Spatial statisti s and image analysis. Le ture 1

Mats Rudemo

Mar h 17, 2020

Mar h�May 2020

Le tures based on notes and books available from ourse home-

pages.

Pra ti al information

Tea hers:

Mats Rudemo: Le tures and Examiner

E-mail: rudemo� halmers.se Tel: +46708626472 Room: H3024

Konstantinos Konstantinou: Computer Exer ises

E-mail: konkons� halmers.se Tel: +46762953386 Room: H3018

S hedule:

Le tures: Mondays and Wednesdays (10.00-11.45), start 23/3

Computer exer i es: Mondays and Wednesdays (13.15-15), start

23/3

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Course Literature

The ourse is mainly based on:

• Le ture notes (Statisti s of Imaging) by Mats Rudemo.

More details are found in:

• Handbook of Spatial Statisti s by Gelfand et. al.

• Elements of Statisti al Learning by Hastie et. al.

• Computer Age Statisti al Inferen e by Efron and Hastie.

The books are available as eBooks, see ourse homepage.

An additional very useful book is

• Glasbey, C.A. and Horgan, G.W. (1995) Image Analysis for

the Biologi al S ien es, Wiley

urrently available hapter-wise on the internet, e.g.

http://www.bioss.a .uk/people/ hris/ h1.pdf

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Examination

There will be two omponents in the examination:

• written exam at the end of the ourse,

• proje t assignment,

and these are weighted equally for the �nal grade.

Su essful ompletion of the ourse will be rewarded by 7.5 hp.

The proje t:

• an be performed in groups of 1-3 students,

• will onsist of three parts: the major part onsists of one

problem you hoose on your own (with approval from me),

and in addition there are two problems introdu ed in the

omputer exer ises

• is presented at a seminar and as a written report at the end

of the ourse.

Lists of earlier proje ts an be found in des riptions of earlier

ourses, e.g.

• A ademi year 16/17. Examiner: Mats Rudemo

• A ademi year 18/19. Examiner: David Bolin

There will also soon be presented some new possible proje ts.

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Course ontents

• Model-based spatial statisti s

• Statisti al and ma hine learning methods for image analysis

• Appli ations

� Imaging

� Remote sensing

� Mi ros opy

� Bioinformati s

� Di�usion

� Transmission ele tron mi rography

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Digital images

A digital image is a matrix of pixels

f = (fij) = (fij, i = 1, . . . ,m, j = 1, . . . , n)

fij ∈ V

Examples:

V = {0, 1}

V = {0, . . . , 255}

V = {0, . . . , 216 − 1}

V = {0, . . . , 255}3

A pixel is spe i�ed by a lo ation (i, j) and a pixel value fij.

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Aerial photographs of a thinning experiment with Nor-

way spru e

Figure 1: Aerial photograph of the thinning experiment KU in northern

Sealand with Norway spru e trees. The position of the airplane at image

a qusition was 560 m above �Nadir�.

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Figure 2: Detail of the previous aerial photograph overing the subplot D with

very heavy thinning.

Figure 3: Detail of the previous aerial photograph showing part of the south-

eastern orner of subplot D.

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Ba klighted trees

Figure 4: Detail of aerial photograph of subplot D of ba klighted Norway

spru e trees a quired from an oblique angle with the airplane lo ated to the

northwest of the experimental area.

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Weed seeds

Figure 5: Seed images, left Rumex rispus, right Rumex thyrsi�orus.

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Weed plants at an early stage.

Figure 6: Above two images of plants of arrot, D. arota, L., and below two images of

plants of ladythumb smartweed P. persi aria, L.

Figure 7: Above two images of plants of fumitory, Fumaria o� inalis, L., and below two

images of plants of orn spurry, Spergula arvensis, L.

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Two-dimensional ele trophoresis images

Figure 8: Images from 2D gel ele trophoresis of baker's yeast grown in a

standard solution, above, and with salt added, below.

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Di�using parti les.

Figure 9: Images 0.2 se onds apart obtained by video mi ros opy showing

di�using parti les. Parti les in pho us are shown as small distin t bla k

obje ts.

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Hnadwritten digits.

The MNIST database of handwritten images onsists of a train-

ing set with 60000 digits and an evaluation set of 10000 digits.

Examples of images from this set is given in Figure 10, a tually

the �rst 100 digits from the training set. The digit images are

28×28 pixel grey level images obtained from 20x20 pixel binary

bla k and white images. The MNIST dataset has been used ex-

tensively as a proving ground for pattern re ognition methods

in luding neural nets.

Figure 10: Examples of 100 handwritten digits from the MNIST database.

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Image �ltering

gij =

p∑

k=−p

p∑

l=−p

wk,lfi+k,j+l. (1)

A 3×3 averaging �lter

w =

w−1,−1 w−1,0 w−1,1

w0,−1 w0,0 w0,1

w1,−1 w1,0 w1,1

=1

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1 1 1

1 1 1

1 1 1

. (2)

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Gaussian �lter

wk,l = c exp(−1

2σ2(k2 + l2)), (3)

where c is hosen su h that

p∑

k=−p

p∑

l=−p

wk,l = 1 (4)

Median �lter

gij = median{fi+k,j+l : |k| ≤ p, |l| ≤ p} (5)

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Verti al edge dete tion

w =

w−1,−1 w−1,0 w−1,1

w0,−1 w0,0 w0,1

w1,−1 w1,0 w1,1

=1

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−1 0 1

−1 0 1

−1 0 1

. (6)

Horizontal edge dete tion

w =

w−1,−1 w−1,0 w−1,1

w0,−1 w0,0 w0,1

w1,−1 w1,0 w1,1

=1

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−1 −1 −1

0 0 0

1 1 1

. (7)

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Aerial photographs of a thinning experiment. Smooth-

ing.

Figure 11: Detail of aerial photograph overing the subplot D with very heavy

thinning (before smoothing).

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Figure 12: Above: Smoothed version of D-plot forest image by ir ular 2D

Gaussian �lter, σ = 4.5 pixel-widths. Below: Same image viewn in perspe -

tive as 3D surfa e, light intensity as verti al oordinate.

Figure 13: Lo ation of maxima in smoothed image.

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Current plan for le tures

1. Introdu tion and ba kground

2. Gaussian random �elds

3. Kriging and parameter estimation

4. Pattern re ognition

5. Ma hine learning

6. Statisti al image modelling

7. Point pro esses

8. Warping, Mi roarrays

9. Ele trophoresis, Remote sensing

10. Di�usion

11. TEM images

12. Re apitulation

13. Seminars

14. Seminars

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