Lecture 3 - Fourier Transform Flashcards
(40 cards)
What is a signal?
A signal is any physical phenomenon that can be represented as a function over time or space and carries information.
* Formally: a function s(t) or s(x,y) that maps from a domain (time or space) to a real or complex range.
* Signals can be 1D (like audio over time) or 2D (like images over space).
What is Analog and Discrete (digital) Signal
- Analog signal: both domain and range are continuous.
- Discrete (digital) signal: both domain and range are sampled into finite steps.
What is a 1D vs 2D Signal
- 1D signal: function of a single variable over time, e.g., audio s(t)
- 2D signal: function of two variables over space, e.g., image s(x,y)
What is a Filter/Kernel/Operator?
- Filter: transforms one signal into another by emphasizing or removing components.
Also called: kernel, mask, or window
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
What is additivity and homogenity?
Additivity: T(s1 + s2) = T(s1) + T(s2)
Homogenity T(alpha * s) = alpha * T(s)
What is Linear and non-linear filtering?
A filter is linear if it satisfies the superposition principle, otherwise it is non-linear filtering
Examples:
Scaling of intensity s(x,y) -> c * s(x,y) is linear
Geometric operations (translation, rotation mirroring) are linear
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But gamma correction s(x,y) -> c * s(x,y)^2 is non-linear
What is Convolution?
Used to measure how much 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
An Example is blurring an image using an average kernel
How to do convolution/What is 2D convolution?
REFER TO SLIDES
What is signal detection?
Signal detection refers to identifying a known pattern (signal) within a larger signal.
* Often implemented using correlation to see if a known signal appears at any point.
What’s correlation of 2 signals?
Correlation measures the similarity between two signals.
- Used for pattern matching and detection.
What’s cross-correlation of 2 signals?
Cross-correlation is correlation of two different signals
- Measures similarity at varying shifts.
- Key in template matching, signal detection.
Convolution vs Correlation
Convolution: Reverses signal
Correlation: Does not reverse signal
What’s auto-correlation?
Auto-correlation is the correlation of a signal with itself
- Used to find repeating patterns or periodicity in a signal.
What is a periodic signal?
A periodic signal is a signal that repeats itself at regular intervals over time
A signal f(t) is periodic if: f(t + T) = f(t) for all t, where T is the period
What is Fourier Series Theorem?
Any periodic function can be expressed as a weighted sum of sines and cosines
Analyse Fourier Series (e.g. rectangular wave)
- Represented as the sum of odd harmonics
- Higher harmonics contribute to sharper edges
REFER TO SLIDES FOR BREAKDOWN
What are the steps to find Fourier Coefficients?
Step 1: Identify the Period T
Determine the duration over which the signal repeats itself. This period defines the basic interval for analysis and sets the fundamental frequency of the signal.
Step 2: Compute the Average Value (DC Component)
Find the average value of the function over one period. This value, often called the DC component, represents the constant (non-varying) part of the signal.
Step 3: Compute the Cosine Components
Extract the parts of the signal that behave like cosine waves at different frequencies. These capture the even symmetry (mirror symmetry around the y-axis) of the signal.
Step 4: Compute the Sine Components
Extract the parts that behave like sine waves at different frequencies. These describe the odd symmetry (mirror symmetry with sign change) in the signal.
Step 5: Reconstruct the Signal Using the Coefficients
Combine the average value, cosine parts, and sine parts together to reconstruct the original signal. The more terms you use, the more accurate the approximation becomes.
What is Discrete vs Continuous Spectrum
A discrete spectrum arises when a signal is periodic. This means the signal repeats itself after a fixed interval (period T).
- Fourier Series → discrete spectrum (periodic signal)
A continuous spectrum arises when a signal is non-periodic. This means the signal does not repeat itself.
- Fourier Transform → continuous spectrum (non-periodic signal)
What is the Fourier Transform?
Used to decompose an image into its sine and cosine components.
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.
Why negative frequencies?
- Negative frequencies represent phase-shifted versions of positive frequencies.
What is the Inverse Fourier Transform (IFT)
Recovers the signal from its frequency representation.
Why are we interested in Fourier Transform (what are the advantages)?
- Separates image content by frequency
- Makes filtering efficient
- Enables compression, denoising
- Highlights patterns (edges, textures)
What is Amplitude and Phase?
Amplitude and Phase Concepts
* Amplitude: magnitude of frequency component.
* Phase: angle, determines position/structure.
In images, phase carries critical structure information.
