Lecture 3 - Fourier Transform Flashcards
What is a signal and what type of signal is a greyscale image?
- A greyscale image can be treated as a 2-D signal.
- A signal is any physical phenomenon that can be modelled as a function of time or position to some real- or vector-valued domain and is used to carry information
What is an analog signal?
Analog (Continuous) Image (Left):
- This image is smooth and does not have visible pixelation.
- It represents a continuous signal where values change smoothly without abrupt jumps.
- The grayscale shading transitions smoothly, without distinct blocks.
What is a digital signal?
Digital (Discrete) Image (Right):
- This image is pixelated and consists of distinct square blocks.
- It represents a discrete signal where values are sampled and quantized into a finite number of levels.
- The transition between shades is not smooth but rather stepped, showing discrete intensity levels.
What are 1D and 2D signals defined as?
- For 1D, the axis will usually be time, the intuitive way to think of the signal s(t) is an audio signal. For discrete signals, we also write sj (subscript j)
. - For 2D, the axes will usually be 2D-space, and\ we will call the signal s(x , y) an image. For discrete signals, we also write si ,j or s [x , y ]. (subscript i and j)
What is filtering?
Filtering is the process of transforming a signal by allowing certain frequencies to pass through while attenuating (reducing) others. It is widely used in audio processing, telecommunications, and signal processing to remove noise, enhance signals, or extract useful information.
- Filter: a tool used to separate the wanted from the unwanted (generic definition)
What is a Low-Pass filter?
Low-Pass Filter (LPF) - Suppressing High Frequencies
* Function: A low-pass filter allows low-frequency signals to pass through while attenuating (removing) higher frequencies.
* Effect on the Signal:
○ As shown in the first part of the image, the original signal contains both high and low frequencies.
○ After filtering, only the slower, low-frequency components remain, while the high-frequency parts are removed.
* Use Cases:
○ Removing high-frequency noise from an audio signal.
○ Smoothing signals in image processing.
○ Filtering out high-frequency interference in communication systems.
REFER TO SLIDES FOR DIAGRAM
What is a High-Pass filter?
High-Pass Filter (HPF) - Suppressing Low Frequencies
* Function: A high-pass filter allows high-frequency signals to pass through while attenuating (removing) lower frequencies.
* Effect on the Signal:
○ The second part of the image shows an original signal with both low and high frequencies.
○ After applying the high-pass filter, the low-frequency components are removed, leaving only the fast-moving high-frequency parts.
* Use Cases:
○ Removing background noise (such as deep hums or rumbles) from audio recordings.
○ Sharpening images by removing low-frequency components in image processing.
○ Enhancing high-frequency signals in communications (e.g., boosting treble in audio equalizers).
REFER TO SLIDES FOR DIAGRAM
What is the superposition principle?
The Superposition Principle applies to linear systems, meaning that the response of the system to multiple inputs can be determined by summing the responses to each input individually.
This includes additivity and homogenity -> REFER TO SLIDES 10 - 12
What is convolution?
REFER TO SLIDES FOR FORMULA
Used to measure how muhc two signals match as one slides over the other
* Convolution gives the area overlap between the two functions as a function of the amount that one of the original functions is translated after reversal
What is 2D convolution
REFER TO SLIDES FOR EXAMPLE
How does spacial filtering work
- The intensity of a pixel at (x,y)(x,y)(x,y) depends not only on itself but also on the neighbouring pixel values.
- A spatial mask (also called a filter, kernel, or template) is used to perform local enhancement.
- The mask slides over the entire image, performing convolution to modify pixel values.
- The size of the mask (e.g., 3×3, 5×5) determines how many neighbouring pixels influence each output pixel.
REFER TO SLIDE 20 FOR EXAMPLE
What problems arise masking near the boarder of an image and how can it be dealt with?
Part of the mask is located outside the image
To deal with it:
- Discard the problem pixel ie 512x512 becomes 510x510 using the formula -> Output Size = Input size - mask size + 1
- Zero padding: add a boarder of zeros around the image before applying the convolution using the formula -> Output Size = Input size + 2P - Mask Size + 1 (where P is the adding added (ie padded by 1 pixel around the entire image)
How is spatial filtering used for smoothing?
For blurring/ noise reduction;
* Blurring is usually used in pre-processing steps, e.g., to remove small details from an image prior to object extraction, or to bridge small gaps in lines or curves
* Equivalent to Low-pass spatial filtering in frequency domain, because smaller (high frequency) details are removed based on, neighborhood averaging (averaging filters)
How do you implement spactial filtering for smoothing?
Implementation: The simplest form of the spatial filter for averaging is a square mask (assume mxm mask) with the same coefficients 1/m2 to preserve the gray levels (averaging).
Applications: Reduce noise; smooth false contours
Side effect: Edge blurring
REFER TO SLIDE 24 - 26 FOR EXAMPLES
What is the time domain representation?
The amplitude (y) measured against time (x) to get kHZ
What is the frequency domain representation?
The amplitude (y) measured against the frequency (x) to visual the frequency spikes of a wave
What is the difference between spatial domain and frequenecy domain?
❖ In spatial domain, we deal with images as it is. The value of the pixels of the image change with respect to scene.
❖ Whereas in frequency domain, we deal with the rate at which the pixel values are changing in spatial domain.
