EE4206/EE5806 Image processing : Differentiating filters
1. A black-and-white picture can be thought of as a two-dimensional "continuous time" signal x(t1,t2), where t1 ad t2 are the horizontal and vertical coordinates, respectively, and x(t1,t2) is the BRIGHTNESS of the image.
2. If the image is repeated periodically in the horizontal and vertical directions, then it can be represented by a two-dimensional Fourier series consisting of SUMS of products of complex exponentials, e^(jw1t1) and e^(jw2t2),
that oscillate at possibly different frequencies in each of the two coordinate directions.
3. SLOW variations in brightness in a particular direction are represented by the LOWER harmonics in that direction. For example, consider an edge corresponding to a sharp transition in brightness that runs vertically in an image.
4. Since the brightness is constant or slowly varying ALONG the edge, the frequency content of the edge in the vertical direction is concentrated at low frequencies.
5. In contrast, since there is an ABRUPT variation in brightness ACROSS the edge, the frequency content of the edge in the horizontal direction is concentrated at higher frequencies.