Image Analysis - week8_… · nPath going from top to bottom nNo sharp turns – smooth nProblem:...

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Image Analysis

Tim B. DyrbyRasmus R. PaulsenDTU Compute

[email protected]

http://www.compute.dtu.dk/courses/02502

mailto:[email protected]

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2020Image Analysis2 DTU Compute, Technical University of Denmark

Lecture 8 – Hough Transformation and Path Tracing

Hough transform

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Breaking NEWS!

One morning I was shopping in Amsterdam with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was

just thinking about whether I could do this, and I then designed the algorithm for the shortest path. As I said, it was a twenty-minute invention.

—Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the ACM, 2001[3]

https://en.wikipedia.org/wiki/Amsterdamhttps://en.wikipedia.org/wiki/Dijkstra%27s_algorithm

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What can you do after today?n Use the Hough transform for line detectionn Describe the slope-intercept, the general form and the normalised

form of linesn Describe the connection between lines and the Hough spacen Use edge detection to enhance images for use with the Hough

transformn Use dynamic programming to trace paths in imagesn Describe how an image can be used as a graphn Describe the fundamental properties of a cost imagen Compute the cost of pathn Compute an accumulator image for path tracingn Compute a back tracing image for path tracingn Choose appropriate pre-processing steps for path tracingn Describe how circular structures can be located using path tracing

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Line Detectionn Find the lines in an image

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What is a line?

n It can be the entire object– Large scale

n Can also be the border between an object and the background– Small scale

n Normally only locally defined

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Enhancing the linesn We want to locate the borders

– Enhance themn Filtering (Prewitt)n Edge detection

Original Prewitt Edge

Prewitt

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What is a line II?n Result of the edge filter is a selection of

white pixelsn Some of them define a line

– Not a perfect straight line– “Linelike”

n How do we find the collection of points that define a line?

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Mathematical line definitionn The classical definition (slope-intercept form)

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x axis

y ax

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y=0.5 * x + 1

y = ax + bSlope Intercept

Can not represent lines that are vertical

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y ax

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Ax + By = C

Mathematical line definitionn General definition

n With

Ax + By = C

A2 + B2 = 1

Line normal

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Mathematical line definitionn Normal parameterisation

n where– is the distance from the

origin– is the angle

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y ax

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Ax + Bx = C

A2 + B2 = 1

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Mathematical line definitionn Normal parameterisation

n Therefore a line can be defined by two values––

n A line can therefore also be seen as a point in a ( , )-space

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Ax + Bx = C

½µ

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Converting linesn From normal to slop-intercept form

𝑝 = 𝑥 cos 𝜃 + 𝑦 sin 𝜃The normal form:

y = 𝑎𝑥 + 𝑏The slope-intercept form:

Slope=a

intercept=b

−𝑥 𝑐𝑜𝑡𝜃 + 𝑝 𝑐𝑜𝑠𝑒𝑐𝜃 = 𝑦

𝑦 = 𝑥 ∗ −𝑐𝑜𝑡𝜃 + 𝑝(𝑐𝑜𝑠𝑒𝑐𝜃)

Slope=a

Intercept=b

-x cos 𝜃 + 𝑝 = ysin 𝜃

Start: 𝑝 = 𝑥 cos 𝜃 + 𝑦 sin 𝜃

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Something about angles

In Matlab and in this presentation

In the course notes

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Hough Space

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r = x cos q + y sin q

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More about angles

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r = x cos q + y sin qWhy?

but Matlab only allows

look at the mirror-projection of the normal

is used to determine if it is the “upper” or “lower line”

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Hough space: Let’s vote for general line …§ Basically a tool to find a line through points.

Hou

gh s

pace

Poin

ts

1) Point coordinates: (x,y)

3) Hough parameters space:(r,𝝷)

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

2) Define origin

4) Map all possible lines through a point for different 𝝷5) “Vote” which line fit best through points

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How do we use the Hough space?

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How do we use the Hough space?n What if every little “line-segment” was plotted in the Hough-

space?

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Filled Hough-Spacen All “line segments” in the image examinedn A “global line” can now be found as a cluster of points

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In practice it is difficult to identify clusters

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Hough transform in practisen Hough Space is represented as an imagen It is quantisized – made into finite boxes

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Pixel

A line segment here

Belongs to this pixel

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Hough transform as a voting schemen The pixels in the Hough space are used to vote for lines. n Each line segment votes by putting one vote in a pixeln The pixels are also called accumulator cells

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Hough transform per pixeln In practise we do not use line segmentsn Each pixel in the input image votes for all potential lines going

through it.

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Hough transform per pixel

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Go through all and calculate (x, y) are fixed

Sinusoid!

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Real Hough Transform

Hough space

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Real Hough Transform IIHough space

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Real Hough Transform and lines

Hough space

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Spot the line!

