image analysis manipulate an image to extract information to help solve a problem. –preprocessing...
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Image AnalysisImage Analysis• Manipulate an image to extract information
to help solve a problem.– Preprocessing - get rid of unnecessary
information– Data reduction - transform the image to a
useable form– Feature analysis - make inferences about the
image.
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Image AnalysisImage Analysis
• Preprocessing– Noise reduction– Gray level quantization – Spatial quantization– Finding regions of interest–
Reducing the number of bits
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• Data reduction process, in which the image(s) are transformed into a more convenient form.– RGB HSL– Image subtraction– Histogram– Feature extraction
Image AnalysisImage Analysis
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• Feature analysis - Specific results– Blood cell counting– Tumor size and location– 3D model–
Image AnalysisImage Analysis
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• Geometric– Resizing– Rotating
• Noise Reduction– Image Smoothing– Median Filtering
• Edge Detection– Searching for discontinuities
• Histogram slicing• Blob detection
Image AnalysisImage Analysis
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Geometric TransformationGeometric Transformation
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InterpolationInterpolation
• Interpolation to determine image values at integer pixel locations of transformed image. – Numerous interpolation schemes exist
• 2 simple methods are
– Nearest neighbor: assign value of nearest known point
– Bilinear interpolation: example on next slide
Interpolation used to simplify data for further processing
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Nearest Neighbor InterpolationNearest Neighbor Interpolation
Interpolate this point To
this value
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Linear InterpolationLinear Interpolation
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Bi-linear InterpolationBi-linear Interpolation
I(x+1,y)
I(x+1,y+1)
I(x,y+1)
I(x,y)
0 (a,b) 1
I(x’,y) = (1-a)I(x,y) + (a)I(x+1,y)
I(x’,y’) = (1-b)I(x’,y) + (b)I(x’,y+1)
I(x’,y+1) = (1-a)I(x,y+1) + (a)I(x+1,y+1)
I(x’,y’) = (1-b)(1-a)I(x,y) + (1-b)(a)I(x+1,y) + (b)(1-a)I(x,y+1) + (a)(b)I(x,y)
I(x’,y’)
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I(x’,y’)
Bi-linear InterpolationBi-linear Interpolation
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Neighborhood Neighborhood OperationsOperations
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ObjectivesObjectives• Why are neighborhoods important?• What is linear convolution?
– discrete– templates, masks or filters– algorithm mechanics– graphical interpretation
• Describe non-linear operators– maximum– minimum– median
• What is tiling?
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Why are neighbourhoods important?Why are neighbourhoods important?
pixel
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Because…Because…
• Provide context for individual pixels.
• Relationships between neighbors determine image features.
• Neighborhood operations:– noise reduction– edge enhancement– zooming
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Noise reduction Edge Enhancement
Zooming
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Neighbourhood OperationsNeighbourhood Operations
• Linear convolution (*)– A*B*C*D = B*C*D*A = ….
• Non-linear operators– median, max, min, ...
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Convolution versus Spectral• We learnt two methods of processing images:
– Convolution– Spectral
• We analyzed and demonstrated how to build a processor (systolic, pipelined, parallel, cellular automaton) for 1D convolution.– 1D convolution is used in speech processing and in polynomial
multiplication.
• We will use visualized animations now to show in more detail how 2D convolution works for images.
• This should convince you how important it is to do convolution quickly in modern Spectral Architectures, especially for 3D etc.
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2D Convolution2D Convolution
• Consists of filtering an image A using a filter (mask, template) B.
• Mask is a small image whose pixel values are called weights.
• Weights modify relationships between pixels.
