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2D Convolution Equation

For an image of size M x N and kernel of size k, a direct implementation would require O(MNk^2) time, which is infeasiblefor a real-time implementation. Instead, it is possible to exploit the separable property of certain convolution matricesh, e.g. that h can be represented as the product of two onedimensional matrices h1 and h2. This effectively reduces ourtwo-dimensional convolution of Equation (1) to two separate instances of one-dimensional convolution:

Separable Convolution

which is implemented as

Separable Convolution Equation

Execution times of direct and separable 2D Convolution

For the same M x N input and square kernel of size k, a separable implementation reduces the computational complexityto O(MNk). By comparison, an implementation using the FFT would cost O(MN logMN). In practice, considerthe separable convolution speed gain evident in the following results for a convolution of a 3 x 3 Gaussian filter kernel withimages of varying sizes (visualized in Figure 2).

Execution Time of Direct 2D Convolution

Execution Time of Separable 2D Convolution

As detailed in later sections, we take advantage of the fast computational complexity of separable convolution in ourimplementation, as all of the filters we apply are separable into one-dimensional matrices.

Noise elimination: two-dimensional gaussian filtering

The first task in our edge detection algorithm is denoising the input in order to reduce the amount of high-frequencycontent in the image by as much as possible without destroying critical information points in the image (e.g. the real edges).We filter out high-frequency noise so that random noise is not mistakenly interpreted as an edge, as edges correspond topoints in the image where the gradient has an above-threshold magnitude.

For example, consider the 5312 x 2988 pixel image (taken at a high ISO to deliberately introduce noise) of Figure 3and a plot of its grayscale intensity versus the image’s spatial dimensions in Figure 4. It is clear from the mesh that thereis much high-frequency noise, which can be removed with a low-pass filter.

Fig. 3. unfiltered noisy image

noisy image

Fig. 4 grayscale intensity of unfiltered noisy image at each pixel

noisy image intensity plot

Fig. 5. fft magnitude of unfiltered noisy image (log scale)

FFT magnitude of unfiltered noisy image (log scale)

To low-pass filter our image, we apply a discrete Gaussian filter. Generally, a Gaussian blur kernel of size 2n+1 x 2n+1(where n is a positive integer, and with parameter sigma) is given by

equation for Gaussian kernel

Our implementation of Gaussian filtering uses the constantsized (k = 3), constant-sigma kernel

Gaussian kernel

This kernel is implemented separably as

Separable Gaussian Kernel

Applying a Gaussian filter with parameters k = 5 and sigma = 5 to the above image significantly denoises the image withoutsacrificing edge precision, which can be seen from the spatial intensity plot in Figure 7.

Fig. 6. gaussian-filtered noisy image

Gaussian-filtered noisy image

Fig. 7. grayscale intensity of gaussian-filtered noisy image

Grayscale intensity of Gaussian-filtered noisy image

Fig. 8. fft magnitude of gaussian-filtered noisy image

FFT magnitude of Gaussian-filtered noisy image

Gradient computation: the sobel operator

Edges in the image correspond to pixel locations at which there is a rapid change in intensity with respect to the image’sspatial dimensions. Thus, edge pixels are defined as those whose gradient magnitude |G| is maximized along thegradient direction ThetaG.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Elec 301 projects fall 2015. OpenStax CNX. Jan 04, 2016 Download for free at https://legacy.cnx.org/content/col11950/1.1
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