optical microscopy: lecture 4 interpretation of the 3 ......sampling and nyquist theorem in order...

GG 711: Advanced Techniques in Geophysics and Materials Science

Pavel Zinin HIGP, University of Hawaii, Honolulu, USA

Optical Microscopy: Lecture 4

Interpretation of the 3-Dimensional Images

and Fourier Optics

www.soest.hawaii.edu\~zinin

Lecture Overview

•Introduction to Fourier Transform

•Fourier Spectrum Approach of Image in 3-D

•Contrast in Reflection and Transmission Microscopy

•Emulated Transmission Confocal Raman Microscopy

Using scanning optical or acoustical microscopes it is possible to

reconstruct structure of three-dimensional objects. How the image we

obtain is related to the structure of the real object? Is it possible to

simulate images of simple-shaped bodies obtained by optical or

acoustical microscopes?

Acoustical images are

difficult for direct

interpretation: Ultrasound

images of a fetus during

seventeen of development

(left) and an artist’s

rendering of the image.

(after Med. Encyclopedia,

2005)

Interpretation of the Acoustical Images in Medicine

“By striving to do the impossible, man has always achieved what is possible.”

Bakunin

The Fourier Transform

What is the Fourier Transform?

The continuous limit: the Fourier

transform (and its inverse)

( ) ( ) exp( )F f t i t dt

1( ) ( ) exp( )

2f t F i t d

Jean Baptiste Joseph Fourier

(1768 – 1830)

Fourier analysis is named after Joseph Fourier,

who showed that representing a function by a

trigonometric series greatly simplifies the study

of heat propagation.

The Fourier Transform

Consider the Fourier coefficients. Let’s define a function F() that incorporates both cosine and sine series coefficients, with the sine series

distinguished by making it the imaginary component:

where t is the time and is the angular frequency; = 2 f, where f is the frequency

( ) ( ) cos( ) ( ) sin( )F f t t dt i f t t dt

( ) ( ) exp( )F f t i t dt

The Fourier

Transform

F() is called the Fourier Transform of f(t). It contains equivalent information to that in f(t). We say that f(t) lives in the “time domain,” and

F() lives in the “frequency domain.” F() is just another way of looking at a function or wave.

The Fourier Transform and its Inverse

Inverse Fourier Transform

FourierTransform ( ) ( ) exp( )F f t i t dt

1( ) ( ) exp( )

2f t F i t d

So we can transform to the frequency domain and back. Interestingly, these

functions are very similar.

There are different definitions of these transforms. The 2π can occur in several

places, but the idea is generally the same.

What do we hope to achieve with the Fourier Transform?

We desire a measure of the frequencies present in a wave. This will

lead to a definition of the term, the spectrum.

From: Prof. Rick Trebino, Georgia Tech

0 0( )cos( ) ( )cos( ) exp( )E t t E t t i t dt

F

0 0

1( ) exp( ) exp( ) exp( )

2E t i t i t i t dt

0 0

1 1( )exp( [ ] ) ( )exp( [ ] )

2 2E t i t dt E t i t dt

0 0 01 1

( )cos( ) ( ) ( )2 2

E t t E E F

0 0 -0

0( )cos( )E t tF

E(t) = exp(-t2) t

Example: 0( )cos( )E t t

( ) exp( )

1[exp( )]

1[exp( ) exp(

2 exp( ) exp(

2

sin(2

F i t dt

i ti

i ii

i i

i

( 2 sinc(

sinc(

F

A

Fourier Transform of a rectangle function: rect(t)

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-F

()

= 1

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

f(t)

t

= 1

( 0 oSinc

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-

F(

)

= 1

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

2

f(t)

t

= 1

1;

2 2

oo of

Sinc(x) and why it's important

• Sinc(x) is the Fourier transform of a rectangle function.

• Sinc2(x) is the Fourier transform of a triangle function.

• Sinc(x) describes the axial field distribution of a lens.

• Sinc2(ax) is the diffraction pattern from a slit.

• It just crops up everywhere...