We will show more examples of convolution now, especially for 2D data
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A1,1 A1,2 A1,3 A1,4
A2,1 A2,2 A2,3 A2,4
A3,1 A3,2 A3,3 A3,4
A4,1 A4,2 A4,3 A4,4
B1,1 B1,2
B2,1 B2,2
BA C
C1,1 C1,2 C1,3
C3,1
C2,1 C2,2 C2,3
C3,2 C3,3
=
Filter,mask or template
Input image
ConvolvedImage
2 2
4 43 3
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A1,1 A1,2
A2,1 A2,2
A1,3 A1,4
A2,3 A2,4
A3,1 A3,2 A3,3 A3,4
A4,1 A4,2 A4,3 A4,4
B1,1 B1,2
B2,1 B2,2 A1,1B1,1
A2,1B2,1
A1,2B1,
2
A2,2B2,
2
A1,1B1,1C1,1= A1,2B1,
2
A2,1B2,1 A2,2B2,
2
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A1,1 A1,2
A2,1 A2,2
A1,3 A1,4
A2,3 A2,4
A3,1 A3,2 A3,3 A3,4
A4,1 A4,2 A4,3 A4,4
B1,1 B1,2
B2,1 B2,2A1,2B1,
1
A2,2B2,1
A1,3B1,
2
A2,3B2,
2
A1,2B1,1C1,2= A1,3B1,
2
A2,2B2,1 A2,3B2,
2
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A1,1 A1,2
A2,1 A2,2
A1,3 A1,4
A2,3 A2,4
A3,1 A3,2 A3,3 A3,4
A4,1 A4,2 A4,3 A4,4
B1,1 B1,2
B2,1 B2,2A1,3B1,
1
A2,3B2,1
A1,4B1,
2
A2,4B2,
2
A1,3B1,1C1,3= A1,4B1,
2
A2,3B2,1 A2,4B2,
2
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A1,1 A1,2
A2,1 A2,2
A1,3 A1,4
A2,3 A2,4
A3,1 A3,2 A3,3 A3,4
A4,1 A4,2 A4,3 A4,4
B1,1 B1,2
B2,1 B2,2
A2,1B1,
1
A3,1B2,1
A2,2B1,
2
A3,2B2,
2
A2,1B1,1C2,1= A2,2B1,
2
A3,1B2,1 A3,2B2,
2
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A1,1 A1,2
A2,1 A2,2
A1,3 A1,4
A2,3 A2,4
A3,1 A3,2 A3,3 A3,4
A4,1 A4,2 A4,3 A4,4
B1,1 B1,2
B2,1 B2,2B1,1 B1,2
B2,1 B2,2B1,1 B1,2
B2,1 B2,2
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Mathematical NotationMathematical Notation
1 1
1,1,,
Mk
ki
Nl
ljljkijilk BAC
NMB
A1,1B1,1C1,1= A1,2B1,
2
A2,1B2,1 A2,2B2,
2
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ConvolutionConvolution
4 4 7 9
4 3 8 9
3 5 9 9
3 6 10 9
-1 2
-1 2
BA C
=
Filter,mask or template
Input image
ConvolvedImage
2 2
4 43 3
6
9
16
23
26
27
21
19
17
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Convolution sizeConvolution sizeImage size = M1 N1
Mask size = M2 N2
Convolution size = M1- M2 +1 N1-N2+1
N1
N2
N1-N2+1
Typical Mask sizes= 33, 5 5, 77,9 9, 1111
What is the convolved image size for a 128 128 image and 7 7 mask?
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*1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
1 1 11 1 1 1 1 1
=
We convolve with 9*9 averaging filter
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Nonlinear Neighbourhood Nonlinear Neighbourhood OperationsOperations
• Maximum
• Minimum
• Median
1,1,
1:,1:,
1,1,1:,1:
,
1,1,1:,1:
,
median
min
max
ljkijiNlljMkki
lk
ljkijiNlljMkki
lk
ljkijiNlljMkki
lk
BAC
BAC
BAC
We discussed already sorter architecture (three variants – pipelined, butterfly combinational and sequential controller). It can be used for all these operations, and also for other non-linear operators
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61 62
57 60
59 65
63 56
59 55 58 57
49 53 55 45
1 1
1 1C1,2=
62
60
59
636359
Max and Min OperationsMax and Min Operations
63=max, 59=min
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61 62
57 60
59 65
63 56
59 55 58 57
49 53 55 45
C1,2=
Median OperationMedian Operation
1 1
1 1
1
1
1 1 1
62
60
59
63
65
56
55 58 57
62
59
65
60
63
56
57
58
55
987654321
59
rank
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9x9 Median
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Edge Detection
• What do we mean by edge detection?
• What is an edge?
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What is Edge Detection?What is Edge Detection?
• Detects large intensity transitions between pixels
• Redraws the image with only the edges showing
0 0 0 33
0 0 45 78
0 45 23 33
0 0 42 76
0 0 0 38
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What is an Edge?
Edge easy to findEdge easy to find
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What is an Edge?
Where is edge? Single pixel wide or multiple pixels?Where is edge? Single pixel wide or multiple pixels?
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What is an Edge?
