Image Transforms

Basic Geometric Transformations

• Transform theory is fundamental in image processing.
• Working with the transform of an image can provide more insight than working with the image itself.
• Applications: image enhancement, restoration, encoding, and description.

Properties of 2D DFT

• Linearity: $af(x,y)+bg(x,y)⟺aF(u,v)+bG(u,v)$
• Shifting: $f(x-x_0,y-y_0)⟺e^{-j2\pi (u{x}_0+v{y}_0)}F(u,v)$
• Modulation: $e^{j2\pi (u_{0}x+v_{0}y)}f(x,y)⟺F(u-u_0,v-v_0)$
• Convolution: $f(x,y)\ast g(x,y)⟺F(u,v)G(u,v)$
• Multiplication: $f(x,y)g(x,y)⟺F(u,v)\ast G(u,v)$
• Separability: $f(x,y)=f(x)f(y)⟺F(u,v)=F(u)F(v)$

Separability in Detail

• 2D Fourier Transforms can be done as a sequence of 1D Fourier Transforms along each axis. $F(u,v)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x,y)e^{-j2\pi (ux+vy)}dxdy$ $={\int }_{-\infty }^{\infty }{e}^{-j2\pi vy}\left[{\int }_{-\infty }^{\infty }f(x,y)e^{-j2\pi ux}dx\right]dy$ $={\int }_{-\infty }^{\infty }F(u,y)e^{-j2\pi vy}dy=F(u,v)$

graph LR
    A[2D DFT] --> B(1D DFT along the rows)
    B --> C(1D DFT along the cols)

• If functions are separable, i.e. $f(x,y)=f(x)g(y)$.
• The FT of a separable function is the product of the FTs of individual functions. $F(u,v)=H(u)G(v)$ if $f(x,y)=h(x)g(y)$.

Discrete Cosine Transform (DCT)

• Separates the image into parts of differing importance (spectral sub-bands).
• Similar to DFT but uses only cosine functions.

1D DCT

$F(u)={\left(\frac{2}{N}\right)}^{\frac{1}{2}}{\sum }_{i=0}^{N-1}\Lambda (i)\cdot \cos \left[\frac{\pi u}{2N}(2i+1)\right]f(i)$

where:

$$\Lambda (i)=\{\begin{array}{ll}\frac{1}{\sqrt{2}}& \text{for }\xi =0\\ 1& \text{otherwise}\end{array}$$

2D DCT

$F(u,v)=(\frac{2}{N}{)}^{\frac{1}{2}}(\frac{2}{M}{)}^{\frac{1}{2}}{\sum }_{i=0}^{N-1}{\sum }_{j=0}^{M-1}\Lambda (i)\Lambda (j)\cos [\frac{\pi u}{2N}(2i+1)]\cos [\frac{\pi v}{2M}(2j+1)]f(i,j)$

where:

\Lambda(\xi) = \begin{cases} \frac{1}{\sqrt{2}} & \text{for } \xi = 0 \\ 1 & \text{otherwise} \end{cases}$$ ### DCT Basic Operation - Input image: *N* by *M*. - $f(i, j)$: intensity of the pixel at row *i* and column *j*. - $F(u, v)$: DCT coefficient at row *u* and column *v*. - Most signal energy lies at low frequencies (upper-left corner of the DCT matrix). - Compression: Achieved because higher frequencies (lower-right values) are often small and can be neglected. - DCT input is typically an 8x8 array of integers, representing pixel gray scale levels (0-255 for 8-bit pixels).