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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

1 f 2 size 12{ { {1} over {f rSup { size 8{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 k x , k y = F k x , k y 2 size 12{P left (k rSub { size 8{x} } ,k rSub { size 8{y} } right )= lline F left (k rSub { size 8{x} } ,k rSub { size 8{y} } right ) rSup { size 8{2} } rline } {}

where the two dimensional Fourier transform is given by:

F k x , k y = x = 0 N 1 y = 0 N 1 f k x , k y 2 e j2π N xk x + yk y size 12{F left (k rSub { size 8{x} } ,k rSub { size 8{y} } right )= Sum cSub { size 8{x=0} } cSup { size 8{N - 1} } { Sum cSub { size 8{y=0} } cSup { size 8{N - 1} } {f left (k rSub { size 8{x} } ,k rSub { size 8{y} } right ) rSup { size 8{2} } } } e rSup { size 8{ { { - j2π} over {N} } left ( ital "xk" rSub { size 6{x} } + ital "yk" rSub { size 6{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 k x , k y size 12{F left (k rSub { size 8{x} } ,k rSub { size 8{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.

Focus analysis of an in-focus image

Focus analysis of 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 1 f 2 size 12{ { {1} over {f rSup { size 8{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 1 f 2 size 12{ { {1} over {f rSup { size 8{2} } } } } {} fall off can be considered the center of focus in the image.

Questions & Answers

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Adaptive region of interest for video. OpenStax CNX. Dec 14, 2010 Download for free at http://cnx.org/content/col11256/1.1
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