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