Lens aberrations


i
would like to now get into a what are called lens aberrations so i talked about a simple
lens and how it works in an ideal setting and that may be familiar to a lot of you already
but there are a lot of problems with lenses especially as we increase the field of view
so if you have a very large lens or lets say if you want the image produced by the lens
to be very large the focal plane to be to be large lets say
then ah there are a lot of imperfections and a lot of problems that arise and in virtual
reality we are highly motivated to do that right if you take a um virtual reality head
mounted display and you put a screen up very close and you put a lens in front of your
eyes and you would like to look through the lens at ah in in different directions you
would like to have a very wide field of view that means that these kind of peripheral effects
of lens and perfections are going to become very important so they are critical to the engineering of
virtual reality systems and thats why i want to go through step by step and talk about
different kinds of aberrations right aberration means what means something ah not right something
different from what we would like to have happen in an ideal setting the professional
optical engineers deal with aberrations of all source right thats their sort of daily
bread and butter lets say all right of characterizing aberrations trying to compensate for them
trying to keep the cost of the lens down in terms of materials and manufacturing all sorts
of issues come into this i want to give you just an idea of this of the kind of things
that can happen so i will talk about lens
lens aberrations so first one i will talk about is spherical
aberration so the cheapest surface to cut for four lenses
is is spherical however it is not ideal ah for for generating a perfect image in the
image plane what tends to happen is the following so you have parallel rays of light coming
in and then um they do not converge at a common focal point so it tends to spread out so that means that there is no place where
i could suppose i wanted to place my projection screen here and i try moving it back and for
some kind of vertical screen i am moving back and for here there is no place where i could
put the screen so that i would have everything perfectly in focus ah if i just use the central
part of the lens then it would be good enough but if i really insist on using all of the
lens and we get all the way out to the exterior and that lens perhaps you know ultimately
i said it could be a big lens very close to your eye if we insist on using these extremal
parts of the lens then the errors tend to get worse and worse so thats one problem um there is a solution so so potential solution
um you can make what are called aspheric lenses right if the sphere is a problem you use something
else so if you look for the word aspheric and lens you can find all kinds of engineering
literature research literature on that um from what i have read um the ideal way to
fix that is that for the lens you make the incoming surface elliptical and the outgoing
wind surface hyperbolic with very carefully chosen parameters obviously so thats harder to manufacture than a spherical
lens so it drives the costs up but it may reduce spherical aberration significantly
in an idealized setting it shouldnt in limit it all together but spherical aberration is
just one of many problems that a lens designer has to focus on so let me give you another
one two optical distortion in this case when we have a high field of
view this might happen if we put um perfectly square grid lines perfectly
straight grid lines ah in front of the the lens and if you do this for example if you
take the ah head mounted display from the lab and you you can do this you can take out
the lens cup and put it over some graph paper it will look something like this right so
so these straight lines become curved this is a kind of distortion called pincushion
distortion and notice that if we were just going to use
the lens in the central part here it might be good enough and you might never notice
you know this is a pretty extreme example but you can see that the further you get away
from the center the stronger the divergence is between being straight and being curved
and since ah because people in modern virtual reality systems want the field of view to
be as wide as possible they have to deal with this problem and it might not have been a problem maybe
ten or fifteen years ago in some other kinds of systems where the field of view is very
narrow where it generally looks like you are just staring at a small screen but if you
want to have your entire field of view filled with the stimulus presented by the virtual
reality system you are going to have to deal with this kind of curvature ah another common case which is essentially
the opposite is ah opposite or inverse is called barrel distortion
the lens ideally is radially symmetric right so it as radial symmetry and so these distortions
tend to be radially symmetric as well so then you can compensate for them by just making
a kind of transformation that adjusts the radius in polar coordinates so the so the so the the felled part doesnt
have any kind of transformation but you just kind of perform from the center if you want
to compensate for this you just perform some kind of radial stretching and the amount of
stretching you do does not depend on theta thats generally how you would compensate for
something like this so these two are inverses and whats actually done in software so i will mention just a little bit this is
done in head mounted displays and now we will little more detail on this ah later on in
the course but ah in software um barrel distortion and i think i called
this t just when i just gave you it as an example when i did the chain of transformations
barrel distortion ah is applied to compensate for the pincushion distortion
of a head mounted display lens so its um you know i like to think of it as
a barrel pincushion annihilation all right so the so the pincushion is one kind of distortion
