sum of multiple simple mappings(sum of sinusoids)
It also offer great convenience and insight when a given mapping can be simply scaled from previously known mappings.
2. Definition of Linearity
The mapping is called additive if the application of this mapping to the sum of two vectors x and y is the same as the sum of the two individual mappings:
The mapping is called homogeneous if the application of this mapping to a scaled vector is the same as the scaled mapping of the original vector:
The mapping L is called linear if the mapping is both additive and homogeneous.
3. A special function - Dirac Distribution
4. 2D Dirac Distribution
5. Shift Invariance and Convolution
Shifting of operation is common in image processing when the same operator kernel is shifted over each pixel position to process the entire image.
Convolution is a special operation for linear and shift invariant systems that can be represented by impulse response of such systems.
6. Definition of Convolution Operation
Impulse Response: The impulse response is defined at the response of a system to the Dirac impulse.
We have established that any function f(x,y) can be represented by linear combination of shifted &(x,y).
Assume that the system is linear and shift invariant, the application of any input f(x,y) to the system, denoted as (f*h)(x,y), can be expressed as:
6. Properties of Convolution
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