Fourier Transform of a rectangle function: rect(t)

-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-

F(

)

= 1

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

2

f(t)

t

= 1

1;

2 2

oo of

-18-16-14-12-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

= 1-

F(

)

= 5

-18-16-14-12-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

= 1-

F(

)

= 10

Example: the Fourier Transform of a Gaussian function

2 2

2

{exp( )} exp( )exp( )

exp( / 4 )

at at i t dt

a

F

t 0

2exp( )at

0

2exp( / 4 )a

∩

The Fourier Transform Properties

■ Linearity

■ Shifting

■ Modulation

■ Convolution

■ Multiplication

■ Separable functions

( , ) ( , ) ( , ) ( , )af x y bg x y aF u v bG u v

( , )* ( , ) ( , ) ( , )f x y g x y F u v G u v

( , ) ( , ) ( , )* ( , )f x y g x y F u v G u v

( , ) ( ) ( ) ( , ) ( ) ( )f x y f x f y F u v F u F v

0 02 ( )

0 0( , ) ( , )j ux vy

f x x y x e F u v

0 02 ( )

0 0( , ) ( , )j u x v y

e f x y F u u v v

Scale Theorem

The Fourier transform

of a scaled function, f(at): { ( )} ( / ) /f at F a aF

{ ( )} ( ) exp( )f at f at i t dt

F

{ ( )} ( ) exp( [ / ]) /f at f u i u a du a

F

( ) exp( [ / ] ) /f u i a u du a

( / ) /F a a

If a < 0, the limits flip when we change variables, introducing a

minus sign, hence the absolute value. (From. Prof. Rick Trebino, Georgia Tech. )

Assuming a > 0, change variables: u = at

Proof:

The Scale

Theorem

in action

f(t) F()

Short

pulse

Medium-

length

pulse

Long

pulse

The shorter

the pulse,

the broader

the spectrum!

This is the essence

of the Uncertainty

Principle!

t

t

t

(From. Prof. Rick Trebino, Georgia Tech. )

Sampling and Nyquist Theorem

2 3 4 5 6

-1

-0.5

0.5

1

2 3 4 5 6

-1

-0.5

0.5

1

Nyquist theorem: to correctly identify

a frequency you must sample twice a

period.

So, if x is the sampling, then π/ x

is the maximum spatial frequency.

Sampling and Nyquist Theorem

In order for a band-limited (i.e., one with a zero power spectrum for frequencies f > B)

baseband ( f > 0) signal to be reconstructed fully, it must be sampled at a rate f 2B . A

signal sampled at f = 2B is said to be Nyquist sampled, and f =2B is called the Nyquist

frequency. No information is lost if a signal is sampled at the Nyquist frequency, and no

additional information is gained by sampling faster than this rate.

The Nyquist–Shannon sampling theorem is a fundamental result in the field of

information theory, in particular telecommunications and signal processing. Sampling is

the process of converting a signal (for example, a function of continuous time or space)

into a numeric sequence (a function of discrete time or space). The theorem states: Theorem: If a function x(t) contains no

frequencies higher than B hertz, it is

completely determined by giving its

ordinates at a series of points spaced 1/(2B)

seconds apart.

OR In order to correctly determine the

frequency spectrum of a signal, the signal

must be measured at least twice per period.

The Nyquist Sampling Theorem states that to avoid aliasing occuring in the sampling of a signal

the samping rate should be greater than or equal to twice the highest frequency present in the

signal. This is refered to as the Nyquist sampling rate.

The Fourier Transform of 2-D objects

Scanning of the intensity along a line

Intensity of the contrast of the image along a line

Fourier Transform of

the intensity of the

contrast of the image

along a line

FT of an Image (Magnitude + Phase)

I log{|F{I}|2+1} [F{I}]


Peters, Richard Alan, II, "The Fourier Transform", Lectures on Image Processing, Vanderbilt University, Nashville, TN, April 2008,

Available on the web at the Internet Archive, http://www.archive.org/details/Lectures_on_Image_Processing.

FT of an Image (Real + Imaginary)

I Re[F{I}] Im[F{I}]


Peters, Richard Alan, II, "The Fourier Transform", Lectures on Image Processing, Vanderbilt University, Nashville, TN, April 2008,

Available on the web at the Internet Archive, http://www.archive.org/details/Lectures_on_Image_Processing.

The Fourier Transform in Space

Gallanger et al., AJR. 190. 2008

Example of the Fourier Transform in Space

Bahadir Gunturk

Light Waves: Plane Wave

Definition: A plane wave in two or three dimensions is like a sine wave in one

dimension except that crests and troughs aren't points, but form lines (2-D) or

planes (3-D) perpendicular to the direction of wave propagation (Wikipedia, 2009).

Definition: A plane wave is a wave in which the wavefront is a plane surface; a wave

whose equiphase surfaces form a family of parallel surfaces (MCGraw-Hill Dict. Of

Phys., 1985).