A maximum where Hough pixel has value 3

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Finding the lines in Hough spacen The lines are found in Hough space where most

pixels have voted for there being a linen Can be found by searching for maxima in Hough

Space Hough transform

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The practical guide to the Hough Transformn Start with an input image

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The practical guide to the Hough Transformn Detect edges and create a

binary image

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The practical guide to the Hough Transformn Compute Hough transform

and locate the maxima

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The practical guide to the Hough Transformn Draw the lines corresponding

to the found maximan Here the cyan line is the

longest

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Path Tracingn The diameter as

function of the distance to the optic cup tells something about the patients health

n We need to find the arteries and veins

n Path tracing is one solution

Fundus imageArteries and veins

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Path tracingn A path is defined as a

curve in an image defined as something that is different from the background

n In this case it is a dark line

n Pre-processing can for example turn edges into dark lines.

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Dynamic Programmingn Break up large problem into

many small sub-problemsn A classic algorithm:

– Dijkstra’s algorithm– One source to all nodes shortest path

n We will look at a simplifiedvariant

Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs". Numerische Mathematik. 1: 269–271.

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Path tracingn A GPS device uses path

tracingn Based on graph

algorithms– A city is a node– A road is an edge. The

weight of the edge is the fuel cost

n How do we come from Copenhagen to Aalborg using the least fuel?

n Dijkstra’s algorithm

Copenhagen

Aalborg

2.3 l

2.1 l

1.3 l

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Images as graphsn Each pixel is a noden Pixel neighbours are

connected by edgesn The edge cost ( )is

the pixel value n Directed graphn Imagine a car driving on

the imagen Called a cost image

𝑐(𝑟, 𝑐)

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Simplified problemn Track dark linesn Path going from top to

bottomn No sharp turns –

smoothn Problem:

– from the top to the bottom

– Sum of pixel values should be minimal

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Simplified problemn Pixel value at (r,c)

equals the cost

n The path P consist of pixels

n The sum of pixel values in the path

𝐶(𝑟, 𝑐)

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Path costn A path is defined as (r,c)

coordinates

P = [(1,3), (2,3), (3, 2), (4,3), (5,4)]

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Quiz 1: Total cost – what is 𝐶!"!?A) 167B) 350C) 403D) 270E) 345

P = [(1,3), (2,3), (3, 2), (4,3), (5,4)]

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Path costn This is NOT the optimal pathn How do we compute the path

P that has minimum 𝐶!"!?

n Test all possible paths?– No! Impossible amount of

posibilities

P = [(1,3), (2,3), (3, 2), (4,3), (5,4)]

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Quiz 2: Path Cost

A path has been found in the image P=[(1,4),(2,4),(3,5),(4,5),(5,5),(6,4)]. A Matlab matrix coordinate system is used. What is the total cost of the path?

A) 196B) 154C) 201D) 185E) 132

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Path restriction: The rulesn Path is only allowed to

– Go down– Move one pixel left or

right

n Longer jumps not allowed

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Accumulator imagen Keeps track of the

accumulated cost for efficient paths finding

n Path ending here has cost 296

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Computing the accumulator image

Step 1: Copy first row of input image

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Step 2: Fill second row

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Step 3: Fill all rows by looking at the previous row

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Quiz 3: Accumulator ImageA) 57B) 167C) 301D) 241E) 145

An optimal path has been found in the image. What is the value of the accumulator image in the marked pixel?

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Using the accumulator image

Step 4: The end of the optimal path can now be found

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The backtracing image

• Keeps track of where the path came from

• Each pixel stores the column number

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Using the backtracing image

Step 5: Trace the path in the backtracing image

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Using the backtracing image

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Quiz 4: BacktracingA) 1B) 2C) 3D) 4E) 5

An optimal path has been found in an image. The backtracing image is seen below and the optimal path ends in the marked pixel. A Matlab coordinate system is used. What is the optimal path?

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Pre-processingn We would like to track

paths that are not dark curves

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Quiz 5 : X-ray preprocessingA) Gaussian smoothingB) 255-IC) Gradient filterD) RegistrationE) Morphological operation

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Pre-processing

Edge filtered image(Gaussian smoothing followed by Prewitt)

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Path tracing on pre-processed image

Paths found on pre-processed image

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Quiz 6: Optimal Path 2

A) 81B) 64C) 11D) 73E) 51

A 5 x 5 image is filled with values given the gray level run length encoding: 2, 180, 1, 15, 3, 112, 1, 8, 4, 177, 1, 20, 4, 195, 1, 12, 3, 242, 2, 25, 3, 9. After that the optimal path is found. What is the total cost?

15+8+20+12+9=64180 180 15 112 112112 8 177 177 177177 20 195 195 195195 12 242 242 24225 25 9 9 9

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Locating Circular Structures

• Define origin inside structure

• Send out spokes

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Locating Circular Structures

• Each spoke is a line in a new image (surface- layer detection)• Prewitt• Dijkstra’s algorithm• Map back the spokes into image

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What did you learn today?n Use the Hough transform for line detectionn Describe the slope-intercept, the general form and the normalised

form of linesn Describe the connection between lines and the Hough spacen Use edge detection to enhance images for use with the Hough

transformn Use dynamic programming to trace paths in imagesn Describe how an image can be used as a graphn Describe the fundamental properties of a cost imagen Compute the cost of pathn Compute an accumulator image for path tracingn Compute a back tracing image for path tracingn Choose appropriate pre-processing steps for path tracingn Describe how circular structures can be located using path tracing

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Next week – Company presentations

Visiopharm

DaluxIH Food

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