st Derivative Filtering (cont ) There is some debate as to how best to calculate these gradients but we will use: z z z z z z z z z z z z which is based on these coordinates z z z 3 z 4 z 5 z 6 z 7 z 8 z 9
48 Sobel Operators Based on the previous equations we can derive the Sobel Operators To ilter an image it is iltered using both operators the results o which are added together
49 Sobel Eample Sobel ilters are typically used or edge detection An image o a contact lens which is enhanced in order to make deects (at our and ive o clock in the image) more obvious
50 st & nd Derivatives Comparing the st and nd derivatives we can conclude the ollowing: st order derivatives generally produce thicker edges nd order derivatives have a stronger response to ine detail e.g. thin lines st order derivatives have stronger response to grey level step nd order derivatives produce a double response at step changes in grey level
51 Combining Spatial Enhancement Methods Successul image enhancement is typically not achieved using a single operation Rather we combine a range o techniques in order to achieve a inal result This eample will ocus on enhancing the bone scan to the right
52 Combining Spatial Enhancement Methods (cont ) (a) Laplacian ilter o bone scan (a) (b) Sharpened version o bone scan achieved (c) by subtracting (a) and (b) Sobel ilter o bone scan (a) (d)
53 Image (d) smoothed with a 5*5 averaging ilter The product o (c) and (e) which will be used as a mask (e) Combining Spatial Enhancement Sharpened image which is sum o (a) and () () Methods (cont ) Result o applying a power-law trans. to (g) (g) (h)
54 Combining Spatial Enhancement Methods (cont ) Compare the original and inal images
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