Fourier Optics Flashcards
What is the 2d fourier transform for image processing
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
What does the origin represent in the fourier expansion image
the DC term, since intensity can’t be negative there is usually a large central DC blob
what is the fourier transform of repeating pure horizontal or vertical cosines
bright spots symmetrically placed about the centre of the FT image
What are the fourier transform images of the intensity
the amplitude images
What happens in the fourier transform of an image with sharp edges (strong constrast)
the grey values change rapidly a bright line in the FT perpendicular to the sharp edge is usually seen
What is required to follow a sharp image edge
high frequency power
What is the idea behind image sharpening
a sharpening or high frequency emphasis filter is needed to preserve some of the low frequency information but to boost the higher frequencies
where does a collimated input beam come to a focus when passed through a lens
at the back focal plane
what is the far field light field
the 2d fourier transform of the aperture field
what is the fraunhofer diffraction formula
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
what are the angular spatial frequencies along x0 and y0
kx = kx1/z and ky = ky1/z
define the spatial frequencies u and v
u = kx/2pi and v = ky/2pi
what happens when a thin transparency is illuminated coherently by a monochromatic plane wave and the light passes through a lens
the field at the back focal plane of the lens is the fourier transform of the transparency times a spherical wavefront
what happens if we place our object transparency in the front focal plane of the FT lens
the FT of the input field is exactly reproduced both in amplitude and phase
describe the features of interest if we place a grating in the object plane, at the ffp of the lens
- 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
How does the image of a grating through a lens arise
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
What is does the lens do when acting as a FT lens
it acts as an angle to position transformer
it brings the far field into a manageable distance to fulfil the Fraunhofer condition
What does the formation of an image involve
the fourier transform of a fourier transform
For what types of light can a lens perform the second fourier transform
visible
What experiments proved the optical image formation ideas
abbe and porter
what does a 4f system do
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
How can masks be used in a 4f system
to filter particular spatial frequencies, masks can be inserted at the fourier plane
If performing porters experiment, what is the significance of finding detail which was not in the original object
hints at the existence of artefacts in high resolution microscopy
how could you remove raster
by filtering with a horizontal slit
what are the effects of a low pass filter (hole)
blurs the edges due to the absence of high spatial frequencies
what are the effects of a high pass filter (mask)
accentuating the edges acting as an edge detector, used for target recognition problems
what are the effects of a vertical pass filter
allows only vetical fourier components corresponding to a horizontal grating
an ideal edge detector would provide the derivative of the function, what will the transparency need to be to achieve this
FT transparency with a linearly increasing transmission in the transform plane will perform the derivative in the image plane
What are pinholes commonly used for
to clean up (homogenise intensity of) gaussian laser beams in holographic imaging
what filter is used in the dark field method
a high pass filter
what is phase contrast microscopy used for
to observe objects which would not normally appear in an image because they are transparent or colourless
What is the idea behind phase contrast microscopy
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
How can we view the phase shift in phase contrast microscopy
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)
What is Schlieren imaging
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
what is used as a schlieren filter
knife edge
