CS558 Computer Vision

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# CS558 Computer Vision - PowerPoint PPT Presentation

CS558 Computer Vision. Lecture IV : Image Filter and Edge Detection. Slides adapted from S. Lazebnik. Recap of Lecture III. Light and Shade Radiance and irradiance Radiometry of thin lens Bidirectional reflectance distribution function (BRDF) Photometric stereo Color What is color?

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### CS558 Computer Vision

Lecture IV: Image Filter and Edge Detection

Recap of Lecture III
• Bidirectional reflectance distribution function (BRDF)
• Photometric stereo
• Color
• What is color?
• Human eyes
• Trichromacy and color space
• Color perception
Outline
• Image filter and convolution
• Convluationand linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector
Outline
• Image filter and convolution
• Convolution and linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector
Motivation: Image denoising
• How can we reduce noise in a photograph?

1

1

1

1

1

1

1

1

1

“box filter”

Moving average
• Let’s replace each pixel with a weighted average of its neighborhood
• The weights are called the filter kernel
• What are the weights for the average of a 3x3 neighborhood?

Source: D. Lowe

Outline
• Image filter and convolution
• Convolution and linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector

f

Defining convolution
• Let f be the image and g be the kernel. The output of convolving f with g is denoted f *g.
• MATLAB functions: conv2, filter2, imfilter

Source: F. Durand

Key properties
• Linearity:filter(f1 + f2) = filter(f1) + filter(f2)
• Shift invariance: same behavior regardless of pixel location: filter(shift(f)) = shift(filter(f))
• Theoretical result: any linear shift-invariant operator can be represented as a convolution
Properties in more detail
• Commutative: a * b = b * a
• Conceptually no difference between filter and signal
• Associative: a * (b * c) = (a * b) * c
• Often apply several filters one after another: (((a * b1) * b2) * b3)
• This is equivalent to applying one filter: a * (b1 * b2 * b3)
• Distributes over addition: a * (b + c) = (a * b) + (a * c)
• Scalars factor out: ka * b = a * kb = k (a * b)
• Identity: unit impulse e = […, 0, 0, 1, 0, 0, …],a * e = a
Annoying details
• What is the size of the output?
• MATLAB: filter2(g, f, shape)
• shape = ‘full’: output size is sum of sizes of f and g
• shape = ‘same’: output size is same as f
• shape = ‘valid’: output size is difference of sizes of f and g

full

same

valid

g

g

g

g

f

f

f

g

g

g

g

g

g

g

g

Annoying details
• What about near the edge?
• the filter window falls off the edge of the image
• need to extrapolate
• methods:
• clip filter (black)
• wrap around
• copy edge
• reflect across edge

Source: S. Marschner

Annoying details
• What about near the edge?
• the filter window falls off the edge of the image
• need to extrapolate
• methods (MATLAB):
• clip filter (black): imfilter(f, g, 0)
• wrap around: imfilter(f, g, ‘circular’)
• copy edge: imfilter(f, g, ‘replicate’)
• reflect across edge: imfilter(f, g, ‘symmetric’)

Source: S. Marschner

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Practice with linear filters

?

Original

Source: D. Lowe

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Practice with linear filters

Original

Filtered

(no change)

Source: D. Lowe

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Practice with linear filters

?

Original

Source: D. Lowe

0

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Practice with linear filters

Original

Shifted left

By 1 pixel

Source: D. Lowe

1

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1

1

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1

1

1

Practice with linear filters

?

Original

Source: D. Lowe

1

1

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Practice with linear filters

Original

Blur (with a

box filter)

Source: D. Lowe

0

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2

1

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Practice with linear filters

-

?

(Note that filter sums to 1)

Original

Source: D. Lowe

0

1

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Practice with linear filters

-

Original

• Sharpening filter
• Accentuates differences with local average

Source: D. Lowe

Sharpening

Source: D. Lowe

=

detail

smoothed (5x5)

original

+

=

original

detail

sharpened

Sharpening
• What does blurring take away?

