image analysis digital image fundamentalsraul/imageanalysis/fundamentals.pdfimage sensing and...

Image Analysis

Digital Image Fundamentals

Raul Queiroz Feitosa

Gilson A. O. P. Costa

8/14/2019 Digital Image Fundamentals 2


Objective

“The purpose of this chapter is to introduce some

basic concepts related to digital images…”



Contents: Elements of Visual Perception

Image Sensing and Acquisition

Image Sampling and Quantization

Image Interpolation

Color

Relationships Between Pixels


Elements of Visual Perception

Structure of the Human Eye

Eye diameter ~ 20mm Fovea diameter ~ 1.5mm




Light receptors in the retina: Rods and Cones




Cones

• Highly sensitive to color

• Photopic (bright light) vision

• 3 types: S(blue); M(green); L(red)

• 6 to 7 mio, located primarily

in the fovea

100

80

60

40

20

400 450 500 550 600 650 700

ab

sorp

tion

wavelength




Cones

• Highly sensitive to color

• Photopic (bright light) vision

• 3 types: S(blue); M(green); L(red)

• 6 to 7 mio, located primarily in the fovea

Rods

• Sensitive to low levels of illumination

• Scotopic (dim-light) vision

• 75 to 150 mio, distributed over the retina



Cone density in fovea area: 150000/mm2 (comparable to artificial sensors)



Image Formation in the Eye



Sensing

sensor array

single sensor

line sensor



Single Sensor



Line Sensor



Sensor Array



Image Formation Model

In this part of the course images will be denoted by a

function of the form

f(x,y):R2→(0,∞)

with two components: illumination and reflectance

f(x,y) = i(x,y) r(x,y)

where

0< i(x,y) <∞ and 0< r(x,y) <1



Image Formation Model

The intensity i of a monochrome image at any coordinate (x ,y) is called the gray level of the image at that point. That is

i(x,y):R2→(0,∞)

Intensity lies in a range

0< Lmin≤ i(x,y) ≤ Lmax <∞

The interval [Lmin, Lmax ] is called gray scale

Common practice is to shift the interval to [0, L-1] where f(x,y)=0 is considered black and f(x,y)=L-1 is considered white.



Sampling: digitizing in space

M colums

N r

ow

s

f(N,M)f(N,2)f(N,1)

f(2,M)f(2,2)f(2,1)

f(1,M)f(1,2)f(1,1)

yxf

),(

Matrix N M

pixel



Quantization: digitizing the amplitude - 2m-1

- 0 - 1

•

•

•



Sampling and quantization



Result of sampling and quantization



400x304 200x152 100x76 50x38

16 gray levels 8 gray levels 4 gray levels 2 gray levels



Spatial resolution

pixels per unit distance

dots per inch (dpi)

pixels per inch (ppi)

Remote Sensing

ground sampling distance (GSD)

pixel resolution, e.g., pixel = 1x1m

Intensity resolution

number of bits used to quantize intensity (= gray levels)

pixel depth, e.g., 8 bits, 11bits, 16bits, etc.


Image Interpolation

What is the intensity at a non-integer pixel coordinate?

(x1 ,y1) (x2 ,y1)

(x1 ,y2) (x2 ,y2)

(x ,y)

I(x ,y) = ax + by + cxy + d

I(x1,y1) = ax1 + by1 + cx1y1 + d

I(x2,y1) = ax2 + by1 + cx2y1 + d

I(x1,y2) = ax1 + by2 + cx1y2 + d

I(x2,y2) = ax2 + by2 + cx2y2 + d

4 unknowns

4

equations

Bilinear interpolation Bicubic interpolation

3

0

3

0

,

i j

ji

ijyxayxI

16 coefficients

16 equations

on 16 nearest neighbors

Image Interpolation

What is the intensity at a non-integer pixel coordinate?

(x2 ,y1) Bilinear interpolation Bicubic interpolation


Image Interpolation

1250 dpi (3692×2812)

↓↑

72 dpi (213×162)

1250 dpi (3692×2812)

↓↑

150 dpi (443×337)

nearest neighbor bilinear bicubic


Image Interpolation

Geometric Transformations (T)

y

x x´

y´

+

+

(xi´,yi´)=T(xi,yi)

(xj,yj)=T-1 (xj´,yj´) +

+

original image transformed image

pixels of the output image

are visited and their values

are estimated upon their

corresponding locations in

the input image

Image Interpolation

Application: image co-registration Fitting of the coordinate system of an image to that of a second image

reference image (orthophoto) unregistered image

Image Interpolation

Application: image co-registration Fitting of the coordinate system of an image to that of a second image

reference image (orthophoto) coregistered image


Color

Multispectral Images are acquired by sensors sensitive to a

limited range of the electromagnetic spectrum

c =

where c = ~3x108m/s


Color

Multispectral Images are acquired by sensors sensitive to a

limited range of the electromagnetic spectrum

cosmic rays gamma

rays

X

rays UV visibel

light

infra

red

termal

radiation

10-5nm 10-3nm 1 nm 0,3 m 0,4 m 0,75 m 3 m 15 m wavelength

Visible range

BLUE GREEN RED

Spectrum in bands

Remote Sensing


Color

RGB and CYM Models

Primary colors

• red

• green

• blue

Secondary colors

• cyan

• yellow

• magenta

A secondary color subtracts or

absorbs a primary color and reflects

or transmits the others.

