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Scaled representations
- Big bars (resp. spots, hands, etc.)
and little bars are both
interesting
- Stripes and hairs, say
- Inefficient to detect big bars with
big filters
- And there is superfluous detail in the filter kernel - Alternative: - Apply filters of fixed size to images of different sizes - Typically, a collection of images whose edge length changes by a factor of 2 (or root 2) - This is a pyramid (or Gaussian pyramid) by visual analogy
A bar in the big images is a hair on the zebraās nose; in smaller images, a stripe; in the smallest, the animalās nose
Resample the
checkerboard by taking
one sample at each circle.
In the case of the top left
board, new representation
is reasonable.
Top right also yields a
reasonable representation.
Bottom left is all black
(dubious) and bottom
right has checks that are
too big.
Constructing a pyramid by
taking every second pixel
leads to layers that badly
misrepresent the top layer
The Fourier Transform
- Represent function on a new
basis
- Think of functions as vectors, with many components
- We now apply a linear transformation to transform the basis - dot product with each basis element - In the expression, u and v select
the basis element, so a function
of x and y becomes a function of
u and v
- basis elements have the form
F g x ( ( , y ))( u , v )= g x ( , y ) e
ā i 2 Ļ ( ux + vy ) dxdy R 2
e ā i 2 Ļ ( ux + vy )
To get some sense of what basis elements look like, we plot a basis element --- or rather, its real part --- as a function of x,y for some fixed u, v. We get a function that is constant when (ux+vy) is constant. The magnitude of the vector (u, v) gives a frequency, and its direction gives an orientation. The function is a sinusoid with this frequency along the direction, and constant perpendicular to the direction.
And larger still...
Phase and Magnitude
- Fourier transform of a real
function is complex
- difficult to plot, visualize
- instead, we can think of the phase and magnitude of the transform
- Phase is the phase of the complex
transform
- Magnitude is the magnitude of
the complex transform
- Curious fact
- all natural images have about the same magnitude transform
- hence, phase seems to matter, but magnitude largely doesnāt
- Demonstration
- Take two pictures, swap the phase transforms, compute the inverse - what does the result look like?
This is the magnitude transform of the cheetah pic
This is the phase transform of the cheetah pic
This is the magnitude transform of the zebra pic
This is the phase transform of the zebra pic
Reconstruction with cheetah phase, zebra magnitude
Various Fourier Transform Pairs
- Important facts
- The Fourier transform is linear
- There is an inverse FT
- if you scale the functionās argument, then the transformās argument scales the other way. This makes sense --- if you multiply a functionās argument by a number that is larger than one, you are stretching the function, so that high frequencies go to low frequencies
- The FT of a Gaussian is a Gaussian. - The convolution theorem - The Fourier transform of the convolution of two functions is the product of their Fourier transforms - The inverse Fourier transform of the product of two Fourier transforms is the convolution of the two inverse Fourier transforms - Thereās a table in the book.
