Now that the relationships between the input and output of a
linear system are known, such a system may be utilized to enhance the
input. For example, assume an image has been degraded by some
distorting function d(x,y). The original image was convolved with the
distorting function, and the spectral contents of the ideal image
Fi(u,v) were attenuated by the frequency response D(u,v) of the
distorting system. By multiplying the degraded image by the inverse
of the D(u,v), the original ideal image is obtained. Any distortion
which can be represented as a linear system might theoretically be
canceled out using the inverse filter. A photograph produced in a
camera with a frequency response which rolls off slowly could be
sharpened by Fourier transforming the image, multiplying by the
inverse filter, and then inverse transforming. In this case, the
inverse filter is one in which the low frequencies are attenuated and
the high frequencies are accentuated (high pass filter). Because the
high frequencies represent the edges in the image, the edges are
accentuated and the photo appears sharper.17 As indicated in the
following diagram, the image is distorted by the function D(u,v) but
in some cases can be restored by multiplying by 1/D(u,v).
fi(x,y)- Fi(uv) X D(u,v):> fd(x,y) = blurred photograph
fd(x,y)-;-Fd(u,v) X 1/D(u,v > f'd(xy) = enhanced photograph
The linear blur of a camera is another classic example. Consider
traveling through Europe on a train with your camera. Upon getting
home and receiving your trip pictures, you find that all of them are
streaked by the motion of the train past the scenes you photographed.
Each point in the scene streaked past the camera, causing a line to be