工程數學(一)期末報告 Group 5
Summary
TLDRThis presentation delves into the realm of image filtering using Fourier series and Fourier transform. It begins with an introduction to time and frequency domains, explaining how frequency analysis can simplify signal processing. The script covers continuous and discrete Fourier transforms, highlighting their applications in image processing, such as reducing computational load in convolution through frequency domain multiplication. It also demonstrates the implementation of image filtering, particularly high-pass filtering, to enhance image edges, showcasing the practical utility of Fourier transforms in digital signal processing.
Takeaways
- 📚 The presentation introduces the concept of image filtering using Fourier series and Fourier transform, starting with the basics of time and frequency domains.
- 🔍 It explains how the time domain represents sound signals with amplitude over time, while the frequency domain shows the frequency components of those signals.
- 👦👧 The script uses the example of male and female voices to illustrate the differences in frequency components, with males having more low-frequency components and females having more high-frequency components.
- 🔧 The purpose of frequency analysis is to convert difficult-to-process time domain information into more analyzable frequency domain information.
- 🔄 The script covers both one-dimensional and two-dimensional signals, explaining how periodic signals in two variables can be represented using Fourier series.
- 🌐 The Fourier transform is introduced as a method to decompose a one-dimensional signal into complex exponential waves, which can be visualized as a combination of cosine and sine waves.
- 📊 The frequency domain is characterized by coordinates representing frequency and amplitude, with the absolute value of the complex number indicating amplitude and the angle representing phase.
- 📈 The presentation moves on to continuous time Fourier transform and discrete time Fourier transform, explaining their formulas and properties.
- 🛠️ It discusses the practical applications of Fourier transform in image processing, particularly in reducing computational resources needed for convolution through the use of frequency domain multiplication.
- 🖼️ The script details the process of implementing an image highpass filter using the Fourier transform, which enhances the edges of an image and aids in feature detection.
- 📉 The results of applying a highpass filter are demonstrated, showing a reduction in low-frequency components and an enhancement of high-frequency signals, indicating successful filtering.
Q & A
What is the main topic of the presentation?
-The main topic of the presentation is image filtering using Fourier series and Fourier transform.
What are the two domains introduced at the beginning of the presentation?
-The two domains introduced are the time domain and the frequency domain.
How do the male and female voices differ in terms of frequency components according to the script?
-The male voice has relatively large low-frequency components, while the female voice has richer high-frequency components.
What is the purpose of converting the time domain signal into the frequency domain?
-The purpose is to locate the signal flow within a specific time range and identify its content and meaning, making it easier to analyze.
What is the difference between Fourier series and Fourier transform in terms of the signals they deal with?
-Fourier series deals with periodic and continuous signals, while Fourier transform deals with non-periodic signals.
What are the three parameters needed to determine a sine wave in the frequency domain?
-The three parameters are frequency, amplitude, and phase.
What is the significance of the formula for the continuous time Fourier transform?
-The formula is used to synthesize signals that are not periodic, using all frequencies to determine the weight of the exponential components.
Why is the discrete Fourier transform (DFT) used instead of the continuous time Fourier transform for digital systems?
-DFT is used because digital systems work with discrete time signals, which are constructed from discrete complex exponentials.
What is the main advantage of using Fourier transform in image processing over convolution?
-Using Fourier transform reduces the computational resources needed for image processing by allowing direct multiplication in the frequency domain instead of convolution in the spatial domain.
How does a high-pass filter affect the appearance of an image?
-A high-pass filter enhances the edges of the image, which can be useful for edge detection.
What is the final step in the image processing procedure using Fourier transform mentioned in the script?
-The final step is performing an inverse Fourier transform on the result to obtain the final output image after applying the filter.
