工程數學(一)期末報告 Group 5

Summary

This 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

00:00

📚 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.

05:01

🔧 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.

10:03

🎉 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.

Keywords

💡Image Filtering

Image filtering is a process used to enhance or modify an image by applying mathematical algorithms to the pixel data. In the context of the video, image filtering is discussed in relation to Fourier series and Fourier transform, which are techniques used to analyze and process signals, including images, in the frequency domain. The script mentions implementing image filters using the Fourier transform to achieve effects like high-pass filtering, which enhances the edges of an image.

💡Fourier Series

A Fourier series is a mathematical tool used to represent periodic functions as an infinite sum of sine and cosine waves. In the video script, the Fourier series is introduced as a method to handle periodic and continuous signals in the time domain. It is used to break down complex signals into simpler, periodic components, which can then be analyzed or manipulated.

💡Fourier Transform

The Fourier transform is a technique that transforms a time-domain signal into the frequency domain, allowing for the analysis of frequency components. The script explains that the Fourier transform can be applied to both continuous time signals and discrete time signals, and it is fundamental in understanding how signals can be represented and processed in terms of their frequency content.

💡Time Domain

The time domain refers to a representation of a signal as a function of time. In the script, the time domain is depicted as a graph with time on the x-axis and amplitude on the y-axis, showing how signals, such as sound waves, vary over time. The time domain is contrasted with the frequency domain, which focuses on the frequency components of the signal.

💡Frequency Domain

The frequency domain is a representation of a signal in terms of frequency rather than time. In the video, the frequency domain is shown as a graph with frequency on the x-axis and amplitude on the y-axis. It is used to analyze the frequency components of a signal, such as the different frequencies present in male and female voices, and is essential for processes like frequency analysis and filtering.

💡Frequency Analysis

Frequency analysis is the process of determining the frequency components of a signal. The script explains that by converting a signal from the time domain to the frequency domain, one can identify the signal's content and meaning within a specific time range. Frequency analysis is crucial for understanding and manipulating signals, and it includes techniques like Fourier series and Fourier transform.

💡Continuous Time Fourier Transform

The continuous time Fourier transform is a specific type of Fourier transform applied to signals that are not periodic and do not have a fundamental frequency or period. The script describes this transform as a way to synthesize signals using all frequencies, with the transform function determining the weight of each frequency component in the signal.

💡Discrete Time Fourier Transform

The discrete time Fourier transform is used for discrete signals, which are composed of discrete complex exponentials. The script mentions that this transform is particularly useful for digital systems where signals are discrete. It allows for the analysis and manipulation of signals in the frequency domain, taking advantage of properties like periodicity in the discrete domain.

💡High-Pass Filter

A high-pass filter is a type of filter that allows high-frequency signals to pass while attenuating low-frequency signals. In the context of the video, a high-pass filter is applied to an image to enhance its edges, which are typically associated with high-frequency components. The script provides an example of implementing a high-pass filter using the Fourier transform to achieve this effect.

💡Spectral Analysis

Spectral analysis involves examining the spectrum of a signal or image to understand its frequency components. The script discusses spectral analysis in the context of image processing, where the Fourier transform is used to decompose an image into its sine and cosine components in the frequency domain. This analysis helps in understanding the frequency content of an image and is crucial for applying filters and other image processing techniques.

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