Fourier Optics

Question 1

What is the 2d fourier transform for image processing

Answer 1

the series expansion of an image function over the 2d space domain in terms of cosine image basis functions
all images are trying to be replicated as a sum of cosine like images

Question 2

What does the origin represent in the fourier expansion image

Answer 2

the DC term, since intensity can’t be negative there is usually a large central DC blob

Question 3

what is the fourier transform of repeating pure horizontal or vertical cosines

Answer 3

bright spots symmetrically placed about the centre of the FT image

Question 4

What are the fourier transform images of the intensity

Answer 4

the amplitude images

Question 5

What happens in the fourier transform of an image with sharp edges (strong constrast)

Answer 5

the grey values change rapidly a bright line in the FT perpendicular to the sharp edge is usually seen

Question 6

What is required to follow a sharp image edge

Answer 6

high frequency power

Question 7

What is the idea behind image sharpening

Answer 7

a sharpening or high frequency emphasis filter is needed to preserve some of the low frequency information but to boost the higher frequencies

Question 8

where does a collimated input beam come to a focus when passed through a lens

Answer 8

at the back focal plane

Question 9

what is the far field light field

Answer 9

the 2d fourier transform of the aperture field

Question 10

what is the fraunhofer diffraction formula

Answer 10

E(kx,ky) is proportional the the double integral from ∞ to -∞ of exp-i(kx times x + ky times y)A(x,y)E(x,y)dxdy

Question 11

what are the angular spatial frequencies along x0 and y0

Answer 11

kx = kx1/z and ky = ky1/z

Question 12

define the spatial frequencies u and v

Answer 12

u = kx/2pi and v = ky/2pi

Question 13

what happens when a thin transparency is illuminated coherently by a monochromatic plane wave and the light passes through a lens

Answer 13

the field at the back focal plane of the lens is the fourier transform of the transparency times a spherical wavefront

Question 14

what happens if we place our object transparency in the front focal plane of the FT lens

Answer 14

the FT of the input field is exactly reproduced both in amplitude and phase

Question 15

describe the features of interest if we place a grating in the object plane, at the ffp of the lens

Answer 15

• the transform lens will form the FT pattern of the bfp
• this would be a line of spots Sn for n= 0, +-1, etc for a grating
• on the image plane the waves form the inverted image of the grating

Question 16

How does the image of a grating through a lens arise

Answer 16

through a double diffraction process
- each point s in the transform plane acts as a point emitter of Huygens wavelets
- wave diffracted by object (grating in this case) then diffracted by the objective lens

Question 17

What is does the lens do when acting as a FT lens

Answer 17

it acts as an angle to position transformer
it brings the far field into a manageable distance to fulfil the Fraunhofer condition

Question 18

What does the formation of an image involve

Answer 18

the fourier transform of a fourier transform

Question 19

For what types of light can a lens perform the second fourier transform

Answer 19

visible

Question 20

What experiments proved the optical image formation ideas

Answer 20

abbe and porter

Question 21

what does a 4f system do

Answer 21

satisfy all condition for fourier transform and correct imaging
- performs a FT followed by an inverse FT so that the image is a perfect replica of the object

Question 22

How can masks be used in a 4f system

Answer 22

to filter particular spatial frequencies, masks can be inserted at the fourier plane

Question 23

If performing porters experiment, what is the significance of finding detail which was not in the original object

Answer 23

hints at the existence of artefacts in high resolution microscopy

Question 24

how could you remove raster

Answer 24

by filtering with a horizontal slit

Question 25

what are the effects of a low pass filter (hole)

Answer 25

blurs the edges due to the absence of high spatial frequencies

Question 26

what are the effects of a high pass filter (mask)

Answer 26

accentuating the edges acting as an edge detector, used for target recognition problems

Question 27

what are the effects of a vertical pass filter

Answer 27

allows only vetical fourier components corresponding to a horizontal grating

Question 28

an ideal edge detector would provide the derivative of the function, what will the transparency need to be to achieve this

Answer 28

FT transparency with a linearly increasing transmission in the transform plane will perform the derivative in the image plane

Question 29

What are pinholes commonly used for

Answer 29

to clean up (homogenise intensity of) gaussian laser beams in holographic imaging

Question 30

what filter is used in the dark field method

Answer 30

a high pass filter

Question 31

what is phase contrast microscopy used for

Answer 31

to observe objects which would not normally appear in an image because they are transparent or colourless

Question 32

What is the idea behind phase contrast microscopy

Answer 32

transparent objects will have an effect on the light passing through them - they alter the phase
diffracted light will have a phase shift with respect to the light passing straight through an objects

Question 33

How can we view the phase shift in phase contrast microscopy

Answer 33

the object approximates to a real DC (1) term and an imaginary phase term (i𝜙(𝑥))
after the FT the DC term is shifted by pi/2 relative to the phase term
ℱ [𝐴] = 𝛿 (𝜈𝑥) + 𝑖ℱ[𝜙 (𝑥) ]
we shift the DC term by pi/2 to see the amplitude using a quarter wave phase plate in the fourier plane of the transform lens ℱ [𝐴 ]= 𝑖𝛿 (𝜈𝑥) + 𝑖ℱ[𝜙 (𝑥) ]
taking the fourier transform again gets the filtered image
ℱ[ℱ 𝐴 ] = 𝑖(1 + 𝜙 −𝑥 ]
its intensity is modulated by the phase shift through the phase object
𝐼 = 1 + 2𝜙 −𝑥 + 𝜙2 −𝑥 ≈ 1 + 2𝜙(−𝑥)

Question 34

What is Schlieren imaging

Answer 34

one half of the diffraction plane is filtered out using an amplitude mask
resulting intensity is proportional to the gradient of the phase
𝐼(𝑥) ∝ |𝜕𝜙 (𝑥)/|𝜕𝑥

Question 35

what is used as a schlieren filter

Answer 35

knife edge