you apply the opposite distortion which is a kind of mathematical inverse and then when
you put the two together you get the identity which hopefully should be everything is perfectly
straight ah its easier said than done its very very hard to tune the parameters of your
barrel distortion in software so that it compensates perfectly there are a couple of interesting
reasons for that um mainly the problem is that human perception is involved in a human
optical system is involved and your eyes move all of this together causes some great trouble
in trying to fix this problem ah which i will give you more understanding of as we go on
questions about this alright so one more i have five aberrations i am going
to cover here so ah its kind of a depressing topic right its just more is it can be a list
of five things that interfere with the performance of these systems and degrade your virtual
reality experience especially if you demand having a high field of view which i think
is reasonable to demand um so another one is chromatic aberration so in order to explain this i should start
talking about ah light waves and frequency decompositions of them so remember that light
one way to look at it is is composed of waves ah varying
between about four hundred nanometers and about seven hundred nanometers um in wavelength
right so we gave the formula for converting between
frequency and you may remember just a simple formula f equals c over lambda frequency is
a speed of light divided by the wavelength so you can convert back and for between these
now you have seen the visible spectrum before perhaps right we talked about the the basic
colors of the visible spectrum um right the color is a red orange yellow
green blue indigo so when people talk about the spectrum of
colors right you]and you have seen light shining through a prism before show examples like
this in a bit um they are talking about um pure sinusoids that correspond to one particular
frequency or one particular wavelength right so um thats a very unusual situation that
i have it may happen if you generate a light using a particular laser for example ah for
light thats bouncing around in this room right now there is um many many different wavelengths
all propagating at the same time right so there is a whole ah mixture of those and thats
called the spectral power of the of the light thats propagating right so those of you with uh signals and systems
background should understand um analysis of signals in the fourier domain and you can
talk about the frequency components of that right so if you have done some fourier analysis
kinds of things before this should not be surprising it applies to light it applies
to sound it applies to all sorts of problems where there are propagating waves so thats
something to to consider the spectral power of light and in terms of the spectrum um say once again
we talk about the visible spectrum and we pick a particular place along it we are just
talking about a single blip in terms of this overall spectral power of light there is one
place along the spectrum where there is light at a specific frequency or wavelength there
is not a mixture but more generally there is a mixture so spectral power of light you
could say that thats like a histogram if you like to think that way its not necessarily
discrete like this but you may you may helpful to think of it as in a continuous way but
um but think of it as kind of a histogram of wavelengths and um we are alternately going
to have light going into our eyes through the pupil and hitting the retina so what is
the spectral power or spectral distribution of that light what does it look like right
are there more greens more blue some reds you know how what exactly do we have ah as
we look at the range of wavelengths so there is two things that affect that one
is going to be the light thats emitted from sources and the second thing is going to effect
that is the various materials in the environment how do they reflect the light thats going
to affect our perception of the colour of something right this board appears to be green
and white in some way i think we all see it like that that depends on the properties of
the board and the properties of the lights that are shining on me now which as i see
them they are mostly white and there is a good reason for that so these are the two
things so i i will write them as there is the um emissions
of the light source so for example maybe i write four hundred here and i write seven
hundred here so these are um in nanometers so i want this to be the wavelengths and then
ah one example i will draw which are just rough plot they give you some kind of idea
on the based on reality in some way here ah this plot can correspond to ah daylight
and this may correspond to an incandescent lamp
all right so we consider the light source and generally when we have a light source
if we are going to engineer one if we want to do a photography or videography then we
would like the light source to contain as much of the spectrum as possible in the visible
range and its also best to have it close to equal you know so that it doesnt overly emphasize
ah lets say red instead of green right so so you would like it to be ah we call white
light which would be a perfect balance across the spectrum with the entire visible spectrum
represented when we get off to the extremes down here maybe whats that called longer shorter
wavelengths is what infrared wait a minute lets see where is the lower frequency long
longer wavelengths is infrared right so thats over here i guess yeah again we can always
change using this f equals c over lambda formula so i guess based on the way this is drawn
this is the infrared part and incandescent bulbs are they tend to get hot right they
generate a lot of heat which should generate a lot of infrared radiation which is why it
seems to be peaking here would be my guess ah so ultraviolet would be on the other side
all right so thats one part is the emission of the source and then there is the spectral
reflectance of the material now both of these subjects emissions of the source in spectral