Noise: have to distinguish noise from actual edgeNoise: have to distinguish noise from actual edge
Noise Noise is hereis here
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What is an Edge?
Is this one edge or two?Is this one edge or two?
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What is an Edge?
Texture discontinuityTexture discontinuity
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Edge Detection – so what is an Edge Detection – so what is an edge to be detected?edge to be detected?
• What is an edge– A large change in image brightness of a short
spatial distance – Edge strength = (I(x,y)-I(x+dx,y))/dx
But this general definition still allows for many theories, software implementation and hardware architectures.
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Now we will Now we will discuss and discuss and illustrate various illustrate various kinds of filter kinds of filter operatorsoperators
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Edge Detection FiltersEdge Detection Filters
•High - Pass Filtering Eliminates Uniform Regions (Low Frequencies)
•edge “detection” or “enhancement”
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Edge Detection FiltersEdge Detection Filters
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Edge Detection ContinuedEdge Detection Continued
•Sum of Kernel Coefficients = 0
•differences in signs emphasize differences in pixel values
•reduces average image intensity
•Negative pixel values in output?
Edge Detection FiltersEdge Detection Filters
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vertical
horizontal diagonal
Edge DirectionEdge Direction
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Directional High Directional High Pass FiltersPass Filters
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Convolution Edge Convolution Edge Detection using Detection using
Sobel and similar Sobel and similar operatorsoperators
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Example of Sobel Operator
Sobel OperatorSobel Operator
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Sobel Edge DetectionSobel Edge Detection
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Convolution Application ExamplesConvolution Application Examples
--Edge DetectionWe apply Sobel Operator
-1 -2 -10 0 01 2 1
-1 0 1
-2 0 2
-1 0 1
Column Mask
Row Mask
as mask to a sub-field of a picture
-1 2 -1
0 0 0
1 2 1
p0, p1, p2
p3, p4, p5
p6, p7, p8 = (p6-p0)+2(p7-p1)+(p8-p2)
We can learn from the result obviously •The result of the above calculation for column mask is horizontal difference•With Row Mask we will get vertical difference
*
The final step of the convolution equation, dividing by the weight , must be ignored
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--Edge Detection with Sobel Operator
The weight of a mask determines the grey level of the image after convolution.
Like the weight of Sobel Mask WW= (-1) + (-2) + (-1) + 0 + 0 + 0 +1 + 2 +1= 0
The resulting image lost its “lightness” to be dark.dark.
Convolution Application ExamplesConvolution Application Examples
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Sobel OperatorSobel Operator
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Sobel OperatorSobel Operator
-1 -2 -1 0 0 0 1 2 1
-1 0 1-2 0 2 -1 0 1
S1= S2 =
Edge Magnitude =
Edge Direction =
22
21 SS
2
11tanS
S
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SobelPrewitt
Ticbetts
Canny
Comparison of Edge Detection Comparison of Edge Detection AlgorithmsAlgorithms
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Edge DirectionEdge DirectionAssymetric kernels detect edges from Assymetric kernels detect edges from specific directionsspecific directions
EastEast
1 1 -1
1 -2 -1
1 1 -1
NorthEaNorthEastst
1 -1 -1
1 -2 -1
1 1 -1
NorthNorth
-1 -1 -1
1 -2 1
1 1 1
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Robinson Robinson OperatorOperator
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Robinson Compass MasksRobinson Compass Masks
-1 0 1-2 0 2 -1 0 1
0 1 2-1 0 1 -2 -1 0
1 2 1 0 0 0 -1 -2 -1
2 1 0 1 0 -1 0 -1 -2
1 0 -1 2 0 -2 1 1 -1
0 -1 -2-1 0 -1 2 1 0
-1 -2 -1 0 0 0 1 2 1
-2 -1 0-1 0 1 0 1 2
Arrows show edge directions
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Roberts Roberts OperatorOperator
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Roberts OperatorRoberts Operator
• Does not return any information about the orientation of the edge
22 ),1()1,()1,1(),( yxIyxIyxIyxI
),1()1,()1,1(),( yxIyxIyxIyxI
or
1 00 -1
0 1-1 0+
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Prewitt OperatorPrewitt Operator-1 -1 -1 0 0 0 1 1 1
-1 0 1-1 0 1 -1 0 1
P1= P2 =
Edge Magnitude =
Edge Direction =
22
21 PP
2
11tanP
P
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Prewitt Row
Edge Detection FiltersEdge Detection Filters
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0 1 2-1 0 1 -2 -1 0
1 2 1 0 0 0 -1 -2 -1Original
and filtered cow
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Edge Detection Edge Detection (continued)(continued)
• First Order (Gradient) Kernels
• Prewitt RowPrewitt Row
1 0 -11 0 -1
1 0 -11 0 -1
1 0 -11 0 -1
• Sobel RowSobel Row
1 0 -11 0 -1
2 0 -22 0 -2
1 0 -11 0 -1
• Combine Row and Column Operators
