Image Filtering Notes

Image Filtering

• Image filtering involves replacing each pixel with a weighted average of its neighborhood to reduce noise.

• The weights are determined by a filter kernel.

1D Case

• A 1D filter is applied to a 1D signal by convolving the filter with the signal.

  • For example, given a filter $[1/3, 1/3, 1/3]$ and a signal $[10, 12, 9, 11, 10, 11, 12]$, the output is calculated by taking the weighted average of the signal values under the filter.

• When applying a filter, you lose pixels at the edges because the filter extends beyond the signal at these points.

• Several strategies exist for handling boundaries:

  2. Zero Padding: Assume values outside the image are zero.

  3. Clamp to Edge: Replicate the edge pixels.

  4. Wrap Around: Treat the image as repeating (useful for periodic patterns).

  5. Crop: Simply crop the output to the valid region.

Image Filtering in 2D

• In 2D, a similar concept applies where a 2D filter kernel is used to compute the weighted average of a pixel's neighborhood.

• Given an image $I$ and a filter kernel $F$, the filtered image $I'$ is given by: $I'(x, y) = \sum<em>{u=-k}^{k} \sum</em>{v=-k}^{k} F(u, v) \cdot I(x+u, y+v)$

• Where $\begin{bmatrix} -k, -k \end{bmatrix}$ to $\begin{bmatrix} k, k \end{bmatrix}$ defines the neighborhood around the pixel $(x, y)$.

Example Filters

2. Box Filter

   • All weights are equal

   • $F = \frac{1}{(2k+1)^2} \begin{bmatrix} 1 &amp; 1 &amp; 1 \ 1 &amp; 1 &amp; 1 \ 1 &amp; 1 &amp; 1 \end{bmatrix}$

   • This computes the average value in the neighborhood.

3. Gaussian Filter

   • Weights are based on a Gaussian distribution.

   • $F(x, y) = \frac{1}{2 \pi \sigma^2} e^{-\frac{x^2 + y^2}{2 \sigma^2}}$

   • This gives more weight to closer pixels.

   • Approximates the solution to the diffusion equation.

4. Derivative Filters

   • Used to compute derivatives of the image.

   • Examples include the Sobel operator:

   • $Dx = \begin{bmatrix} -1 &amp; 0 &amp; 1 \ -2 &amp; 0 &amp; 2 \ -1 &amp; 0 &amp; 1 \end{bmatrix}, Dy = \begin{bmatrix} -1 &amp; -2 &amp; -1 \ 0 &amp; 0 &amp; 0 \ 1 &amp; 2 &amp; 1 \end{bmatrix}$

   • These filters are used in edge detection.

Properties of Linear Filters

2. Linearity: Filtering is a linear operation.

   • if $I3 = I1 + I2$, then $I3 * F = I1 * F + I2 * F$

3. Shift Invariance: The result of filtering does not depend on the position in the image.

   • if $I2(x, y) = I1(x - x0, y - y0)$, then $I2 * F = (I1 * F)(x - x0, y - y0)$

4. Commutativity: The order of filtering does not matter when multiple filters are applied.

   • $I * (F1 * F2) = (I * F1) * F2 = (I * F2) * F1$

5. Associativity: Filters can be applied in stages.

   • $I * (F1 * F2) = (I * F1) * F2$

6. Separability: Some filters can be separated into two 1D filters, which reduces computational complexity.

   • For example, a Gaussian filter is separable:

   • $F(x, y) = e^{-\frac{x^2 + y^2}{2 \sigma^2}} = e^{-\frac{x^2}{2 \sigma^2}} \cdot e^{-\frac{y^2}{2 \sigma^2}} = Fx(x) \cdot Fy(y)$

   • This means filtering can be done by first filtering all rows with $Fx$ and then filtering all columns with $Fy$

Applications of Image Filtering

2. Noise Reduction: Averaging filters (like the box filter) reduce noise but also blur the image.