1.3 Images: 2d signals

Image processing

Linear shift invariant systems

$H$ is LSI if:

• $$H({}_1{f}_1(x, y)+{}_2{f}_2(x, y))=H(f_1(x, y))+H(f_2(x, y))$$ for all images $f_1$ , $f_2$ and scalar.
• $$H(f(x-x_0, y-y_0))=g(x-x_0, y-y_0)$$

2d fourier analysis

$$(u, v)=\int \,d y$$∞ ∞ x ∞ ∞ f x y u x v y where $$ is the 2D FT and $u$ and $v$ are frequency variables in $x(u)$ and $y(v)$ .

2D complex exponentials are eigenfunctions for 2D LSI systems:

2d sampling theory

We can sample the height of the surface using a 2D impulse array.

$$f_s(x, y)=S(x, y)f(x, y)$$ where $f_s(x, y)$ is sampled image in frequency

2D FT of $s(x, y)$ is a 2D impulse array in frequency $S(u, v)$

$$\text{multiplication in timeconvolution in frequency}$$ $$F_s(u, v)=(S(u, v), (u, v))$$

Nyquist theorem

Assume that $f(x, y)$ is bandlimited to $(B_x)$ , $(B_y)$ :

If we sample $f(x, y)$ at spacings of $(x)< \frac{\pi }{B_x}$ and $(y)< \frac{\pi }{B_y}$ , then $f(x, y)$ can be perfectly recovered from the samples by lowpass filtering: