# 0.5 Focus detection

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Focus detection

Focus Detection:

One important aspect of images is focus. While qualitatively deciding whether an image is in focus or not is relatively easy, quantitatively it can be quite difficult. One way to detect whether or not an image is in focus is by examining its power spectrum.

## Power spectrum and focus

It is generally assumed that natural images are made up of fractals, and it can be shown that the power spectrum (power as a function of frequency) of a natural image should fall off as

$\frac{1}{{f}^{2}}$

where f is the frequency.

As an image goes out of focus, it becomes blurred. That is to say that the edges are less sharp. If an image contains less sharp edges, its power spectrum will contain less high-frequency power. The power spectrum of an out-of-focus image should, therefore, fall off faster than an in-focus image.

So by calculating the power spectrum and examining its linear regression on a loglog plot (log[power] vs log[frequency]), we can get an indicator of focus.

## Calculating the power spectrum

The power specturm is simply the square of the two dimensional Fourier transform:

$P\left({k}_{x},{k}_{y}\right)=\mid F{\left({k}_{x},{k}_{y}\right)}^{2}\mid$

where the two dimensional Fourier transform is given by:

$F\left({k}_{x},{k}_{y}\right)=\sum _{x=0}^{N-1}\sum _{y=0}^{N-1}f{\left({k}_{x},{k}_{y}\right)}^{2}{e}^{\frac{-\mathrm{j2\pi }}{N}\left({\text{xk}}_{x}+{\text{yk}}_{y}\right)}$

Note that denotes an individual image pixel. You may have noticed that the above equations define a square image. While a non-symmetric two dimensional Fourier transform exists, using square images eases the process.

Because whether or not an image is in focus depends on the magnitude of power as a function of frequency, once the two dimensional power spectrum is computed as above, we radially average the spectrum. That is, the average of the values which lie on a circle a distance R from the origin is taken. Because frequency increases linearly in all directions from the origin, radially averaging the power spectrum gives the average power at one frequency , effectively collapsing the two dimensional spectrum to one dimension. It should be noted that $F\left({k}_{x},{k}_{y}\right)$ has been centered around baseband, meaning the frequency of the rotionally averaged power spectrum extends from 0 to N/2 -1.

The power spectrum’s falloff on a loglog plot can now be examined to determine focus.

## Illustrative example of focus analysis on entire image

The following images show the results of a linear regression of the power spectrum on a loglog plot for an in-focus image and an out-of-focus image.

As expected, the out-of-focus image yielded a linear regression with a slope of -3.3, while the in-focus image yielded a linear regression with a slope of -2.3, indicating that the out-of-focus image has fewer high frequency components.

## Determining regions of focus

Because frequency and power should be related exponentially as stated before, the loglog plot should display a linear relationship. Taking the linear regression of the loglog plot leads to an estimate of the frequency fall off. For example, if the linear regression where to return a slope of -2, we know that the power spectrum falls off as $\frac{1}{{f}^{2}}$ .

The same principles used to determine whether or not an image is in focus can be used to determine what region of an image is in focus. Because cameras can only focus on one spatial plane, in a single picture certain objects will be more in focus than others. To determine which region of an image is in focus, one simply has to divide the image into separate spatial region and then use the methods described above on each region. The region whose power spectrum conforms most closely to the $\frac{1}{{f}^{2}}$ fall off can be considered the center of focus in the image.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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