❖ The term frequency in an image tells about the rate of change of pixel values.
What is process of spatial domain processing?
spatial domain (input image matrix) -> processing -> output image matrix
What is process of frequency domain processing?
spatial domain (input image matrix) -> Fourier transform -> frequency domain (complex image) -> manipulation > frequency domain (complex image) -> inverse fourier transform -> output image matrix
REFER TO SLIDE 34
What are the application of frequency domain?
❖ Applications
Preprocessing
▪ Filtering
▪ Enhancement, etc.
Data Compression
Feature Extraction
▪ Edge Detection
▪ Corner detection, etc.
Spacial Frequencies and Discrete Fourier Transforms (DFT)
REFER TO SLIDES 37 - 44
Define transforms
A signal can be converted from time domain into frequency domain using mathematical operators
For example: fourier series and fourier transformation
What is the Fourier series and what does it claim?
Periodic signals can be represented into sum of Sines and Cosines when multiplied with a certain weight.
❖ What is the Fourier series claim?
‘Any univariate function can be rewritten as a weighted sum of sines and cosines of different frequencies.’
REFER TO SLIDE 48 FOR EXAMPLE
What is the Fourier Transform?
The non periodic signals whose area under the curve is finite can also be represented into integrals of the sines and cosines after being multiplied by a certain weight.
NOTE: The DFT is the sampled Fourier Transform and therefore does not contain all frequencies forming an image, but only a set of samples which is large enough to fully describe the spatial domain image.
What are the applications for the Fourier Transform?
❖ The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent.
❖ The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression.
What is Discrete Fourier Transform (DTF)
❖ Discrete Fourier Transform (DFT) transforms uniformly spaced real signals from the spatial domain into complex numbers in the frequency domain
❖ DFTs are performed basically in any device which processes signals (e.g. cell phones, CDs, DVDs and so forth), using the Fast Fourier Transform (FFT) as a time-efficient algorithm
Discrete Fourier Transform (DTF) / Inverse DTF - 1D and 2D Formulas
REFER TO SLIDES
What are some important notes about DFT?
The value of the transform at the origin of the frequency domain, at F(0,0), is called the dc component
the values of the Fourier transform are complex, meaning they have real and imaginary parts.
▪ Complex number z = a + 𝑖 b, with 𝑖 ^2 = −1, has real part a and imaginary part b
we visually analyze a Fourier transform by computing a Fourier spectrum (the magnitude of F(u,v)) and display it as an image.
How can images be represented?
❖ An image can be represented as two components: high frequencies and low frequencies
Low frequencies make up the bulk of the information (areas of low variation in intensity) such as smooth regions in image
High frequencies make up the edges and fine detail (areas of high variation in intensity) such as edges in an image
Where are the low and high frequencies?
❖ Due to periodicity, low frequencies are shown in the four corners of the matrix I; the highest frequencies are at the center of the matrix I - REFER TO SLIDE 59
What is shifting and how does it affect the frequencies?
Shifting is moving the low frequenecies to the center and the high frequencies to the edges.
❖ After shifting, the Fourier Spectra shows both low and high frequency components
Low frequencies are near the origin
High frequencies are away from the origin
REFER TO SLIDE 60
What are the properties of DFT
Periodicity: The spectrum repeats itself endlessly in both directions with period N
Amplitude and Phase
REFER TO SLIDES 61 - 62
What is a complex plane?
❖ Complex Plane: A two-dimensional graph where the horizontal axis maps the real part and the vertical axis maps the imaginary part of any complex number or function. REFER TO SLIDE 63 FOR FORMULA
What is a polar form?
❖ Polar Form: Rather than using a and b, the polar coordinates use r and θ in their ordered pairs. REFER TO SLIDE 63 FOR FORMULA
What are the formulas for magnitude, power spectrum and phase angle?
REFER TO SLIDES 65
Why do we detect signals and how can we detect signals?
Often, we have two signals: (1) “real-life” that we are measuring, (2) ideal signal we are looking for.
REFER TO SLIDE 74 FOR FORMULAS
What is correlation in signals?
How similar two signals are by integrating their product overtime
REFER TO SLIDE 75 - 76 FOR FORMULA AND EXAMPLES
What is cross-correlation?
A way to find the phase (position) within a measurement (where the shape is known but the position isnt)
REFER TO SLIDE 77 FOR FORMULA
What are the properties of fourier transform?
- If 𝐹 𝜔 is the Fourier Transform of 𝑓 𝑥 , 𝐺 𝜔 is the Fourier Transform of 𝑔 𝑥 , and 𝑎 and 𝑏 are constants
– The Fourier Transform is linear
– Changing the spatial scale inversely affects frequency and amplitude
– Shifting the function only changes the phase of the spectrum
NOTE: that the amplitude spectrum is invariant to spatial shift.
REFER TO SLIDES FOR FORMULAS
What is convolution theorem?
Convolution in the spatial domain corresponds to multiplication in
the frequency domain and vice versa.
* For efficiency purpose, convolution in one domain is often implemented as multiplication in the other domain.
* Division in the frequency domain corresponds to deconvolution in the spatial domain. This can be the basis by which blurred images can be restored.
REFER TO SLIDE 82 FOR FORMULAS
