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3.4 Fourier transforms 121

• Tent: The piecewise linear tent function,

tent(x) = max(0, 1 −|x|), (3.57)
2
has a sinc Fourier transform.
• Gaussian: The (unit area) Gaussian of width σ,
1 x 2
G(x; σ)= √ e − 2σ 2 , (3.58)
2πσ
has a (unit height) Gaussian of width σ −1 as its Fourier transform.

• Laplacian of Gaussian: The second derivative of a Gaussian of width σ,
x 2 1
LoG(x; σ)=( − )G(x; σ) (3.59)
σ 4 σ 2
has a band-pass response of
√
2π 2 −1
− ω G(ω; σ ) (3.60)
σ
as its Fourier transform.
• Gabor: The even Gabor function, which is the product of a cosine of frequency ω 0 and
a Gaussian of width σ, has as its transform the sum of the two Gaussians of width σ −1
centered at ω = ±ω 0 . The odd Gabor function, which uses a sine, is the difference
of two such Gaussians. Gabor functions are often used for oriented and band-pass
filtering, since they can be more frequency selective than Gaussian derivatives.

• Unsharp mask: The unsharp mask introduced in (3.22) has as its transform a unit
response with a slight boost at higher frequencies.

• Windowed sinc: The windowed (masked) sinc function shown in Table 3.2 has a re-
sponse function that approximates an ideal low-pass filter better and better as additional
side lobes are added (W is increased). Figure 3.29 shows the shapes of these such fil-
ters along with their Fourier transforms. For these examples, we use a one-lobe raised
cosine,
1
rcos(x)= (1 + cos πx)box(x), (3.61)
2
also known as the Hann window, as the windowing function. Wolberg (1990) and
Oppenheim, Schafer, and Buck (1999) discuss additional windowing functions, which
include the Lanczos window, the positive first lobe of a sinc function.
We can also compute the Fourier transforms for the small discrete kernels shown in Fig-
ure 3.14 (see Table 3.3). Notice how the moving average filters do not uniformly dampen
higher frequencies and hence can lead to ringing artifacts. The binomial filter (Gomes and
Velho 1997) used as the “Gaussian” in Burt and Adelson’s (1983a) Laplacian pyramid (see
Section 3.5), does a decent job of separating the high and low frequencies, but still leaves
a fair amount of high-frequency detail, which can lead to aliasing after downsampling. The
Sobel edge detector at first linearly accentuates frequencies, but then decays at higher fre-
quencies, and hence has trouble detecting fine-scale edges, e.g., adjacent black and white
columns. We look at additional examples of small kernel Fourier transforms in Section 3.5.2,
where we study better kernels for pre-filtering before decimation (size reduction).