Smoothing with box filter revisited
• What’s wrong with this picture?
• What’s the solution?

Source: D. Forsyth

Smoothing with box filter revisited
• What’s wrong with this picture?
• What’s the solution?
• To eliminate edge effects, weight contribution of neighborhood pixels according to their closeness to the center

“fuzzy blob”

Outline
• Image filter and convolution
• Convluationand linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector
Gaussian Kernel
• Constant factor at front makes volume sum to 1 (can be ignored when computing the filter values, as we should renormalize weights to sum to 1 in any case)

0.003 0.013 0.022 0.013 0.003

0.013 0.059 0.097 0.059 0.013

0.022 0.097 0.159 0.097 0.022

0.013 0.059 0.097 0.059 0.013

0.003 0.013 0.022 0.013 0.003

5 x 5,  = 1

Source: C. Rasmussen

Gaussian Kernel
• Standard deviation : determines extent of smoothing

σ = 2 with 30 x 30 kernel

σ = 5 with 30 x 30 kernel

Source: K. Grauman

Choosing kernel width
• The Gaussian function has infinite support, but discrete filters use finite kernels

Source: K. Grauman

Choosing kernel width
• Rule of thumb: set filter half-width to about 3σ
Gaussian filters
• Remove “high-frequency” components from the image (low-pass filter)
• Convolution with self is another Gaussian
• So can smooth with small- kernel, repeat, and get same result as larger- kernel would have
• Convolving two times with Gaussian kernel with std. dev. σis same as convolving once with kernel with std. dev.
• Separable kernel
• Factors into product of two 1D Gaussians

Source: K. Grauman

*

=

=

*

Separability example

2D convolution(center location only)

The filter factorsinto a product of 1Dfilters:

Perform convolutionalong rows:

Followed by convolutionalong the remaining column:

Source: K. Grauman

Why is separability useful?
• What is the complexity of filtering an n×n image with an m×mkernel?
• O(n2 m2)
• What if the kernel is separable?
• O(n2 m)
Noise
• Salt and pepper noise: contains random occurrences of black and white pixels
• Impulse noise: contains random occurrences of white pixels
• Gaussian noise: variations in intensity drawn from a Gaussian normal distribution

Source: S. Seitz

Gaussian noise
• Mathematical model: sum of many independent factors
• Good for small standard deviations
• Assumption: independent, zero-mean noise

Source: M. Hebert

Reducing Gaussian noise

Smoothing with larger standard deviations suppresses noise, but also blurs the image

Reducing salt-and-pepper noise

3x3

5x5

7x7

• What’s wrong with the results?
Alternative idea: Median filtering
• A median filter operates over a window by selecting the median intensity in the window
• Is median filtering linear?

Source: K. Grauman

Median filter
• What advantage does median filtering have over Gaussian filtering?
• Robustness to outliers

Source: K. Grauman

Median filter

Median filtered

Salt-and-pepper noise

• MATLAB: medfilt2(image, [h w])

Source: M. Hebert

Gaussian vs. median filtering

3x3

5x5

7x7

Gaussian

Median

Sharpening revisited

Source: D. Lowe

=

detail

smoothed (5x5)

original

+ α

=

original

detail

sharpened

Sharpening revisited
• What does blurring take away?

unit impulse

Gaussian

Laplacian of Gaussian

image

unit impulse(identity)

blurredimage

Outline
• Image filter and convolution
• Convolution and linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector
Application: Hybrid Images
• A. Oliva, A. Torralba, P.G. Schyns, “Hybrid Images,” SIGGRAPH 2006
Application: Hybrid Images

Gaussian Filter

• A. Oliva, A. Torralba, P.G. Schyns, “Hybrid Images,” SIGGRAPH 2006

Laplacian Filter

Outline
• Image filter and convolution
• Convoluation and linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector
Edge detection
• Goal: Identify sudden changes (discontinuities) in an image
• Intuitively, most semantic and shape information from the image can be encoded in the edges
• More compact than pixels
• Ideal: artist’s line drawing (but artist is also using object-level knowledge)