The large arrow is a vector called

the wave vector, which defines (a)

the direction of wave propagation

by its orientation perpendicular to

the wave fronts, and (b) the

wavenumber by its length. We can

think of a wave front as a line

along the crest of the wave.

x y zi k x k y k z tAe

The Fourier Transform in Space

Time Fourier Transform ( ) ( ) exp( )F f t i t dt

x y zi k x k y k z tAe

Fourier Transform in Space?

dxdyykxkiZyxZkkU yxyx exp),,(),,(

As must satisfy the Helmholtz equation, every solution can be decomposed into

plane waves exp(ikxx + ikyy + ikzz) with wave vector k = (kx, ky, kz), |k|=/c, and c being

the velocity of sound in the coupling liquid. If kx2 + ky

2 k², then . Such waves are

denoted as homogeneous waves.

( , , ) ( , , )expx y x y x yx y Z U k k Z i k x k y dk dk

The Fourier Transform in Space: Angular Spectrum

Conversely, the potential can then be written as the inverse Fourier transform of

the angular spectrum.

yxyxyx dkdkykxkiZkkUZyx exp),,(2

1),,(

2

The inverse Fourier transform represents the wavefield in the plane Z as a

superposition of plane waves exp(ikxx + ikyy).

dxdyykxkiZyxZkkU yxyx exp),,(),,(

Properties of the Fourier Transform

The Fourier spectrum of the field at the plane z = Z, U(kx,ky,Z) can be

expressed through the Fourier spectrum of the field at the plane z = 0,

U(kx,ky,0).

( , , ) ( , ,0)expi x y i x y zU k k Z U k k ik Z

Properties of the Fourier Transform

The Fourier spectrum of the circular Function

.

1, 1( )

0, 1

rP r

r

J. W. Goodman. “Introduction to Fourier Optics”, , McGraw-Hill, (1996).

Is the jinc function:

( , ) ( , )expx y x yU k k x y i k x k y dxdy

2 2

r x yk k k

-15 -10 -5 0 5 10 15

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

1.22 (

)

kr

-1.22

r

rr

k

kJAkU

)()( 1

sin 0.5 (1 cos )( )

0.5 (1 cos )

kzz B

kz

Axial Intensity Distribution

Debye integral can be simplified for = 0.

0

( ) exp( ) exp( cos )sin

exp( )exp( ) exp( cos )

o

o

z u f ikf ikz d

u f ikfikz ikz

kz

For small angles cos ~ 1 – 0.5 sin2, then

2 2

2 2

sin sin / 4) sin / 4)( )

sin / 4 / 4

kz kz NAz B B

kz kz NA

PSF: Point Spread Function

Field in the Focus and Fourier Spectrum Approach

2

0 0

( ) ( , )exp( (cos cos sin sin cos( )) sinP P Pr A P ikr d d

In the limit f → ∞ (Debye approximation) this equation is valid throughout the

whole space. It expresses the field in the focal region as a superposition of plane

waves whose propagation vectors fall inside a geometrical cone formed by

drawing straight lines from the edge of the aperture through the focal point,

which, in contrast to the usual Debye integral, are weighted with P(,). Where

P(,) is the Pupil Function.

We may say that each point on P(,) is responsible for the emission (and, by reciprocity,

for the detection) of the plane wave component

emitted along the line from the point on P(,)

through the focal point


To find the angular spectrum of the emitted field in the focal plane, we set the third

coordinate of r equal to zero and substitute Cartesian coordinates for the angular

integration variables. For the vector components of k hold: kx = k sinθ cosφ,

ky = k sinθ sinφ, kz = cosφ = . The Jacobi determinant corresponding yields. 222

yx kkk

yx

z

yx dkdkkk

dkdkk

dd1

cos

1sin

2

Denoting the Cartesian coordinates of r by x, y, z, Equation can be rewritten:

z

yx

yxyxikk

dkdkykxkikkPzyx exp),(

2

1)0,,(

2

.

Up to a constant P(kx,ky) is defined as

elsewhere

kkkPkkP yxyx

,0

),,(),(

222

(1)


.

( , ,0) ( , ,0)expx y x y x yx y U k k i k x k y dk dk

1, sin( )

0,

r

r

k kP k

otherwise

The Pupil Function of an ideal lens can

be described by a circle function

1 sin( )

sin

J kA

k

Fourier transform of the circle function

J. W. Goodman. “Introduction to Fourier Optics”, , McGraw-Hill, (1996).


.

Equation (1) also tells us that the lens performs Fourier transform on the incident

electromagnetic.

),(),( vuvuh

),( yxP),()],([ vuhyxPF

Plane wave passing the lens

with a pupil function P(x,y)

transforms into jinc function in

focal plane of the lens.

