Medical Image Analysis · 2020. 4. 11. · In Matlab and in this presentation In the course notes....

DTU Compute

Medical Image Analysis

Rasmus R. PaulsenDTU Compute

[email protected]

http://www.compute.dtu.dk/courses/02512

mailto:[email protected]://www.compute.dtu.dk/courses/02512

DTU Compute

2020Image Analysis2 DTU Compute, Technical University of Denmark

Lecture 10 – Hough Transformation and Path Tracing

Hough transform

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

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

transform Use dynamic programming to trace paths in images Describe how an image can be used as a graph Describe the fundamental properties of a cost image Compute the cost of path Compute an accumulator image for path tracing Compute a backtracing image for path tracing Choose appropriate pre-processing steps for path tracing Describe how circular structures can be located using path tracing

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Line Detection Find the lines in an image

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What is a line?

It can be the entire object– Large scale

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

Normally only locally defined

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Enhancing the lines We want to locate the

borders– Enhance them

Filtering Edge detection

Original Prewitt Edge

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

white pixels Some of them define a line

– Not a perfect straight line– “Linelike”

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

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

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y = ax + bSlope Intercept

Can not represent lines that are vertical

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Ax + By = C

Mathematical line definition General definition

With

Ax + B y = C

A2 + B 2 = 1

Line normal

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Mathematical line definition Normal parameterisation

where– is the distance from the

origin– is the angle

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Ax + Bx = C

A2 + B 2 = 1

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Mathematical line definition Normal parameterisation

Therefore a line can be defined by two values––

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

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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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More about angles

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ρ = x cos θ + y sin θWhy?

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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How do we use the Hough space?

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

space?

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Filled Hough-Space All “line segments” in the image examined 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 practise Hough Space is represented as an image 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 scheme The pixels in the Hough space are used to vote for lines. Each line segment votes by putting one vote in a pixel The pixels are also called accumulator cells

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Hough transform per pixel In practise we do not use line segments 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 space The lines are found in Hough space where most

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

Space Hough transform

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

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

binary image

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The practical guide to the Hough Transform Compute Hough transform

and locate the maxima

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The practical guide to the Hough Transform Draw the lines corresponding

to the found maxima Here the cyan line is the

longest

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Path Tracing The diameter as

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

We need to find the arteries and veins

Path tracing is one solution

Fundus imageArteries and veins

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Path tracing A path is defined as a

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

In this case it is a dark line

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

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Dynamic Programming Break up large problem into

many small sub-problems A classic algorithm:

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

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 tracing A GPS device uses path

tracing Based on graph

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

weight of the edge is the fuel cost

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

Dijkstra’s algorithm

Copenhagen

Aalborg

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2.1 l

1.3 l

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Images as graphs Each pixel is a node Pixel neighbours are

connected by edges The edge cost is the

pixel value Directed graph Imagine a car driving on

the image Called a cost image

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Simplified problem Track dark lines Path going from top to

bottom No sharp turns –

smooth Problem:

– from the top to the bottom

– Sum of pixel values should be minimal

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Simplified problem Pixel value at (r,c)

equals the cost

The path P consist of pixels

The sum of pixel values in the path

𝐶𝐶(𝑟𝑟, 𝑐𝑐)

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

coordinates

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

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Total cost – what is 𝐶𝐶𝑡𝑡𝑡𝑡𝑡𝑡?A) 167B) 350C) 403D) 270E) 345

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

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Path cost This is NOT the optimal path How do we compute the path

P that has minimum 𝐶𝐶𝑡𝑡𝑡𝑡𝑡𝑡?

Test all possible paths?– No! Impossible amount of

posibilities

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

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Path Cost

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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?

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Path restriction Path is only allowed to

– Go down– Move one pixel left or

right Longer jumps not

allowed

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Accumulator image Keeps track of the

accumulated cost for possible paths

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

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

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

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Pre-processing We would like to track

paths that are not dark curves

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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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OptimalPath 2

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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?

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Next week

Statistical shape analysis

Medical Image AnalysisLecture 10 – Hough Transformation and Path TracingWhat can you do after today?Line DetectionWhat is a line?Enhancing the linesWhat is a line II?Mathematical line definitionMathematical line definitionMathematical line definitionMathematical line definitionSomething about anglesHough SpaceMore about anglesHow do we use the Hough space?How do we use the Hough space?Filled Hough-SpaceHough transform in practiseHough transform as a voting schemeHough transform per pixelHough transform per pixelReal Hough TransformReal Hough Transform IIReal Hough Transform and linesFinding the lines in Hough spaceThe practical guide to the Hough TransformThe practical guide to the Hough TransformThe practical guide to the Hough TransformThe practical guide to the Hough TransformPath TracingPath tracingDynamic ProgrammingPath tracingImages as graphsSimplified problemSimplified problemPath costTotal cost – what is 𝐶 𝑡𝑜𝑡 ?Path costPath CostPath restrictionAccumulator imageComputing the accumulator imageComputing the accumulator imageComputing the accumulator imageAccumulator ImageUsing the accumulator imageThe backtracing imageUsing the backtracing imageUsing the backtracing imageBacktracingPre-processingX-ray preprocessingPre-processingPath tracing on pre-processed imageOptimal� Path 2Next week

Medical Image Analysis · 2020. 4. 11. · In Matlab and in this presentation In the course notes....

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