image enhancement spatial filtering

 Image enhancement in the spatial domain can be represented as: gm,n = Tfm,n  The transformation T maybe linear or nonlinear..  The mask operation can be implemented in MATLAB using

  Prof Duong Anh Duc

 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

Enhanced Image

Transformation

Given Image

 If the operator T is linear and shift invariant (LSI), characterized by the point-spread sequence (PSS) h(m,n) , then (recall convolution)

 In practice, to reduce computations, h(m, n) is of “finite extent:

h(k, l) = 0 for (k, l)  where  is a small set (called neighborhood)  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

 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

 Examples of some common filters (1-D case):

Lowpass filter Highpass filter

 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:

o The mask is truncated at the border (free boundary)

o 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)

o The boundaries “wrap around” (periodic boundary)

 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

 Image smoothing refers to any image-to-image transformation

designed to “smooth” or flatten the image by reducing the rapid to-pixel variation in grayvalues

pixel- Smoothing filters are used for:

o Blurring: This is usually a preprocessing step for removing small

(unwanted) details before extracting the relevant (large) object, bridging gaps in lines/curves,

o Noise reduction: Mitigate the effect of noise by linear or nonlinear

operations

 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

Original Image Avg Mask

N = 3 N = 7

N = 11 N = 21

Noise-free Image

Zero-mean Gaussian noise, Variance = 0.01

Zero-mean Gaussian noise, Variance = 0.05

 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

 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

Gaussian noise: s = 0.2 Salt & Pepper noise: prob = 0.2

MSE = 0.0075 MSE = 0.0125

Output of 3x3 Averaging filter

MSE = 0.0089 MSE = 0.0042

Output of 3x3 Median filter

 This involves highlighting fine details or enhancing details that have been blurred

 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

 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))

Highpass filtering

Original Image

 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) = A f(m,n) – lowpass(f(m,n))

 This is also referred to as unsharp masking

Highpass filtering

Original Image

 Averaging tends to blur details in an image Averaging involves

 The magnitude of the gradient is:

 Discrete approximations to the magnitude of the gradient is normally used

 Consider the following image region:

 We may use the approximation:

z1 z2 z3

 This can implemented using the masks:

 As follows:

 Alternatively, we may use the approximation:

 This can implemented using the masks:

 As follows:

 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

 Better approximations to the gradient can be obtained by:

 This can be implemented using the masks:

as follows:

 Another approximation is given by the masks:

 The resulting masks are called Sobel operators