reflectance become very important in computer graphics when if you want to make a completely
artificial scene a kind of virtual scene and you want to render how that might look you
need to make models of these things right we make artificial or virtual light sources
we will make virtual materials and then decide what its spectral reflectance property should
be and then hopefully it will look or could convince our brains that it looks reasonable
much like it would look in the real world so even if to graphics this is very important
but its also going to be important just in our understanding of how human vision works
in the real world because its part of this optical system i am going into all of this
explanation because i need to explain chromatic aberration but this is also going to be useful
in many of the things we we do in the course here its useful generally for virtual reality
so spectral reflectance will be another kind of plot maybe i have at the top of the plot
one for total reflectance and zero for no reflectance so this is the amount of reflectance
maybe i could call it the coefficient of reflectance if you like
so for example ah up here for still let me write my
so have my wavelengths here and ah one example of put along here is a snow how is that right so snow generally looks very very white um
it reflects pretty much reflects everything very nicely right but it is not ah it is not
a specular reflection and that when i look at the snow it does not look like a mirror
ah its a diffuse reflection but it tends to not look like its some special color right
so so thats because it has this kind of very ideal spectral reflectance ideal in terms
of looking perfectly white or just reflecting back whatever the light sources if i shine up perfectly red light with only
one wavelengths in it on the snow what color will the snow be was that green red red it will be exactly the light that i shined
on it right thats how it should look ideally you know there may be some distortions and
things um if i take something else like ah say i take a look at grass i realize maybe
this isnt the idealized color here for grass but a should show up a bit um may tend to
peak somewhere and ah it may peak right in the area where ah that wavelength should correspond
to green if the grass is green the grass turns yellow because its too dry then i guess it
will move somewhere else but it may have a more distinct signature if its something we
perceived to be a particular color so that even when you shine white light on it you
still get you may still get a very distinctive amount of emphasis along a particular part
of this visible spectrum so those are the things we perceived as having
a particular color you shine a white light on them and not everything is reflected back
its very specific the more specific it is the more we perceive a particular pure color
does that make sense all right so we have this spectra now so now think about it so
there is some light sources the lights been bouncing around and it could have multiple
inter reflections off of different objects based on the spectral reflectance and how
much power gets dissipated remember i said some of the light gets absorbed as well so there is less and less as it goes along
well by the time its all done lets suppose now that light decides to hit a lens right
so if the light hits a lens then here we have parallel rays coming in and lets do it here
off of the extreme ah you may have seen a picture like this for a prism before so as
the light comes in it turns out that the speed of propagation of the waves through the medium
depends on the frequency right or depends on the wavelength if you like either way which
ones going faster here reds going faster through it and blue is tending
to get more stuck is that right or is it the other way around right okay so red red seems
to be going faster through here um if if it were going straight through then it would
be essentially no effect right ah so the ah the shorter the wavelengths the slower it
goes through the material for the way this picture is drawn and so thats the case then you will have a focal
length for pure blue you will have a focal lengths for pure green you will have focal
lengths for pure red and if you generally have some kind of distribution some distribution
that corresponds to the spectral power of light or some histogram of various wavelengths
then um the focal plane will really be distributed in some kind of way right you may want to
put it in one place to really focus the reds well and in another place to focus the greens
well that doesnt sound very satisfying right but thats what you have to deal with um what
are some potential solutions to that so they are still under chromatic aberration i i shown
you a picture there of it possible solutions so
one um find a lens material find and use the lens material with a high whats called abbe
number ah what that means is that its low dispersion what that means is ah little dependency
well lets see let me put it this way um the refractive index n depends little on wavelength so there are some materials that refract the
same way the speed of light through them does not vary much um based on the frequency or
wavelength so thats one way the trouble with that is that these materials tend to be very
expensive so in mass produced consumer products this is not very reasonable unless someone
find some new material thats all of a sudden cheap easy to manufacture and all of a sudden
saves us number two um form a compound lens
with ah two materials all right two different materials or media
so this is commonly done in lens design so there may be ah one material which is causing
separation the crown glass and then another kind of glass called flint glass which is
put right up against the lens to try to bring them back together again so you can use two
materials play some kind of tricks its a kind of delicate art for the design of the ah lens so optical engineering becomes very difficult
because of these and again if you can get the costs right and the materials correct
not too brittle and you know whatever other things you need the optical properties you