Outlines
📚 Introduction to Image Filtering with Fourier Series and Transforms
The video script introduces the concept of image filtering using Fourier series and transforms. It begins with an explanation of the time and frequency domains, using a practical example of male and female voice signals to illustrate the differences in frequency components. The script then delves into the purpose of frequency analysis, which is to convert difficult-to-process time domain information into more analyzable frequency domain information. It also covers the one-dimensional Fourier series and moves on to two-dimensional signals, explaining the periodic nature of these signals and how to obtain Fourier coefficients through integration. The script touches on the distinction between Fourier series, which handles periodic and continuous signals, and Fourier transform, which deals with non-periodic signals. It concludes with a brief introduction to the properties of continuous time Fourier transform and discrete time Fourier transform, setting the stage for further exploration in image processing.
🔧 Applications and Implementations of Fourier Transform in Image Processing
This paragraph discusses the application of Fourier transform in image processing, highlighting its efficiency over traditional convolution methods. It emphasizes the computational savings achieved by using Fourier transform to analyze and synthesize signals in the frequency domain. The script explains the process of image processing using Fourier transform, which involves performing a Fourier transform on the input image, applying a filter in the frequency domain, and then obtaining the final output image through an inverse Fourier transform. The paragraph provides an example of implementing a high-pass filter in image processing, showing how the edges of an image become enhanced after applying the filter. It also compares the frequency spectra before and after the application of the high-pass filter, demonstrating the effective removal of low-frequency signals. The script concludes with a brief mention of the assignment, which includes creating a presentation and recording a video, and thanks the audience for their attention.
🎉 Conclusion and Acknowledgment
The final paragraph serves as a conclusion to the video script, acknowledging the audience for their time and attention. It succinctly wraps up the presentation without adding further content, providing a polite and professional closing to the video.
Mindmap
Keywords
💡Image Filtering
💡Fourier Series
💡Fourier Transform
💡Time Domain
💡Frequency Domain
💡Frequency Analysis
💡Continuous Time Fourier Transform
💡Discrete Time Fourier Transform
💡High-Pass Filter
💡Spectral Analysis
Highlights
Introduction to image filtering with Fourier series and Fourier transform.
Explanation of time domain and frequency domain in signal processing.
Demonstration of how male and female voices differ in frequency domain representation.
Purpose of converting time domain to frequency domain for signal analysis.
Frequency analysis includes Fourier series and Fourier transform.
Introduction to two-dimensional signals and their periodicity in the T1-T2 plane.
Description of how to obtain Fourier series coefficients through integration.
Difference between Fourier series for periodic signals and Fourier transform for non-periodic signals.
Explanation of one-dimensional Fourier transform and its decomposition into complex exponential waves.
Properties of continuous time Fourier transform, including derivatives, post function, time shifts, and convolution.
Importance of discrete time signals in digital systems and the use of discrete Fourier transform.
Properties of discrete time Fourier transform and difference equations.
Application of Fourier transform in image processing to decompose images into sine and cosine components.
Efficiency of Fourier transform in image processing over convolution due to reduced computational resources.
Implementation steps of image processing using Fourier transform, including filtering and inverse transform.
Demonstration of high-pass filter application in image processing to enhance edges.
Comparison of frequency spectra before and after high-pass filtering to show the reduction of low-frequency components.
Assignment details including PPT production, video recording, and post-production.
