4/30/2020

Digital Image Processing

Image Enhancement Spatial Filtering

Duong Anh Duc - Digital Image Processing 1

Image Enhancement: Spatial Filtering

 Image enhancement in the spatial domain can be represented as: Transformation Enhanced Image

g(m,n) = T(f)(m,n)

Given Image  The transformation

T

maybe linear or nonlinear. We will mainly study linear operators

T

but will see one important nonlinear operation. 4/30/2020 Duong Anh Duc - Digital Image Processing 2

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) 4/30/2020 Duong Anh Duc - Digital Image Processing 3

How to specify T

   In practice, to reduce computations,

h(m,n)

of “finite extent: where 

h

(

k,l

)

= 0

for (

k,l

)   is a small set (called neighborhood).  is is also called as the support of

h

. In the frequency domain, this can be represented as:

G

(

u,v

) =

H e

(

u,v

)

F e

(

u,v

) where

H e

(

u,v

) and

F e

(

u,v

) are obtained after appropriate zeropadding.

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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. 4/30/2020 Duong Anh Duc - Digital Image Processing 5

How to specify T

 Examples of some common filters (1-D case): Lowpass filter Highpass filter 4/30/2020 Duong Anh Duc - Digital Image Processing 6

 If

h

(

m

,

n

) is a 3 by 3 mask given by

w 1 w 2 w 3 h

=

w 4 w 5 w 6 w 7 w 8 w 9

then 4/30/2020 Duong Anh Duc - Digital Image Processing 7

  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

). 4/30/2020 Duong Anh Duc - Digital Image Processing 8

 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. 4/30/2020 Duong Anh Duc - Digital Image Processing 9

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. 4/30/2020 Duong Anh Duc - Digital Image Processing 10

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. 4/30/2020 Duong Anh Duc - Digital Image Processing 11

Example of Image Blurring

4/30/2020 Original Image Duong Anh Duc - Digital Image Processing Avg. Mask 12

Example of Image Blurring

4/30/2020

N

= 3 Duong Anh Duc - Digital Image Processing

N

= 7 13

Example of Image Blurring

4/30/2020

N

= 11 Duong Anh Duc - Digital Image Processing

N

= 21 14

Example of noise reduction

4/30/2020 Noise-free Image Duong Anh Duc - Digital Image Processing 15

Example of noise reduction

4/30/2020 Zero-mean Gaussian noise, Variance = 0.01

Duong Anh Duc - Digital Image Processing 16

Example of noise reduction

4/30/2020 Zero-mean Gaussian noise, Variance = 0.05

Duong Anh Duc - Digital Image Processing 17

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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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

={

a 1

,

a 2

, …,

a k

} are the pixel values in a neighborhood of a given pixel with

a 1



a 2

 … 

a k

. Then   Note: Median of a set of values is the “center value,” after sorting.

For example: If

A

6.

={0,1,2,4,6,6,10,12,15} then median(

A

) = 4/30/2020 Duong Anh Duc - Digital Image Processing 19

Example of noise reduction

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

4/30/2020 MSE = 0.0337 MSE = 0.062 Duong Anh Duc - Digital Image Processing 20

Example of noise reduction

Output of 3x3 Averaging filter 4/30/2020 MSE = 0.0075

MSE = 0.0125

Duong Anh Duc - Digital Image Processing 21

Example of noise reduction

Output of 3x3 Median filter 4/30/2020 MSE = 0.0089

MSE = 0.0042

Duong Anh Duc - Digital Image Processing 22

Image Sharpening

 This involves highlighting fine details or enhancing details that have been blurred.

4/30/2020 Before After Sharpening Duong Anh Duc - Digital Image Processing 23

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.

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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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Example

4/30/2020 Original Image Highpass filtering Duong Anh Duc - Digital Image Processing 26

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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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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Example

Original Image 4/30/2020 Highpass filtering High-boost filtering Duong Anh Duc - Digital Image Processing 29

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: 4/30/2020 Duong Anh Duc - Digital Image Processing 30

Derivative filter

 The magnitude of the gradient is:  Discrete approximations to the magnitude of the gradient is normally used. 4/30/2020 Duong Anh Duc - Digital Image Processing 31

Derivative filter

 Consider the following image region:

z 1 z 2 z 3 z 4 z 5 z 6



z 7 z 8 z 9

We may use the approximation: 4/30/2020 Duong Anh Duc - Digital Image Processing 32

Derivative filter

 This can implemented using the masks:  As follows: 4/30/2020 Duong Anh Duc - Digital Image Processing 33

Derivative filter

 Alternatively, we may use the approximation:  This can implemented using the masks:  As follows: 4/30/2020 Duong Anh Duc - Digital Image Processing 34

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.

35 4/30/2020 Duong Anh Duc - Digital Image Processing

Example: Roberts cross-gradient operator

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Example: Roberts cross-gradient operator

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Prewitt operators

 Better approximations to the gradient can be obtained by:  This can be implemented using the masks: as follows: 4/30/2020 Duong Anh Duc - Digital Image Processing 38

Sobel operators

 Another approximation is given by the masks:  The resulting masks are called

Sobel operators

. 4/30/2020 Duong Anh Duc - Digital Image Processing 39

Example

4/30/2020 Prewitt Duong Anh Duc - Digital Image Processing Sobel 40