Sampling

Reference: Digital Image Processing from Prof. Ghassan Alregib

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Analog vs. Digital

 

Why 'Rectangular' Lattice/Sampling?

• m = horizontal sampling intervals of period (X, 2X, 3X, ...)
• n = vertical sampling intervals of period (Y, 2Y, 3Y, ...)
• Here, X and Y refers to the sampling interval in the x/y direction each.
• Rectangular sampling is defined by the periodicity of the sampling interval in the x/y direction.

 

• pi, omega is two frequencies?
• m,n is an integer, finite values. -> f[m,n] shall be sampling.
• Fourier transform(F(u,v)), however, is continuous in terms of u and v.

 

 

• The right image is sampled image of the left image.
• Low rate of sampling: Still maintain the structure & details of the image.

 

 

• Below are the definitions of the fourier transform in analog signal.
• image in (6,4) coordinate would be similar to f_a(12,12).

 

• What is the relation between the fourier transform of our continuous signal / digital signal?
• When can we reconstruct our analog signal from digital signal, covered by the sampling theorem?

 

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• Here, X and Y are scaling factors.

 

• F_a and F shall be similar, just scaling matters.

• right image: spatial domain (circle repeated)
• bandlimited study: choice of X and Y can interfere the other circle(or maybe W).
• In order not to interfere, the maximum frequency X <= 1/W

 

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