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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

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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
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Cied
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Sanket Reply
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Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
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Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
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Azam
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Uday
I'm interested in Nanotube
Uday
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Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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