Note that f(x,y) is the image and is REAL, but F(u,v) (abbreviate as F) is the FT and is, in general, COMPLEX.
Generally, F is represented by its MAGNITUDE and PHASE rather that its REAL and IMAGINARY parts, where:
MAGNITUDE(F) = SQRT( REAL(F)^2+IMAGINARY(F)^2 )
PHASE(F) = ATAN( IMAGINARY(F)/REAL(F) )
Briefly, the MAGNITUDE tells "how much" of a certain frequency component is present and the PHASE tells "where" the frequency component is in the image."
This is interesting... I've never thought of FT's in terms of Magnitude and Phase before.... I wonder if these two aspects of FT equations could have any correspondences with other "aspect pairs" (for lack of a better term) in Physics, for example, Amperage and Voltage, Speed and Acceleration, Space and Time, Wavelength and Frequency, etc., etc. (in other words, a thorough comparison would need to be done between Magnitude and Phase and other "aspect pairs" in Physics, and see what's the same, see what's different, etc.
There might be something to discover there... or at least (re)understand a little bit better...
Anyway, excellent article (I learned stuff I didn't know about the Fourier Transform) -- thanks to the author for writing it, upvoted and favorited!
Multiplying Mag/Phase representations is easy; adding them is hard. The opposite is true for the Re/Im representation.
So some of your representation choice depends on what you're doing. If you're multiplying two FTs (i.e. to perform convolution in the spatial or time domain) then Mag/Phase is easier. If you're adding signals together, Re/Im is easier.
[...]
Note that f(x,y) is the image and is REAL, but F(u,v) (abbreviate as F) is the FT and is, in general, COMPLEX.
Generally, F is represented by its MAGNITUDE and PHASE rather that its REAL and IMAGINARY parts, where: MAGNITUDE(F) = SQRT( REAL(F)^2+IMAGINARY(F)^2 ) PHASE(F) = ATAN( IMAGINARY(F)/REAL(F) )
Briefly, the MAGNITUDE tells "how much" of a certain frequency component is present and the PHASE tells "where" the frequency component is in the image."
This is interesting... I've never thought of FT's in terms of Magnitude and Phase before.... I wonder if these two aspects of FT equations could have any correspondences with other "aspect pairs" (for lack of a better term) in Physics, for example, Amperage and Voltage, Speed and Acceleration, Space and Time, Wavelength and Frequency, etc., etc. (in other words, a thorough comparison would need to be done between Magnitude and Phase and other "aspect pairs" in Physics, and see what's the same, see what's different, etc.
There might be something to discover there... or at least (re)understand a little bit better...
Anyway, excellent article (I learned stuff I didn't know about the Fourier Transform) -- thanks to the author for writing it, upvoted and favorited!