Fuzzy Logic Based Edge Detection

Digital Image ProcessingEdge Detection using Dual FIS Optimization

Ishaan Gupta

03914802810

7E123 – E2

Electronics and Communications

MAIT

Mentored By:Prof. Nitin SharmaAssistant ProfessorElectronics and Communications DeptMAIT

What is D.I.P. ?

• Processing of digital images by means of a digital computer.• Output can be image and/or values.• Deals with spatial coordinates, amplitude of ‘f’ at any pair coordinates

(x,y) called gray levels or intensity values which are finite and discrete.

Steps in DIPImage acquisition

Image filtering and enhancement

Image restoration

Color Image Processing.

Wavelets and multi-resolution

Processing.

Compression

Morphological Processing.

Segmentation

Representation & Description.

Object Recognition

Image acquisition

• Done via:• Camera – Visible spectrum• Image matrix (Draw Functions / Convert intensity values -> Image)• Techniques :

• Xray• Gamma• Ultrasound• IR• Satellite (Multi-resolution)

Image filters & enhancement

• Filters – LPF, HPF, BPF, Gaussian Filters, etc.• Depth – Field view, panorama, IR based, Laser Based• Enhancement – Brightness, Contrast, Smoothening, Equalization,

Saturation [RGB Master, R master, G Master, B Master] , etc.

Detection

• Edge• Color• Intensity / gray level - Binary and grayscale Images• Objects and Object description.

Edge detection algos

• Gaussian - Canny’s Algo, Shen-Castan, etc• LoG (Laplacian of Gaussian) – Marr-Hildreth – Second Derivative• Zero Crossing – LoG based• Classical - Prewitt• Classical - Sobel

ComparisonsOperator Advantages Disadvantages

Classical (Sobel,prewitt, Kirsch,…)

Simplicity,Detection of edges and their

orientations

Sensitivity tonoise, Inaccurate

ZeroCrossing(Laplacian, Second

directional derivative)

Detection ofedges and their orientations. Having fixed characteristics in all directions

Responding tosome of the existing edges,

Sensitivity to noise

Laplacian ofGaussian(LoG) (Marr-Hildreth)

Finding thecorrect places of edges, Testing wider

area around the pixel

Malfunctioningat the corners, curves and where the gray level intensity function varies. Not finding the orientation of edge

because of using the Laplacian filter

Gaussian(Canny,Shen-Castan)

Usingprobability for finding error rate,

Localization and response. Improving signal to noise ratio, Better detection

specially in noise conditions

ComplexComputations, False zero crossing,

Time consuming

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Edge Detection

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• Convert a 2D image into a set of curves• Extracts salient features of the scene• More compact than pixels

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Origin of Edges

• Edges are caused by a variety of factors

depth discontinuity

surface color discontinuity

illumination discontinuity

surface normal discontinuity

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Profiles of image intensity edges

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Edge detection

1. Detection of short linear edge segments (edgels)

2. Aggregation of edgels into extended edges• (maybe parametric description)

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Edgel detection

• Difference operators• Parametric-model matchers

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Edge is Where Change Occurs

• Change is measured by derivative in 1D• Biggest change, derivative has maximum magnitude• Or 2nd derivative is zero.

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Image gradient• The gradient of an image:

• The gradient points in the direction of most rapid change in intensity

The gradient direction is given by:

• how does this relate to the direction of the edge?

The edge strength is given by the gradient magnitude

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The discrete gradient• How can we differentiate a digital image f[x,y]?

• Option 1: reconstruct a continuous image, then take gradient• Option 2: take discrete derivative (finite difference)

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The Sobel operator• Better approximations of the derivatives exist

• The Sobel operators below are very commonly used

-1 0 1

-2 0 2

-1 0 1

1 2 1

0 0 0

-1 -2 -1

• The standard defn. of the Sobel operator omits the 1/8 term– doesn’t make a difference for edge detection– the 1/8 term is needed to get the right gradient value, however

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Gradient operators

(a): Roberts’ cross operator (b): 3x3 Prewitt operator(c): Sobel operator (d) 4x4 Prewitt operator

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Effects of noise• Consider a single row or column of the image

• Plotting intensity as a function of position gives a signal

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Solution: Smooth first

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Derivative theorem of convolution• This saves us one operation:

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Laplacian of Gaussian• Consider

Laplacian of Gaussianoperator

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2D edge detection filters

• is the Laplacian operator:

Laplacian of Gaussian

Gaussian derivative of Gaussian

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Optimal Edge Detection: Canny

• Assume: • Linear filtering• Additive iid Gaussian noise

• Edge detector should have:• Good Detection. Filter responds to edge, not noise.• Good Localization: detected edge near true edge.• Single Response: one per edge.

