Administrivia Homework 2 due Tue., Feb. 23 before class Linearity - - PowerPoint PPT Presentation

CMPSCI 370: Intro to Computer Vision

Image processing: #3 linear filtering continued …

University of Massachusetts, Amherst February 18, 2016 Instructor: Subhransu Maji 1

• Homework 2 due Tue., Feb. 23 before class
• Linearity of light
• Color constancy
• Hybrid images
• Today’s lecture
• Review of last lecture
• Edge detection

Administrivia

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Joys of computer vision research

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http://xkcd.com/1425/

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• How can we reduce noise in a photograph?

Motivation: Image de-noising

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• 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?

Moving average

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1 1 1 1 1 1 1 1 1 “box filter”

Source: D. Lowe

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• Let f be the image and g be the kernel. The output of

convolving f with g is denoted f * g.

∑

− − = ∗

l k

l k g l n k m f n m g f

,

] , [ ] , [ ] , )[ (

Convolution

6 Source: F. Durand

• MATLAB functions: conv2, filter2, imfilter

Convention: 
 kernel is “flipped” for convolution

f

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Gaussian vs. box filtering

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Gaussian smoothing fails to get rid of salt-and-pepper noise

Reducing salt-and-pepper noise

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3x3 5x5 7x7

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• A median filter operates over a window by selecting the

median intensity in the window

Alternative idea: Median filtering

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The median filtering is not linear

Source: K. Grauman

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• What advantage does median filtering have over Gaussian

filtering?

• Answer: Robustness to outliers

Median filter

10 Source: K. Grauman

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MATLAB: medfilt2(image, [h w])

Salt-and-pepper noise Median filtered

Source: M. Hebert

Median filter

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Gaussian vs. median filtering

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3x3 5x5 7x7 Gaussian Median

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What does blurring take away?

Sharpening revisited

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• riginal

smoothed (5x5)

–

detail

=

sharpened

=

Let’s add it back:

• riginal

detail

+ α

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Sharpening filter

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Gaussian unit impulse Laplacian of Gaussian

I = blurry(I) + sharp(I) sharp(I) = I − blurry(I) = I ∗ e − I ∗ gσ = I ∗ (e − gσ)

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• A. Oliva, A. Torralba, P.G. Schyns, 


“Hybrid Images,” SIGGRAPH 2006

Application: Hybrid Images

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Gaussian Filter Laplacian Filter

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motorcycle and bicycle

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dolphin and car

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Homework 2, part 3

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Next: edge detection

20 Winter in Kraków photographed by Marcin Ryczek

But first, any questions?

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• 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)

Edge detection

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Source: D. Lowe

Attneave's Cat (1954)

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Edges are caused by a variety of factors:

Origin of edges

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depth discontinuity surface color discontinuity illumination discontinuity surface normal discontinuity

Source: Steve Seitz

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

function

Edge detection

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image intensity function
 (along horizontal scanline) first derivative edges correspond to
 extrema of derivative

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One dimensional derivatives

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y = f(x) m = ∆y ∆x m = f(x + h) − f(x) (x + h) − x = f(x + h) − f(x) h Gradient

https://en.wikipedia.org/wiki/Derivative

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For 2D function f(x), one can compute a derivative for each direction v Directional derivatives of the function along the axes are called partial derivatives. For example the partial derivative with respect to x is:

Two dimensional derivatives

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ε ε

ε

) , ( ) , ( lim ) , ( y x f y x f x y x f − + = ∂ ∂

→

Source: K. Grauman

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For 2D function f(x,y), the partial derivative is: For discrete data, we can approximate using finite differences: To implement the above as convolution, what would be 
 the associated filter?

Partial derivatives with convolutions

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ε ε

ε

) , ( ) , ( lim ) , ( y x f y x f x y x f − + = ∂ ∂

→

1 ) , ( ) , 1 ( ) , ( y x f y x f x y x f − + ≈ ∂ ∂

Source: K. Grauman

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Partial derivatives of an image

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Which one shows changes with respect to x?

• 1

1 1

• 1
• r
• 1 1

x y x f ∂ ∂ ) , ( y y x f ∂ ∂ ) , (

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Other approximations of derivative filters exist:

Finite difference filters

28 Source: K. Grauman

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The gradient points in the direction of most rapid increase in intensity
 
 
 The gradient of an image:

Image gradient

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The gradient direction is given by

Source: Steve Seitz

The edge strength is given by the gradient magnitude

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

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Edge detection example

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https://en.wikipedia.org/wiki/Prewitt_operator

edge magnitude image

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Edge detection in Matlab

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Consider a single row or column of the image

Effects of noise

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Where is the edge?

Source: S. Seitz

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Solution: smooth first

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• To find edges, look for peaks in

) ( g f dx d ∗ f g f * g

) ( g f dx d ∗

Source: S. Seitz

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Smoothed derivative removes noise, but blurs edge. Also finds edges at different “scales”

1 pixel 3 pixels 7 pixels

34 Source: D. Forsyth

Scale of smoothing

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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

• Prewitt filter
• 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

Smoothing vs derivative filters

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