30-11-2012, 05:08 PM
Discrete Cosine Transform
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Discrete Fourier Transform
• last classes, we have studied the DFT
• due to its computational efficiency the DFT is very
popular
• however, it has strong disadvantages for some
applications
– it is complex
– it has poor energy compaction
• energy compaction
– is the ability to pack the energy of the spatial sequence into as
few frequency coefficients as possible
– this is very important for image compression
– we represent the signal in the frequency domain
high only have transmit 2
– if compaction is high, we to a few coefficients
– instead of the whole set of pixels
Discrete Cosine Transform
• a much better transform,
from this point of view, is the DCT
– in this example we see the
amplitude spectra of the image above
– under the DFT and DCT
– note the much more
concentrated histogram
obtained with the DCT
• why is energy compaction
important?
– the main reason is
image compression
– turns out to be beneficial
in other applications
Image compression
• the transform throws away correlations
– if you make a plot of the value of a pixel as a function of one of
its neighbors
• a second advantage of working in the
frequency domain
– is that our visual system is less sensitive
to distortion around edges
– the transition associated with the edge
masks our ability to perceive the noise
– e.g. if you blow up a compressed picture,
it is likely to look like this
– in general, the
compression
errors are more
annoying in the
smooth image
regions
Quantizer
• note that we can quantize some frequency coefficients
more heavily than others by simply increasing Q
• this leads to the idea of a quantization matrix
• we start with an image block (e.g. 8x8 pixels)