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Optimal Edge Detection: Canny (continued)

• Optimal Detector is approximately Derivative of Gaussian.• Detection/Localization trade-off

• More smoothing improves detection• And hurts localization.

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The Canny edge detector

• original image (Lena)

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The Canny edge detector

norm of the gradient

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The Canny edge detector

thresholding

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The Canny edge detector

thinning(non-maximum suppression)

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Non-maximum suppression

• Check if pixel is local maximum along gradient direction

• requires checking interpolated pixels p and r

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Predictingthe nextedge point

Assume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s).

(Forsyth & Ponce)

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Hysteresis

• Check that maximum value of gradient value is sufficiently large• drop-outs? use hysteresis

• use a high threshold to start edge curves and a low threshold to continue them.

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Effect of (Gaussian kernel size)

Canny with Canny with original

The choice of depends on desired behavior• large detects large scale edges• small detects fine features

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Scale• Smoothing• Eliminates noise edges.• Makes edges smoother.• Removes fine detail.

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fine scalehigh threshold

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coarse scale,high threshold

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coarsescalelowthreshold

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Scale space

• Properties of scale space (w/ Gaussian smoothing)• edge position may shift with increasing scale ()• two edges may merge with increasing scale • an edge may not split into two with increasing scale

larger

Gaussian filtered signal

first derivative peaks

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Edge detection by subtraction

original

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Edge detection by subtraction

smoothed (5x5 Gaussian)

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Edge detection by subtraction

smoothed – original(scaled by 4, offset +128)

Why doesthis work?

filter demo

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Gaussian - image filter

Laplacian of Gaussian

Gaussian delta function

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An edge is not a line...

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Finding lines in an image

• Option 1:• Search for the line at every possible position/orientation• What is the cost of this operation?

• Option 2:• Use a voting scheme: Hough transform

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Finding lines in an image

• Connection between image (x,y) and Hough (m,b) spaces

• A line in the image corresponds to a point in Hough space• To go from image space to Hough space:

• given a set of points (x,y), find all (m,b) such that y = mx + b

x

y

m

b

m0

b0

image space Hough space

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Finding lines in an image

• Connection between image (x,y) and Hough (m,b) spaces• A line in the image corresponds to a point in Hough space• To go from image space to Hough space:

• given a set of points (x,y), find all (m,b) such that y = mx + b

• What does a point (x0, y0) in the image space map to?

x

y

m

b

image space Hough space

– A: the solutions of b = -x0m + y0

– this is a line in Hough space

x0

y0

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Corners contain more edges than lines.

• A point on a line is hard to match.

Corner detection

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Corners contain more edges than lines.• A corner is easier

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Edge Detectors Tend to Fail at Corners

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Finding Corners

Intuition:

• Right at corner, gradient is ill defined.

• Near corner, gradient has two different values.

Fuzzy Logic&

Fuzzy Inference System (FIS)

Introduction to Fuzzy Sets

• Introduced by A. L. Zadeh (1965)• Fuzzy sets provide the mechanism for dealing with imprecise

information• Based and related closely to usage of probability in crisp information.• Provides margin for error and its correction possibilities in both input

and output values.• Takes into account full or partial membership and relationship

between one value to another.

FuzzyInferenceSystem (FIS)

FIS

Types

Mamdani

Sugeno

Components

Membership Functions

IF-THEN Rules

Logical Operations

Applications

DIP

Localizations

Network Analysis

Steps in FIS

FIS Toolbox in MATLAB

Conventional FIS usage in DIP

FIS

(4 In

put)

Image processor Edge

My usage

Image Restoration

Image Enhancement

Noise Removal using

LoGFIS OutDIP2Edge

Detection2DIP 1Edge detection1

Mat2Gray out1

FIS Image processor FIS Image

Processor

Edge Out

Mat2Gray out Filtration Noise

RemovalImage

enhancementImage

restoration

FIS 1

Membership Function of Input

Membership of Output

IF-THEN Rule set = 16

FIS2

Membership Function of Input

Membership of Output

IF-THEN Ruleset = 28 +10

Final Outputs Compared

Thank You