Digital Image Processing Image Enhancement Part IV

Nonlinear Filtering Nonlinear Filters –Cannot be expressed as convolution –Cannot be expressed as frequency shaping “Nonlinear” Means Everything (other than linear) –Need to be more specific –Often heuristic –We will study some “nice” ones

Impulsive (Salt & Pepper) Noise with probability p a with probability p b with probability 1 - p a - p b noisy pixels clean pixels X : noise-free image, Y : noisy image Definition –Each pixel in an image has a probability p a or p b of being contaminated by a white dot (salt) or a black dot (pepper) add salt & pepper noise

Order-Statistics Filters nonlinear spatial filters response is based on ordering (ranking) the pixels contained in the image area encompassed by the filter replacing of the center pixel with the value determined by the ranking result

Median Filtering Median filter replaces the pixel value by the median value in the neighborhood The principal function is to force distinct gray level points to be more like their neighbors. excellent noise-reduction capabilities with less blurring than linear smoothing filters effective for impulse noise (salt-and-pepper noise)  Min: Set the pixel value to the minimum in the neighbourhood  Max: Set the pixel value to the maximum in the neighbourhood cf.) max filter → R = max {z k | k = 1,2,…,9} min filter → R = min {z k | k = 1,2,…,9}

Order-Statistics Filters –Given a set of numbers Denote the OS as such that Applying Median Filters to Images –Use sliding windows (similar to spatial linear filters) –Typical windows: 3x3, 5x5, 7x7, other shapes …… middle value max value min value

Median Filters originalnoisy ( p a = p b = 0.1 ) median filtered 3x3 windowmedian filtered 5x5 window From MATLAB sample images

Iterative Median Filters From [Gonzalez & Woods] Idea: repeatedly apply median filters 1 time 2 times 3 times

Switching Median Filters From [Wang & Zhang] Motivation –Regular median filters change both “bad” and “good” pixels Idea –Detect/classify “bad” and “good” pixels –Filter “bad” pixels only

originalnoisy ( p a = p b = 0.1 ) regular 5x5 median filteredswitching 5x5 median filtered Switching Median Filters From MATLAB sample images

Order Statistics (OS) Filters Recall Order Statistics: For OS such that OS filter: General Form Special Cases where ( M +1)-th

Order Statistics (OS) Filters Note: An OS Filter is Uniquely Defined by {w i } Example 1: Example 2: ( M +1)-th M -th( M +2)-th then

Examples A 4x4 grayscale image is given by 1)Filter the image with a 3x3 median filter, after zero- padding at the image borders zero-padding median filtering impulse?

Examples 2)Filter the image with a 3x3 median filter, after replicate- padding at the image borders replicate -padding median filtering impulse cleaned!

Examples 3)Filter the image with a 3x3 OS filter, after replicate- padding at the image borders. The weighting factors of the OS filter are given by {w i | i = 1, …, 9} = {0, 0, 0, ¼, ½, ¼, 0, 0, 0} replicate -padding OS filtering

Gradient  the term gradient is used for a gradual blend of colour which can be considered as an even gradation from low to high values  At each image point, the gradient vector points in the direction of largest possible intensity increase,  the length of the gradient vector corresponds to the rate of change in that direction.  Two types of gradients, with blue arrows to indicate the direction of the gradient

Sobel operators Represents a rather inaccurate approximation of the image gradient The operator calculates the gradient of the image intensity at each point Giving the direction of the largest possible increase from light to dark and the rate of change in that direction The result therefore shows how "abruptly" or "smoothly" the image changes at that point

Sobel Example

Grayscale image of a brick wall & a bike rack scale image of a brick wall & a bike rack

Combining Spatial Enhancement Methods Successful image enhancement is typically not achieved using a single operation Rather we combine a range of techniques in order to achieve a final result This example will focus on enhancing the bone scan to the right

Combining Spatial Enhancement Methods Laplacian filter of bone scan (a) Sharpened version of bone scan achieved by subtracting (a) and (b) Sobel filter of bone scan (a) (a) (b) (c) (d)

Combining Spatial Enhancement Methods Images taken from Gonzalez & Woods, Digital Image Processing (2002) The product of (c) and (e) which will be used as a mask Sharpened image which is sum of (a) and (f) Result of applying a power-law trans. to (g) (e) (f) (g) (h) Image (d) smoothed with a 5*5 averaging filter

Combining Spatial Enhancement Methods  Compare the original and final images Images taken from Gonzalez & Woods, Digital Image Processing (2002)

assignments  Chapter 3  1, 6, 10, 12, 18, 22, 23.