Edge Detection Filters: compare Edge Detection Filters: compare Prewitt Prewitt and Sobeland Sobel
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1D Laplacian Operator
)(xf
x
xf
)(
2
22 )(
x
xfxf
first derivative
second derivative
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2D Laplacian Operator2D Laplacian Operator
2
2
2
22 ,,
),(y
yxf
x
yxfyxf
0 -1 0-1 4 -1 0 -1 0
1 -2 1-2 4 -2 1 -2 1
-1 -1 -1-1 8 -1-1 -1 -1
Convolution masks approximating a Laplacian
This is just one example of Laplacian, we can use much larger window
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0 -1 0-1 4 -1 0 -1 0
Input Mask Output
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Image Processing Operations for Early Vision:
Edge Edge DetectionDetection
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Reminder: Effect of FiltersReminder: Effect of Filters
low
high
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EdgesEdges… are the important part of images
intensity
color
edges
textures
contours
condensation...
simplest, least robust
most difficult, most robust There are many letters B occluded
by black shape here. How to find them?
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Edge Detection
• Edges are curves in the image plane across which there is a “significant” change in image brightness.
• The goal of edge detection is the construction of an idealized line drawing
Image Processing OperationsImage Processing Operations
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Pixels on edgesPixels on edges
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Edges foundEdges found
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Edge effects: rarely ideal edgesEdge effects: rarely ideal edges
Not all information is created equal...
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Causes of edgesCauses of edges
• Depth discontinuity– One surface occludes another
• Surface orientation discontinuity– the edge of a block
• reflectance discontinuity– texture or color changes
• illumination discontinuity– shadows
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Edges: causesEdges: causesWhat are they? Why?
four physical events that cause image edges...
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What are they? Why?
discontinuities in
• surface color/intensity• surface normal• depth• lighting (specularities)
four physical events that cause image edges...
Edges: causesEdges: causes
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Edges are image locations with a local maximum in image gradient in the direction of that gradient
(steepness)
Edges: causesEdges: causes
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Edge DetectionEdge Detection
• Finding simple descriptions of objects in complex images– find edges
– interrelate edges
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Examples of edgesExamples of edges
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•One idea to detect edges is to differentiate the image and look for places where the brightness undergoes a sharp change•Consider a 1-D example. Below is an intensity profile for a 1-D image.
Edges are not ideal... Edges are not ideal...
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•Below we have the derivative of the previous graph.
•Here we have a peak at x=18, x=50 and x=75.•These errors are due to the presence of noise in the image.
Edge DetectionEdge Detection
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Finding EdgesFinding EdgesImage Intensity along a line
First derivative of intensity
Smoothed via convolving with gaussian
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• This problem is countered by convolving a smoothing function along with the differentiation operation.
• The mathematical concept of convolution allows us to perform many useful image-processing operations.
Edge DetectionEdge Detection
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• One standard form of smoothing is to use a Gaussian function.
• Now using the idea of convolving with the Gaussian functionwe can revisit the 1-D example.
Image Processing OperationsImage Processing Operations
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• With the convolving applied we can more easily see the edge at x=50.
Using convolving we are able to discover where edges are located and this allows us to make an accurate line drawing.
Convolving to find edgesConvolving to find edges
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•Here is an example of using convolving in an 2-D picture of Mona Lisa
Edge Detection by ConvolutionEdge Detection by Convolution
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edge
Second derivative
derivative
Zero CrossingZero Crossing
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EdgeEdge
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Edge ParametersEdge Parameters
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Remainder: How the Point Remainder: How the Point Detection Mask operates on one Detection Mask operates on one
color imagecolor image
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Goal: to find regions of an image with locally maximal gradient magnitude.(1) Smooth the image to reduce the effects of noise
Replace each pixel by a weighted sum of its neighbors.
