Linear Filters

Image Filtering: Linear Filters

Convolution

The fundamental operation in image filtering is convolution. It’s represented as:

f' = g * f

Where:

• f’: The output (filtered) image.
• g: The filter (also known as the filter kernel).
• f: The input image.
• *: The convolution operator.

Linearity

A crucial property of many filters is linearity. A linear filter satisfies the following condition:

$g(\alpha x+\beta x)=\alpha g(x)+\beta g(y)$

Where:

• g(): The filter function.
• α, β: Scalar constants (scaling factors).
• x, y: Input signals (e.g., pixel intensities or image patches).

Interpretation: The filter’s response to a linear combination of inputs is the same as the linear combination of the filter’s responses to each input individually.

Time and Space Invariance

Space Invariance (or Translational Invariance):

An operation or system is space-invariant if its response is the same regardless of the object’s location in the image. If you shift the input image, the output is simply shifted by the same amount.

Example: A convolutional filter for edge detection is space-invariant. It will detect an edge pattern whether it’s in the top-left or bottom-right of the image.

Time Invariance (or Temporal Invariance):

In video processing, a system is time-invariant if a time delay in the input sequence results in only a corresponding time delay in the output, with no other changes in the output. The system’s behavior doesn’t change over time itself.

Example: A frame differencing algorithm for motion detection is time-invariant. If you apply it to a video starting at time T=0, it detects motion. If you start the same video at T=5, it will still detect the same motion events, just 5 time units later.

Types of Linear Filters

Based on these properties, we can define:

• Linear Function Filter: A filter that satisfies the linearity property.
• Linear Time-Invariant (LTI) Filter: A filter that is both linear and time-invariant (shift-invariant).
• Linear Space Invariant (LSI) Filter: Equivalent to an LTI filter, emphasizing spatial invariance.

Summary

Linear, shift-invariant (LSI or LTI) filters are fundamental in image processing. Their properties allow for efficient implementation and analysis. They are widely used for various tasks, including:

• Smoothing/Blurring: Reducing noise and fine details.
• Sharpening: Enhancing edges and details.
• Edge Detection: Identifying boundaries between regions.
• Feature Extraction: Extracting meaningful information from images.

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Linear Transformation for Pixel Values

This derivation explains a common linear transformation used to map pixel intensity values from an input range to a desired output range.

Formula:

$$\frac{f^{\prime }-f_{\text{min}}}{f_{\text{max}}-f_{\text{min}}}=\frac{g-g_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}$$

Where:

• f’: The output pixel value after transformation.
• f: The input pixel value.
• f_min: The minimum input pixel value.
• f_max: The maximum input pixel value.
• g_min: The desired minimum output pixel value.
• g_max: The desired maximum output pixel value.
• g: The input pixel value after applying the filter function.

Derivation:

The formula is derived from the general equation of a straight line:

$$f^{\prime }=m\cdot g+c$$

By substituting the known input and output range boundaries (f_min, f_max, g_min, g_max) and solving for the slope (m) and y-intercept (c), we arrive at the given formula.

1. Finding ‘m’ and ‘c’:

We have two known points:

• When g = g_min, we want f' = f_min
• When g = g_max, we want f' = f_max

Substituting these into the linear equation, we get two equations:

$$f_{\text{min}}=m\cdot g_{\text{min}}+c(\text{Equation 1})$$ $$f_{\text{max}}=m\cdot g_{\text{max}}+c(\text{Equation 2})$$

2. Solving for ‘m’ (Slope):

Subtract Equation 1 from Equation 2:

$$f_{\text{max}}-f_{\text{min}}=m\cdot (g_{\text{max}}-g_{\text{min}})$$

Solve for m:

$$m=\frac{f_{\text{max}}-f_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}$$

3. Solving for ‘c’ (Y-intercept):

Substitute the value of m back into Equation 1:

$$f_{\text{min}}=\left[\frac{f_{\text{max}}-f_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}\right]\cdot g_{\text{min}}+c$$

Solve for c:

$$c=f_{\text{min}}-\left[\frac{f_{\text{max}}-f_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}\right]\cdot g_{\text{min}}$$

4. Substituting ‘m’ and ‘c’ back into the linear equation:

$$f^{\prime }=\left[\frac{f_{\text{max}}-f_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}\right]\cdot g+f_{\text{min}}-\left[\frac{f_{\text{max}}-f_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}\right]\cdot g_{\text{min}}$$

5. Rearranging to get the equation in the notes:

Subtract f_min from both sides:

$$f^{\prime }-f_{\text{min}}=\left[\frac{f_{\text{max}}-f_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}\right]\cdot (g-g_{\text{min}})$$

Finally, divide both sides by (f_max - f_min):

$$\frac{f^{\prime }-f_{\text{min}}}{f_{\text{max}}-f_{\text{min}}}=\frac{g-g_{\text{min}}}{g_{\text{max}}-g_{\text{min}}}$$

Purpose:

This linear transformation is often used for:

• Contrast Stretching: Expanding the range of pixel values to improve the contrast of an image. For instance, mapping the input range [50, 150] to the output range [0, 255].
• Normalization: Scaling pixel values to a standard range (e.g., [0, 1]).
• Intensity Adjustment: Shifting the brightness of an image.

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