EE409_4_Contrast_Enhancement_c

EE409 – Digital Image Processing 2017-2018 Spring

4. Image Enhancement in Spatial Domain c) Image Derivatives, Edge Detection & Sharpening

by: Dr. Gökhan Koray GÜLTEKİN

Edges  Edges are those places in an image that correspond to object boundaries.  Edges are pixels where image brightness changes abruptly.

Brightness vs. Spatial Coordinates

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Edges 

Ideal edge

A(x) x

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It is usually ramped because of sensor processing during capture Noisy edge

A(x) x

A(x) x

Edge Detection 

Edge information in an image is found by looking at the relationship

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a pixel has with its neighborhoods. 

If a pixel’s gray-level value is similar to those around it, there is probably not an edge at that point.

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If a pixel’s has neighbors with widely varying gray levels, it may present

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an edge point.

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Edges in 2-D Images 

Direction of gradient at any point is the direction of maximum change.

Edge Detection Methods 

Many are implemented with convolution mask and based on discrete approximations to differential operators.

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Differential operations measure the rate of change in the image brightness function.

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Some operators return orientation information.

Other only return information about the existence of an edge at each point.

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Image Derivatives:

1st Order Derivative (Gradient) and

2nd Order Derivative (Laplacian)

Derivative(Gradient) of a Function  Derivative gives the rate of change(slope) of a function  Definition of 1 dimensional, discrete, 1st order derivative:  df / dx = f(x+1) – f(x)  Derivative operator: -1

f(x+1) f(x)

1 x

x+1

 The 2nd order derivative is the derivative of the 1st order derivative…

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Derivatives of a 1D Signal  

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First derivative: dA /dx

A x 

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Second derivative, d2A/dx2,generates two peaks at beginning and end of edge

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dA dx

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d2A dx 2

Called ‘ringing’

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Discrete Derivatives 200

1D Signal:

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1st derivative:

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2nd derivative:

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Spatial Differentiation  Differentiation measures the rate of change of a function  Let’s consider a simple 1 dimensional differentiation example using this image:

Spatial Differentiation A

B

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First and Second Order Derivative

A

B

First and Second Order Derivative

 1st order generally produces thicker edges  2nd order shows stronger response to detail  1st order generally response stronger to gray level step  2nd order produce double (pos/neg) response at step change

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Horizontal and Vertical Gradients(derivative) -1 -1

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Roberts (±45°) Edge Detector:

Sobel (0°,90°) Edge Detector:

Sobel Edge Detector Detects vertical edges

To 1.

2.

3.

Detects horizontal edges

find edge strength(magnitude) and direction(angle); Find the horizontal gradient (derivative) Find the vertical gradient (derivative) Find magnitude and angle from gradients

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Edge Strength & Direction To 1.

2.

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find edge strength(magnitude) and direction(angle); Find the horizontal gradient (derivative) Find the vertical gradient (derivative) Find magnitude and angle from gradients

Edge Detection

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2nd Order Derivative (Laplacian) The 2nd order derivative is the derivative of the 1st order derivative‌ F(x-1)

pixels

F(x)

F(x+1)

F(x+1) - F(x)

F(x+2) - F(x+1)

derive F(x) - F(x-1) 1st derivative derive [ F(x+1)-F(x) ] – [ F(x)-F(x-1) ] 2nd derivative F(x+1) -2F(x)+F(x-1)

2nd Order Derivative (Laplacian) One dimensional 2nd order derivative in x direction: F(x+1)-F(x) – (F(x)-F(x-1))= F(x+1) -2F(x) +F(x-1) 1

-2

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One dimensional 2nd order derivative, in y direction:

1 -2 1

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2nd Order Derivative (Laplacian) Two dimensional 2nd order derivative: 0

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=

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This mask is called the ‘LAPLACIAN’ (remember calculus ?)

Laplacian Masks

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Calculating Laplacian in Matlab imd = im2double(im); % create laplacian h=[1 1 1;1 -8 1; 1 1 1]; % Filter: imlap=conv2(imd,h); % Normalize and show imshow(imlap,[]);

Notice: Derivatives may generate negative values

Laplacian Example

In Matlab: ‘saturn.png’

 1 1 1  1 8 1    1 1 1

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What is Edge-enhancement? 

Physcophysical experiments indicate that an image with sharp edges is often more subjectively pleasing than the original image

How can we enhance edges?  

Q: What is a measure of the strength of an edge?

A: How steep it is

A(x)

x

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Now we will see methods to enhance (sharpen) edges

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Sharpening in General:

High Boost Filtering and

Unsharp Masking

Sharpening (High Pass) Spatial Filters 

Previously we have looked at smoothing filters which remove fine detail

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Sharpening spatial filters seek to highlight fine detail

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Highlight edges (High Boost)

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Remove blurring from images (Unsharp Masking)

Sharpening filters are based on spatial differentiation

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High Boost Sharpening Edges detected by the Laplacian can be used to sharpen the image:

Image

+ Laplacian

Sharpened Image

In Matlab: ‘moon.tif’

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High Boost Sharpening

High Boost Sharpening Filters Laplacian

Sobel

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High Boost Sharpening Example: High boost sharpening in MATLAB imd=im2double(im); h=[1 1 1;1 -8 1; 1 1 1]; imlap=imfilter(imd,h); figure; imshow(imd) figure; imshow(imd+imlap) Note: imlap=imfilter(imd,h, ‘replicate‘) This command will apply border padding instead of zero padding

High Boost Sharpening Sharpening can be done in 1 pass: 0

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Original Image

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Sharpened Image

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High Boost Sharpening High Boost Sharpening Masks:

High Boost Sharpening

 1 1 1  1 8 1    1 1 1

 1 1 1  1 9 1    1 1 1

A high-pass filter (Laplacian)

A high-boost filter 36

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High Boost Sharpening (Using Sobel) imd=im2double(im); h1=[1 2 1;0 0 0; -1 -2 -1];

% horizontal sobel mask

h2=h1';

% vertical sobel mask

p1=imfilter(imd,h1); % apply horiz. sobel filter on image p2=imfilter(v,h2); % apply vert. sobel filter on image p3=abs(p1)+abs(p2); % Add abs. Sobel edges imshow(imd),figure,imshow(p3+imd); % Add sobel edges to image

Unsharp Masking Basic Idea (unsharp masking): Subtract a BLURRED version of an image from the image itself !

Fsharp = F – Fblurred

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Unsharp Masking Attenuate low frequencies Fsharp = a*F – Fblurred , a>=1

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Unsharp Masked (sharpened) Image

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Original Image (multiplied by a)

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Averaging Filtered Image

Unsharp Masking Example: Unsharp masking sharpening in MATLAB imd=im2double(im); h=ones(3,3); imavg=imfilter(imd,h); figure; imshow(imd) figure; imshow(2*imd-imavg)

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Unsharp Masking Different Examples of Sharpening Filters: 0

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Laplacian + Image (High Boost) 0

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Notice: Normalization may be needed after these operations

Questions

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1e7

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a=1e7+1

?

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