Processing for compression - The MPEG - page 75

Figure 3.33 also shows that when a plane input picture is rotated about a horizontal axis, the distance from the top
of the picture to the eye is no longer the same as the distance from the bottom of the picture to the eye. The result
is that the top and bottom edges of the picture subtend different angles to the eye, and where the rays cross the
target plane, the image has become trapezoidal. There is now no such thing as the magnification of the picture.
The magnification changes continuously from top to bottom of the picture, and if a uniform grid is input, after a
perspective rotation it will appear non-linear as the diagram shows.
Figure 3.33: In a planar rotation effect the source plane ABCD is the rectangular input picture. If it is rotated
through the angle, ray tracing to a single eye at left will produce a trapezoidal image A 'B' C 'D' on the target.
Magnification will now vary with position on the picture.
Figure 3.34 shows that when a picture is warped, this causes the pixels in the picture to fail to register with the
standard pixel spacing. The solution is two-dimensional interpolation. One pixel value actually represents the peak
brightness of a two-dimensional intensity function, which is the effect of the modulation transfer function of the
system on an infinitely small point. In order to compute an interpolated value, it is necessary to add together the
contribution from all relevant samples, at the point of interest. Each contribution can be obtained by looking up the
value of a unity impulse curve at the distance from the input pixel to the output pixel to obtain a coefficient, and
multiplying the input pixel value by that coefficient.
Figure 3.34: When a picture is warped, the pixels no longer register with the standard locations. Interpolation in
two dimensions is used to create pixels on the standard grid.
The process of taking several pixel values, multiplying each by a different coefficient and summing the products
can be performed by the FIR (finite impulse response) configuration described earlier. The impulse response of the
filter necessary depends on the magnification. Where the picture is being enlarged, the impulse response can be
the same as at normal size, but as the size is reduced, the impulse response has to become broader
(corresponding to a reduced spatial frequency response) so that more input samples are averaged together to
prevent aliasing. The coefficient store will need a three-dimensional structure, such that the magnification and the
interpolation phase in each axis must both be supplied to obtain a set of coefficients. The magnification can easily
be obtained by comparing successive outputs from the address generator. Alternatively, the coefficients could be
computed dynamically.
[ 12 ] Newman, W.M. and Sproull, R.F., Principles of Interactive Computer Graphics , Tokyo: McGraw-Hill (1979)
[ 13 ] Gernsheim, H., A Concise History of Photography , London: Thames and Hudson, 9-15 (1971)
3.11 Transforms and duality
The duality of transforms provides an interesting insight into what is happening in common processes. Fourier
analysis holds that any periodic waveform can be reproduced by adding together an arbitrary number of
harmonically related sinusoids of various amplitudes and phases. Figure 3.35 shows how a square wave can be
built up of harmonics. The spectrum can be drawn by plotting the amplitude of the harmonics against frequency. It
will be seen that this gives a spectrum which is a decaying wave. It passes through zero at all even multiples of the