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Neighbourhood operations simply operate on a, Neighbourhoods are mostly a rectangle around a, Any size rectangle and any shape filter are, For each pixel in the origin image, the outcome, Min Set the pixel value to the minimum in the, Max Set the pixel value to the maximum in the, One of the simplest spatial filtering operations, Especially useful in removing noise from images, Also useful for highlighting gross detail, The image at the top left is an original image, The subsequent images show the image after, More effective smoothing filters can be generated, Pixels closer to the central pixel are more, Often referred to as a weighted averaging, By smoothing the original image we get rid of, Filtering is often used to remove noise from, Sometimes a median filter works better than an, There are a few approaches to dealing with, Can add extra code and slow down processing, Typically with either all white or all black, The filtering we have been talking about so far, Convolution is a similar operation, with just one, For symmetric filters it makes no difference, Previously we have looked at smoothing filters, Sharpening spatial filters seek to highlight fine, Differentiation measures the rate of change of a, Lets consider a simple 1 dimensional example, The formula for the 1st derivative of a function, Its just the difference between subsequent, The formula for the 2nd derivative of a function, Simply takes into account the values both before, The 2nd derivative is more useful for image, We will come back to the 1st order derivative, The first sharpening filter we will look at is, where the partial 1st order derivative in the x, So, the Laplacian can be given as follows, We can easily build a filter based on this, Applying the Laplacian to an image we get a new, The result of a Laplacian filtering is not an, We have to do more work in order to get our final, Subtract the Laplacian result from the original, In the final sharpened image edges and fine, The entire enhancement can be combined into a, This gives us a new filter which does the whole, There are lots of slightly different versions of, Using sequence of linear spatial filters in order, Implementing 1st derivative filters is difficult, For a function f(x, y) the gradient of f at, For practical reasons this can be simplified as, There is some debate as to how best to calculate, Based on the previous equations we can derive the, To filter an image it is filtered using both, Sobel filters are typically used for edge, Comparing the 1st and 2nd derivatives we can, 1st order derivatives generally produce thicker, 2nd order derivatives have a stronger response to, 1st order derivatives have stronger response to, 2nd order derivatives produce a double response, Successful image enhancement is typically not, Rather we combine a range of techniques in order, This example will focus on enhancing the bone. 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