Transcripts
hello everyone today our group will
introduce image filtering with foral
series and foral transform we will Begin
by introducing time domain frequency
domain and forer series then we will
cover continuous time forer transform
and discre time for transform finally we
will explain applications and
implementations so what are time domain
and frequency domain imagine a scene
where one female and one male record
their voice separately converts the male
and female voices into signal and saf
them we can see that it is time domain
above x-axis is time y AIS represent
sound with amplitude which means value
below is a frequency domain the xaxis is
in decb and the Y AIS is frequency it is
not difficult to see that the male voice
has relatively large low frequency
components and the female voice has
richer high frequency components than
the male
voice so what is the purpose of this
process by converting the time domain
into frequency domain we can locate the
signal flowing within a specific time
range and identify its contents and the
meaning of the content the process we
explained before is frequency analysis
which aims to convert the original time
domain information which is difficult to
process in frequency domain information
is easier to analyze frequency analysis
includes both foral series and foral
transform which we will introduce
later the one dimensional for Series has
already been introduced in class so we
won't elaborate on it here consider a
two dimensional signal as T1 T2 where T1
and T2 are two independent variables
this signal satisfies the foll following
equation where K and L are integrals in
other words this signal is periodic in
the T1 T2 plan the period in the T1
direction is capital T1 and the period
of in the t2 direction is capital T2 so
here is the for series of the
signal the
coefficients a MN can be obtained
through
integration in the time domain for
series deal with periodic and continuous
signals while in the frequency domain it
deal with a periodic discrete signals
for transform on the other hand
transform a periodic and continuous
signal in the time domain into a
periodic and continuous signal in the
frequency
domain generally speaking the one
dimensional fot transform decomposes a
one dimensional signal into several
complex EXP expansional waves EJ Omega X
because of the Olas formula each complex
expansional wave EJ Omega X can be
regarded as a combination of cosine wave
as J * sine wave for a sine wave three
parameters are needed to determine it
frequency amplitude and pH in frequency
domain one dimensional coordinates
represents frequency and the function
value corresponding to its coordinat is
is f of Omega which is a complex number
is amplitude the absolute value of f of
Omega is the amplitude a of the sine
wave of this frequency and its face
represented as a angle of f of Omega is
5 what is shown on the right side of the
figure below is only the ude diagram
which is also used more in Signal
processing part two is Introduction of
continuous time fre transform and
discreete time fre
transform first let's look at the
formula in the presentation below now we
consider functions X of T that are not
parad in this case there are not
necessarily such things as a fundamental
period and a fundamental frequency
therefore to synthesiz a signals X of T
we need all frequencies we see that X of
Omega determines the weight of the
exponential J Omega T in the Sy size of
the signal X of T that function Omega is
called the frent transform as of X of T
and will denoted by S of Omega and f of
x of T the further transform the
analyzes equation is given by the
formula in the presentation above
there are some properties of continuous
time fre transform include
derivatives post function timeing
shifts
convolution real
signals however in our lives we may use
more discrete time signals in digital
system so we should use discrete F
transform to solve this
problem a discrete sign signals is
contrust from discreete complex
exponentials not that a discreete
complex exponential J Omega n is
perioded in Omega with Period 2 pi that
is why the inore over Omega is reduced
to one period of 2
pi next there are some important
properties of the discre time for
transform and
difference
equations th the fation form is an
important image processing tool that is
used to decompose and image into its s
and cosine components in a f domain
image each point represent a
partic frequency contained in a special
domain
image because we are only concerned with
digital image
so we will restrict this discussion to
the discre frent
form so we come through the disre time
for transform to analyze and produce our
topic part three is about the
application and implementation of the
iMed filter with f transform and FAL
series why do we need for transform to
image processing
[Music]
although we can use convolution to do it
but convolution will was a significant T
amount of computation
resources now we recall fre transform
property is L when we want to convert
two signals we can first use f transform
to obtain the values of the signals in
the frequency domain and then Direct
multiply them
[Music]
together finally the result is the same
as convolution but we use l computation
resources now let's Implement image in
processing including F
transform before that we need to
understand the procedure of image
processing first we performed a foral
transform on the input image then we
apply the transform signal to the
filtering
equation finally we uptain the final
output image by performing an inverse
forer transform on the
result so our goal is to use this
picture to implement image highpass
filter the left side is our code and the
right side is the spectrum of the
original image
let's see part of
coat firstly we inut the image F
represents the result of the 2D F
transform then we need to put a filter
mask in the center of the
picture make the mass round and do high
pass filter
this is an illustrative diagram of
executing a
mask finally we do an inverted F
transform to get the
results let's see the
results after applying the high P filter
the edges of the photo image become
inhanced this also tell us that highp
filters can help us in in
detection here we compare the Spectra of
two graphs in a new picture of the low
frequency components are reduced this
indicates that the high PA filter we
applied successfully and effectively
filtered out the low prancy
signals this is our
reference there is our assignment
includes PPT production and video
recording and post
production thanks for your listening
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