I(x,y) = wij I(x+i,y+j) i,j = -1
1
old Imageweight “mask” new
Image
oldnewweights
4 22
2 11
2 11
x =
(scaled)
34 37 137 148
29 46 141 140
34 130 149 131
41 142 152 144
35 210 210 210
43 210 210 210
60 210 210 210
72 210 210 210
Edge Detection uses ConvolutionEdge Detection uses Convolution
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Goal: to find regions of an image with locally maximal gradient magnitude.(1) Smooth the image to reduce the effects of noise
It’s possible to do this one dimension at a time...
Replace each pixel by a weighted sum of its neighbors.
I(x,y) = wij I(x+i,y+j) i,j = -1
1
old Imageweight “mask” new
Image
oldnewweights
4 22
2 11
2 11
x =
(scaled)
34 37 137 148
29 46 141 140
34 130 149 131
41 142 152 144
35 62 116 143
43 76 122 141
60 102 136 140
72 112 145 143
smoothed vertically
smoothed horizontally
original image
Edge Detection uses ConvolutionEdge Detection uses Convolution
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Goal: to find regions of an image with locally maximal gradient magnitude.(1) Smooth the image to reduce the effects of noise
(2) Estimate the gradient at each pixel
I(x,y) = wij I(x+i,y+j) i,j = -1
1
oldnewweights
Replace each pixel by a weighted sum of its neighbors.
Use another mask of weights -- this time to approximate taking derivatives.
In the x direction
In the y direction
-1 1
-1
1
with new weights
same procedure
(convolution)
= Iy= Ixddx
I ddy
I
Edge Detection uses ConvolutionEdge Detection uses Convolution
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Goal: to find regions of an image with locally maximal gradient magnitude.(1) Smooth the image to reduce the effects of noise
(2) Estimate the gradient at each pixel
(3) Find the gradient magnitude and thin the resulting lines.
I(x,y) = wij I(x+i,y+j) i,j = -1
1
oldnewweights
Replace each pixel by a weighted sum of its neighbors.
Use another mask of weights -- this time to approximate taking derivatives.
-1 1-1
1
same procedure with new weights (convolution)
x
y
I(x,y) = sqrt(Ix + Iy ) 22
Seek out maxima in the gradient direction .
= atan2(Ix, Iy)
Edge Detection uses ConvolutionEdge Detection uses Convolution
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Goal: to find regions of an image with locally maximal gradient magnitude.(1) Smooth the image to reduce the effects of noise
(2) Estimate the gradient at each pixel
(3) Find the gradient magnitude and thin the resulting information.
(4) Choose a threshold -- any gradients above it classify as edges.
I(x,y) = wij I(x+i,y+j) i,j = -1
1
oldnewweights
Replace each pixel by a weighted sum of its neighbors.
Use another mask of weights -- this time to approximate taking derivatives.
I(x,y) = sqrt(Ix + Iy ) 22
-1 1-1
1
same procedure with new weights (convolution)
x
y
In the gradient direction, look for local maxima: = atan2(Ix, Iy)
Edge Detection uses ConvolutionEdge Detection uses Convolution
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Theory of Theory of Gradient Gradient
Based Edge Based Edge DetectionDetection
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Formal Model of Edge Formal Model of Edge
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Formal Model of Edge Formal Model of Edge
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Formal Model of Edge (cont)
Formal Model of Edge Formal Model of Edge
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Formal Model of Edge (cont) Formal Model of Edge Formal Model of Edge
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Formal Model of Edge (cont) Formal Model of Edge Formal Model of Edge
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Formal Model of Edge (cont) Formal Model of Edge: Roberts Formal Model of Edge: Roberts
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Formal Model of Edge (cont) Formal Model of Edge: Laplacian and Marr-Formal Model of Edge: Laplacian and Marr-Hildreth Hildreth
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Formal Model of Edge (cont) Formal Model of Edge Formal Model of Edge
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Formal Model of Edge (cont) Formal Model of Edge Formal Model of Edge
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ThresholdsThresholds
very high thresholdoriginal image
Thresholds are important, done before or during edge detection.
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Thresholds
very high thresholdoriginal image
ThresholdsThresholds
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very high thresholdoriginal image
ThresholdsThresholds
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very high threshold
reasonable
original image
ThresholdsThresholds
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very high threshold
too low !reasonablethis all takes time...
original image
ThresholdsThresholds