Lens as a Fourier transform

.

Fourier transform of letter “E”

Lens as a Fourier transform

.

6 mm

1/e2

r -0.04 -0.02 0.00 0.02 0.04Radius, mm

20 m

6 mm

1/e2

Input laser beam

Gaussian

Output laser beam

super-Gaussian

Focused laser beam

Flat top

a b c

Focal πShaper

6 mm

1/e2

r -0.04 -0.02 0.00 0.02 0.04Radius, mm

20 m

6 mm

1/e2

Input laser beam

Gaussian

Output laser beam

super-Gaussian

Focused laser beam

Flat top

a b c

Focal πShaper

Courtesy to Vitali. Prakapenka, University of Chicago


ZikkkUkkU zyxiyxi 'exp)','()','(''

For the reflection microscope we have , since the backscattered wave propagates opposite to the

z-direction. Combining all above we obtain the output signal of the reflection microscope as

' '

( , , ) ( , ) ( ' , ' ) ( , , ' , ' ) exp ' ' ''

x y x y

x y x y s x y x y x x y y z z

z

dk dk dk dkV X Y Z P k k P k k g k k k k i k k X k k Y k k Z

kk

Image Formation of Spherical Particle in Reflection Microscope

X-Z scan through a steel sphere for a reflection microscope(OXSAM) at 105 MHz. The

radius of the sphere was 560 μm. The semi-aperture angle of the microscope lens was

26.5°. Z=0 corresponds to focussing to the centre of the sphere. (b) X-Z scan through a

steel sphere for reflection microscope calculated with the same parameter as in From

Weise, Zinin et al., Optik 107, 45, 1997.

Acoustical images are

difficult for direct

interpretation: Ultrasound

images of a fetus during

seventeen of development

(left) and an artist’s

rendering of the image.

(after Med. Encyclopedia,

2005)

Elasticity in Medicine

“By striving to do the impossible, man has always achieved what is possible.”

Bakunin

Some Fourier transform pairs (graphical illustration)

Important conclusion from the theory we developed is that size of the spherical particle

can be determined only from image taken by a transmission microscope. The size of the

image of the spherical particle in reflection microscope is less than the real size of the

particle and is equal to a sin(α), where a is the radius of the particle, and α is

semiaperure angle of the lens. This theory has found several direct application in

practical microscopy: surface imaging and in developing Emulated Transmission

Confocal Raman Microscopy.

X-z scan of 3-D image of

steel sphere. X-z scan of 3-D image of

a liquid (water) drop. X-z scan of 3-D image of

a Plexiglas sphere.

Optical images of yeast cells

Optical images of yeast cells on glass (100x objective) in transmission mode (a), in

reflection mode (b). The red circles mark the position of the laser beam.


Calculated vertical scans through a transparent sphere with refraction index of

1.05 and refraction index of surrounding liquid of 1.33: (a) reflection microscope

with aperture angle 30 o; (a) transmission microscope with aperture angle 30o.


Sketch of the optical rays when cell is (a) attached to the glass substrate or (b)

to mirror.

Emulated Transmission Confocal Raman Microscopy

500 1000 1500 2500 3000

0

100

200

300

400

1316

1661

2933

751

1004

1453

Counts

Raman shift (cm-1)

(b)

1590

Optical image of the yeast bakery cells in the reflection confocal microscope. Rectangle shows the area of

the Raman mapping. (a) Raman spectra of the cell α measured with green laser excitation (532 nm,

WiTec system).

Map of the Raman peak

intensity centered at 2933

cm-1. The intensity of the

2933 cm-1 peak is shown in

a yellow color scale. (b) Map

of the Raman peak intensity

centered at 1590 cm-1. The

intensity of the 1590 cm-1

peak is shown in a green

color scale.

Summary Lecture 6

• Sinc function is a Fourier Transform of the rectangula

Function.

• Distribution of the field in the focal plane is the spacial

Fourier Transform of the Pupil Function of the Lens

• Fourier Spectrum Approach of Image in 3-D

• Contrast in Reflection and Transmission Microscopy

• Emulated Transmission Confocal Raman Microscopy

Home work

1. Present a definition of the Fourier transform (SO).

2. Describe difference in image formation of transmission and reflection

optical microscope of 3-D objects. (KK).

3. Formulate and explain Nyquist Sampling Theorem (SO).

4. Derive a Fourier transform of the function f(t)={1, if -1/2

optical microscopy: lecture 4 interpretation of the 3 ......sampling and nyquist theorem in order...

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