Any color can be expressed as

additive combinations of the

primary colors.

B

G

R

Y

M

C

1

1

1

Convertion operation

cyan

yellow

magenta

red green

blue


Color

HSV Model Any color is defined by three values: • Hue (H): associated with the dominant

wavelength.

• Saturation (S): refers to the purity, or the

amount of white light mixed with a hue.

• Value (V): associated to brightness.

Advantages: • Brightness is expressed by intensity,

while chromaticity by hue and

saturation.

• Intimately related to the human

perception of colors.

Conversion Operation See text book


Color

CIE Lab Model Any color is defined by three values: • L: associated with the brightness.

• a : - green / + magenta .

• b : - blue / + yellow.

Advantages: • Perceptually uniform: equal

distances on the CIELab space

correspond to equal perceived color

differences.

• Larger gamut (the number of

colors that can be accurately

represented).

Conversion Operation See text books



Neighbors of a Pixel

Neighborhood N4 Neighborhood ND Neighborhood N8

N8 = N4 ND

(x,y-1)

(x,y +1)

(x+1,y) (x-1,y) (x,y)

pixel p (x-1,y-1) (x+1,y-1)

(x+1,y+1) (x-1,y+1)

(x,y)

(x+1,y-1)

(x+1,y+1)

(x+1,y)

(x-1,y-1)

(x-1,y+1)

(x,y -1)

(x-1,y)

(x,y+1)

(x,y)

pixel p



Adjacency

Let V be the set of gray-levels used to define adjacency.

Two pixels p and q with values in V are adjacent if:

4-adjacency : q N4 (p),

8-adjacency : q N8 (p),

m-adjacency : (i) q N4 (p), OR

(ii) q ND (p) AND N4 (p) N4 (q) = .

q

p q p

q

p

q

p

q

p

adjacent adjacent adjacent non adjacent non adjacent



Why is this important?

How many objects are there in this picture?



Why is this important?

Are the object borders connected?

No if 4-adjacency is considered Yes for all adjacency types



Adjacency

Two image subsets S1 and S2 are adjacent if some pixel in S1 is adjacent to some pixel in S2.

Path

A (digital) path (or curve) from pixel p with coordinates (x,y) to pixel q with coordinates (s,t) is a sequence of distinct pixels with coordinates

(x0,y0), (x1,y1) , ... , (xn,yn)

where (x0,y0) = (x,y) , (xn,yn) = (s,t),

and (xi,yi) and (xi-1,yi-1) are adjacent for 1≤ i ≤ n.



Connectivity between pixel sets

Two pixels p and q are said to be connected in a subset S

of pixels in an image, if there exists a path between them

consisting entirely of pixels in S.

pixels in S

p q

p and q are connected in S ?



Connected components

For any pixel p in S, the set of pixels that are connected

to it in S is called a connected component of S.

Pixels in S are colored white

How many connected components in S ?



Connected components

If it only has one connected component, then the set S is

called a connected set.

Pixels in S are colored white

Is S a connected set?



Region

A subset of pixels R in an image is called a region if it

is a connected set.

How many regions (objects) in this image?



Boundary (border or contour)

The boundary of a region R is the set of pixels in the

region that have one or more neighbors that are not in

R.



Distance Measures

For pixels p, q and z with coordinates (x,y), (s,t),

and (u,v), respectively, D is a distance function or

metric if:

(a) D(p,q) 0 (D(p,q ) = 0, iff , p = q)

(b) D(p,q) = D(q,p ), and

(c) D(p,z) D(p,q ) + D(q,z ).



Commonly used distance functions:

Euclidean distance:

De(p,q) = [( x - s )2 + ( y - t )2]1/2

City-block (Manhatan) distance:

D4(p,q) = |( x - s )| + |( y - t )|

Chessboard distance:

D8(p,q) = max( |( x - s )| ,|( y - t )| )

D4

De D8

p

q

4 3 2 3 4

3 2 1 2 3

2 1 0 1 2

3 2 1 2 3

4 3 2 3 4

2 2 2 2 2

2 1 1 1 2

2 1 0 1 2

2 1 1 1 2

2 2 2 2 2


Next Topic

Image

Enhancement in the

Spatial Domain

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