Edges & the Hough Transform Instructor - Simon Lucey 16-423 - - - PowerPoint PPT Presentation

Edges & the Hough Transform

Instructor - Simon Lucey

16-423 - Designing Computer Vision Apps

Today

• Motivation.
• What is an Edge?
• Oriented Filters.
• Hough Transform.
• Advance Methods.

Today

• Motivation.
• What is an Edge?
• Oriented Filters.
• Advance Methods.
• Hough Transform.

1968

• riginal image

Canny edge detector human annotator

Taken from Isola et al. “Crisp Boundary Detection Using Pointwise Mutual Information”

Edges are Semantic

• No mathematical definition of an edge, contour of boundary.
• Classic problem in computer vision.
• By definition they are semantic.
• Early work focussed on oriented filters.
• e.g. Canny edge detectors.
• Developed by John F. Canny in 1986.

“John Canny”

Today

• Motivation.
• What is an Edge?
• Oriented Filters.
• Hough Transform.
• Advance Methods.

D.H. Hubel & T.N. Wiesel. Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The Journal of Physiology, 160(1):106, 1962.

D.H. Hubel & T.N. Wiesel. Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The Journal of Physiology, 160(1):106, 1962.

D.H. Hubel & T.N. Wiesel. Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The Journal of Physiology, 160(1):106, 1962.

D.H. Hubel & T.N. Wiesel. Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The Journal of Physiology, 160(1):106, 1962.

D.H. Hubel & T.N. Wiesel. Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The Journal of Physiology, 160(1):106, 1962.

D.H. Hubel & T.N. Wiesel. Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The Journal of Physiology, 160(1):106, 1962.

Olshausen & Field 1996

x1

x1 x2

x1

. . .

x2 xN−1

x1

. . .

x2 xN−1 xN

x1

. . .

x2 xN−1 xN

  . . . . . .   . . . . . .

X =

M × N

0.05 0.1 0.15 0.2 0.25

• 6
• 4
• 2

2 4 6

x

p(x)

3.19 bits

Olshausen & Field 1996

H(x) = −

n

X

n=1

p(xn) · log[p(xn)] =

x

p(x)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

• 6
• 4
• 2

2 4 6

1.41 bits

Olshausen & Field 1996

H(x) = −

n

X

n=1

p(xn) · log[p(xn)] =

=

X

D

Z

×

+

✏

• H. Lee, A. Ng, et al. 2007

Not Always Zero Always Zero

Olshausen & Field 1996

• c)

=

X

D

Z

×

+

✏

• H. Lee, A. Ng, et al. 2007

Not Always Zero Always Zero

Olshausen & Field 1996

• c)

1D Filter

∗

1D Filter

∗

1D Filter

∗

2D Filter

∗

  1, 0, −1 1, 0, −1 1, 0, −1  

(Prewitt)

2D Filter

∗

  1, 0, −1 2, 0, −2 1, 0, −1  

(Sobel)

2D Filter

2D Filter

+

=

2D Filter

• c)

99.6% sparse per patch

99.6% sparse per patch

Not Always Zero Always Zero

=

+

✏

• M. Zeiler, et al. 2010

x

∗

z1

+ . . . + ∗

zK

dK

d1

Not Always Zero Always Zero

=

+

✏

• M. Zeiler, et al. 2010

x

∗

z1

+ . . . + ∗

zK

dK

d1

• c)

Edges

Adapted from: Elder “Are Edges Incomplete?” IJCV 1999.

Naive Approach?

• c)

∗

How do we recover edges?

Canny Edge Detector

Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince

Canny Edge Detector

Compute horizontal and vertical gradient images h and v

Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince

Oriented Filters

• Seems inefficient to have to search all possible orientations,

+

=

• Instead one can express all orientations as a linear

combination of x- and y- gradient filters.

Canny Edge Detector

Quantize to 4 directions

Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince

Canny Edge Detector

Non-maximal suppression

Adapted from: Computer vision: models, learning and inference. Simon J.D. Prince

Non-Max Suppression

• Interesting, view of non-max suppression in terms of sparse

coding.

=

X

D

Z

×

• Non-max suppression attempts to enforce that Z is sparse.

Canny Edge Detector

Hysteresis Thresholding

Classic Edge Detector - Problems

Simple Step Edge Filtering Response Texture Edge Filtering Response

Original Color Image

(a)

Appearance Edges Found by Linear Filtering

(b)

Taken from: “Occlusion Boundaries: Low-Level Detection to High-Level Reasoning” - A. Stein (Ph.D. Thesis)

Today

• Motivation.
• What is an Edge?
• Oriented Filters.
• Hough Transform.
• Advance Methods.

• A line is classically represented as - .
• Not good, in case of vertical lines (i.e. ).
• Instead common to represent lines using ,

Hough Transform

y = m · x + c

m = ∞ (θ, ρ)

y = −x tan θ + ρ cos θ

θ → angle from horizontal axis to perpendicular ρ → perpendicular distance between the origin and the line

Hough Transform

Adapted from: Robotics, Vision and Control. Peter Corke.

x y

Exhaustive Search

“Hough Transform Parameters”

ρ θ

Hough Transform

Adapted from: Robotics, Vision and Control. Peter Corke.

x (pixels) y (pixels) y (pixels) x (pixels) ρ (pixels) ρ (pixels)

θ (radians) θ (radians)

Hough Transform

• Elegant in principle - in practice it can work well or

infuriatingly badly.

• Performs poorly when the scene contains a lot of texture or

edges are indistinct.

• Lots of experimentation required to find strong edges AND

thresholds within the Hough peak detector.

Original Color Image

(a)

Appearance Edges Found by Linear Filtering

(b)

State of the Art

Sobel & Feldman 1968 Arbeláez et al. 2011 (gPb) ez et al. Dollár & Zitnick 2013 (SE) Our method Human labelers

Taken from Isola et al. “Crisp Boundary Detection Using Pointwise Mutual Information”

Isola et al. 2014

Things to try in iOS - GPUImage

More to read…

• Prince et al.
• Chapter 13, Sections 1-2.
• Corke et al.
• Chapter 13, Section 2.
• P. Isola et al. “Crisp Boundary Detection Using

Pointwise Mutual Information”, ECCV 2014.