may be able to make a compound lens like this called an achromatic doublet and you see even
in this picture it doesnt show it being perfectly compensated but it may greatly reduce the
um the chromatic aberration a third trick um which would be a computer science kind
of solution just fix it in software right and that is being done right now in current
ahead mounted displays so compensate just as we talked about the
optical distortion the barrel distortion compensating for pincushion distortion you can also compensate
in software by shifting or distorting based on the lets say sub pixel wavelengths no quite want to put pixel there because the
if you if you hold a magnifying lens up to a screen which i suggest you do and then you
can study the pixel structure very carefully you see that what we call one pixel in computer
graphics actually corresponds to several sub pixel elements that are lighting up right
in you know all the all correspond perfectly they are not just it isnt such that r g and
br just all superimposed in one place its a kind of pattern a tiling pattern um for
example the the the screens in the lab use a pentile display you may you may observe
all right questions about this so i think that all of them have some kind of flaws the
software compensation is not perfect on these other solutions are costly and again not perfect
its something that we unfortunately face yes in the human eye do we have ah this chromatic
aberration or is it thats a wonderful question yeah ah yes we
do so in the human eye we have chromatic aberration but your brain learns to compensate for it
so you dont see it and this is one kind of theme that will happen over and over again
there are all kinds of problems that our eyes have that our brain is just fixing the most
interesting one or most well known one probably being the the blind spot thats due to the
optic nerve so that part of your your retina is essentially
missing ah but do we see a blind spot ah i can show you
some experiments where you can try to find it but it’s a interesting but our brains are
repairing all of these flaws so that would be analogous to number three here so we the
software can fix it your brain also fixes it no its very very nice question sure all
right so that was also these one two and threes are inside of the big number three which is
the third aberration which was chromatic aberration so i probably should make these you know should
have been little little one twos and threes in that sense um for the record here was they
are just inside of my main numberings so number four is astigmatism
which corresponds to a lifter elliptical eccentricity all the lens i will just show a quick picture of this um
so instead of having perfect radial symmetry dont do that ah instead of having perfect
radial symmetry when you have light waves propagating in the horizontal plane going
through the lens there will be one focal point but when you have light waves propagating
in a vertical plane there will be another focal point and so this will mean that there
is no place where you can get a perfectly focused two dimensional image those of you in the audience who have who
are wearing a corrective eyewear some of you may have an astigmatism and that cannot be
fixed by just changing the diopter right so any problems of nearsightedness farsightedness
you can just play around with the diopter do some adding and subtracting and fix it
however you like and then in in a in a head mounted display if it has adjustments you
could move the lens forward or backwards away from the screen to compensate for your nearsightedness
or farsightedness however you cannot a compensate by just moving
lenses back and forth for a stigmatism you have to design some kind of corrective asymmetry
into the lenses to fix this um so ah so so there is some examples you may have seen before
where some some letters may look sharp along one direction then blurry along the other
or vice versa and there is no way to your brain may try to find an intermediate focal
length your eyes may try to find an intermediate focal length to try to bring everything roughly
into focus so a stigmatism is the fourth one and the
fifth one is called coma sometimes called co matic aberration instead of chromatic aberration
i will just call it coma thats derived from the word comet because it appears as a kind
of comet image here and this tends to happen when the imp when the part of the image you
care about is very far away from the optical axis so this is a central optical axis here and
these rays are coming in at an angle we are still looking at parallel rays for this and
so at the focal plane you get these kinds of patterns that emerge this but gets particularly
worse for thicker lenses because the waves are getting offset and shifted as they travel
through the lens and you get these kinds of this kind of repeating patterns you might
have if you have ever played with a magnifying glass on the sun you may have seen some kind
of repeated patterns like this you may also see it sometimes in photography or in movies
you may see some um what appears to be a bright spot but then a bunch of smaller bright spots
trailing off from it so that is an example of um you know this
does coma pattern you get its an example of whats called an airy pattern a i r y pattern
this will also happen um in a diffraction so you may have seen very simple examples
in physics classes where you you you form a slit in a material and then look at how
the light diffracts as it goes through that and the wave fronts will tangle in some kind
of complicated interference pattern and it will generate stripes so thats the kind of thing thats going on
here and thats an example of an airy pattern this coma is an example of an airy pattern
and you will indeed see these kinds of things if there is a very bright distinctive say
one pixel is lit up ah and its over at the periphery it may appear as some kind of comet
pattern so so its harder to find those but i have seen it happen before so thats all
of the optical aberrations that i want to cover any questions about that

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