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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
what is Nano technology ?
Bob Reply
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?
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
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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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