Source: D. Lowe

Origin of edges
• Edges are caused by a variety of factors:

surface normal discontinuity

depth discontinuity

surface color discontinuity

illumination discontinuity

Source: Steve Seitz

intensity function(along horizontal scanline)

first derivative

edges correspond toextrema of derivative

Characterizing edges
• An edge is a place of rapid change in the image intensity function

image

Outline
• Image filter and convolution
• Convoluation and linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector
Derivatives with convolution

For 2D function f(x,y), the partial derivative is:

For discrete data, we can approximate using finite differences:

To implement above as convolution, what would be the associated filter?

Source: K. Grauman

Partial derivatives of an image

-1 1

1 -1

-1 1

or

Which shows changes with respect to x?

Finite difference filters
• Other approximations of derivative filters exist:

Source: K. Grauman

• The gradient of an image:

The gradient points in the direction of most rapid increase in intensity

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

The gradient direction is given by

The edge strength is given by the gradient magnitude

Source: Steve Seitz

Outline
• Image filter and convolution
• Convolution and linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector

Where is the edge?

Effects of noise
• Consider a single row or column of the image
• Plotting intensity as a function of position gives a signal

Source: S. Seitz

g

f * g

• To find edges, look for peaks in
Solution: smooth first

f

Source: S. Seitz

f

Derivative theorem of convolution
• Differentiation is convolution, and convolution is associative:
• This saves us one operation:

Source: S. Seitz

Derivative of Gaussian filter
• Are these filters separable?

y-direction

x-direction

Derivative of Gaussian filter
• Which one finds horizontal/vertical edges?

y-direction

x-direction

Scale of Gaussian derivative filter
• Smoothed derivative removes noise, but blurs edge. Also finds edges at different “scales”

1 pixel

3 pixels

7 pixels

Source: D. Forsyth

Review: Smoothing vs. derivative filters
• Smoothing filters
• Gaussian: remove “high-frequency” components; “low-pass” filter
• Can the values of a smoothing filter be negative?
• What should the values sum to?
• One: constant regions are not affected by the filter
• Derivative filters
• Derivatives of Gaussian
• Can the values of a derivative filter be negative?
• What should the values sum to?
• Zero: no response in constant regions
• High absolute value at points of high contrast
Outline
• Image filter and convolution
• Convolution and linear filter
• Gaussian filter, box filter, and median filter
• Application: Hybrid images
• Edge detection
• Image derivatives
• Gaussian derivative filters
• Canny edge detector
The Canny edge detector

original image

The story of Lena

Slide credit: Steve Seitz

The Canny edge detector

The Canny edge detector

How to turn these thick regions of the gradient into curves?

thresholding

Non-maximum suppression

Check if pixel is local maximum along gradient direction, select single max across width of the edge

requires checking interpolated pixels p and r

The Canny edge detector

Problem: pixels along this edge didn’t survive the thresholding

thinning

(non-maximum suppression)

Hysteresis thresholding
• Use a high threshold to start edge curves, and a low threshold to continue them.

Source: Steve Seitz

high threshold

(strong edges)

low threshold

(weak edges)

hysteresis threshold

Hysteresis thresholding

original image

Source: L. Fei-Fei

Recap: Canny edge detector
• Filter image with derivative of Gaussian
• Find magnitude and orientation of gradient
• Non-maximum suppression:
• Thin wide “ridges” down to single pixel width
• Define two thresholds: low and high
• Use the high threshold to start edge curves and the low threshold to continue them
• MATLAB: edge(image, ‘canny’);

J. Canny, A Computational Approach To Edge Detection, IEEE Trans. Pattern Analysis and Machine Intelligence, 8:679-714, 1986.

Edge detection is just the beginning…

human segmentation

image