Ter Haar Romeny, ISS 2014 Prof. Bart M. ter Haar Romeny, PhD Biomedical Image Analysis & Interpretation TU/e, Department of Biomedical Engineering [email protected]

ter Haar Romeny, ISS 2014

Prof. Bart M. ter Haar Romeny, PhD

Biomedical Image Analysis & Interpretation

TU/e, Department of Biomedical Engineering

[email protected]

Multi-scale segmentation

Edge focusing, watershed, winding numbers


MR slice hartcoronair

scale

• toppoints

• graphtheory

Edge focusing


Structures exist at their own scale:

Original = e0 px = e1 px = e2 px = e3 px

Noise edges


100 200 300 400 500

4

2

2

4

6

8

10

The graph of the sign-change of thefirst derivative of a signal as a function of scale is denoted the scale-space signature of the signal.

Zero-crossingsof the secondorder derivative= max of firstorder derivative,

as a functionof scale


The notion of longevity can be viewed of a measure of importance for singularities [Witkin83]. The semantical notions of prominence and conspicuity now get a clear meaning in scale-space theory.

In a scale-space we see the emergence of the hierarchy of structures. Positive and negative edges come together and annihilate in singularity points.


Example:

Lysosomesegmentationin noisy 2-photonmicroscopy3D images ofmacrophages.


Marching-cubes isophote surface ofthe macrophage.

slice 24 slice 21 slice 25 slice 18 slice 22 slice 21

slice 24 slice 23 slice 24 slice 20 slice 18 slice 24

Preprocessing:- Blur with = 3 px- Detect N strongest maxima


We interpolatewith cubic splinesinterpolation35 radial tracksin 35 3Dorientations


The profiles are extremely noisy:

Observation: visually we can reasonably point the steepest edgepoints.


Edge focusingover all profiles.

Choose a startlevel based onthe task, i.e. finda single edge.


Detected 3D points per maximum.

We need a 3D shape fit function.


1

2

,1

23

2 sin, 1

23

cos,

1

23

2 sin, 1

42 15

2 sin2,

1

215

2 cossin, 1

45

3 cos2 1,

1

215

2 cossin, 1

42 15

2 sin2

The 3D points are least square fit with 3D spherical harmonics:


Resulting detection:


An efficient way to detect maxima and saddlepoints is found inthe theory of vector field analysis (Stoke’s theorem)


Topological winding numbers

p Li1dLi2. . .dLin i1i2...inL jpL jpn2p L1dL2 L2dL1

L12 L22

N-D

2-D

is the wedge product (outer product for functionals)


In 2D: the surrounding of the point P is a closed path around P.

The winding number of a point P is defined as the number of times the image gradient vector rotates over 2 when we walk over a closed path around P.

maximum: = 1minumum: = 1regular point: = 0saddle point: = -1monkey saddle: = -2


The notion of scale appears in the size of the path.

Winding number = +1 extremum

Winding number = -1 saddle


Generalised saddle point (5th order): (x+i y)5

The winding numbers add within a closed contour, e.g.A saddle point (-1) and an extremum (+1) cancel, i.e. annihilate.

Catastrophe theory

Winding number = - 4 monkey saddle



2 4 6 8 1010200

300

500

700

1000

Slopefor MR image: 1.66555

2 4 6 8 1010

1000

1500

2000

Slopefor white noise: 1.91549

The number of extrema and saddlepoints decrease as e-N over scale

Decrease of structure over scale scales with the dimensionality.


Fertility Prospects

In most developed countries a postponement of childbearing is seen.

E.g. in the Netherlands: Average age of bearing first child is 30 years.

Computer-Assisted Human Follicle Analysis for Fertility Prospects with

3D Ultrasound

ter Haar Romeny et al., IPMI 1999

Application:


The number of follicles decreases during lifetime


Resting 0.03 mm initiation of growth> 120 days?

Early growing 0.03 - 0.1 mm

Preantral 0.1 - 0.2 mm basal growth~ 65 days

Antral 0.2 - 2 mm

Selectable 2 - 5 mm rescued by FSH window~ 5 days

Selected 5 - 10 mm

Dominance 10 - 20 mm maturation~ 15 days

Ovulation

A follicle’s life


3D Ultrasound is a safe, cheap and versatile appropriate modality

Kretz Medicor 530D


Two 3D acquisition strategies:

1. Position tracker on regular probe

2. Sweep of 2D array in transducer

Trans-vaginal probe

Regular sampling from irregularly space slices



Manual counting is very cumbersome Automated follicle assessment

• 2-5 mm hypodense structures• structured noise• vessels may look like follicles• ovary boundary sometimes missing


Detection of a singularity (i.e. a minimum)

From theory of vector fields several important theorems (Stokes, Gauss) exist that relate something happening in a volume with just its surface.We can detect singularities by measurements around the singularity.

P

1-D: difference of signs of the gradient i zero crossing or extremum

yxi

,

i i

The surrounding of the point P are just 2 pointsleft and right of P 1D sphere.


. iji jWd

1 2 2 1

2 21 1

.d d

d

. iji jd d In subscript notation:

where ij is the antisymmetric tensor.

01

10ij


Example of a result:

1 cm

Dataset 2563, radius Stokes’ sphere 1 pixel, blurring scale 3 pixels


Detection of follicle boundaries:

• generation of 200 - 500 rays in a homogeneous orientation distribution• determine most pronounced edge along ray by winding number focusing• fit spherical harmonics to get an analytical description of the shape• calculate volume and statistics on shape

Distance along rayDistance along ray

Sca

le

US inte

nsi

ty

Sca

le

Distance along ray


3D scatterplot ofdetected endpoints

3D visualisation of fittedspherical harmonics function


Validation with 2 bovine ovaria

• anatomincal coupes• high resolution MR• 3D ultrasound

Follicle#

xcenter

ycenter

zcenter

distance toneighbor(pixels)

volume fromspherical harmonics

(mm3)

volumefrom MRI

volumefrom

anatomyv00 99 51 35 25.4 259.7 250.0 262.1

93 28 44 45.0 28.4 27.0 29.8113 55 74 41.6 56.2 54.9 59.3

v01 33 41 66 25.3 242.347 22 75 44.5 34.064 49 44 38.9 54.7

v44 72 49 84 25.4 239.769 28 70 45.2 28.432 51 82 40.1 59.3


Conclusions:

• 3D ultrasound is a feasible modality for follicle-based fertilitiy state estimation• automated CAD is feasible, more clinical validation needed• winding numbers are robust (scaled) singularity detectors• a robust class of topological properties emerges


Multi-scale watershed segmentation

Watershed are the boundaries of merging water basins, when theimage landscape is immersed by punching the minima.

At larger scale the boundaries get blurred, rounded and dislocated.


Regions of different scales can be linked by calculating thelargest overlap with the region in the scales just above.


The method is often combined with nonlinear diffusion schemes

E. Dam, ITU


Nabla Vision is an interactive 3D watershed segmentation tooldeveloped by the University of Copenhagen.

Sculpture the 3D shape by successively clicking precalculatedfiner scale watershed details.

Documents

Ter Haar Romeny, ISS 2014 Prof. Bart M. ter Haar Romeny, PhD Biomedical Image Analysis & Interpretation TU/e, Department of Biomedical Engineering [email protected]