Linear filtering is a fundamental operation in image processing and computer vision. It is used to modify or enhance an image by applying a filter (kernel) over a local neighborhood of pixels. The result is a new image in which each pixel value is a weighted sum of its neighbors, with weights defined by the filter kernel.
1. Definition of Linear Filtering
A linear filter transforms an input image f(i, j) into an output image g(i, j) by computing:
g(i, j) = Σ Σ f(i + k, j + l) × h(k, l)
where:
h(k, l)is the filter kernel (weights)- The operation is performed over a small window (neighborhood)
This process is also called convolution when the filter is applied with flipped indices:
g(i, j) = Σ Σ f(i − k, j − l) × h(k, l)
In compact form: g = f * h
2. Properties of Linear Filters
- Shift Invariant: Same operation is applied throughout the image.
- Linear: Follows the principle of superposition.
- Efficient: Can be computed using convolution or correlation.
3. Common Types of Linear Filters with Examples
a) Box Filter (Moving Average Filter)
– All pixel weights are equal (e.g., 3×3 matrix of 1s). – Averages pixel values in a K×K window. – Used for basic blurring.
1/9 × [1 1 1 1 1 1 1 1 1]
Example: Used to smooth noisy images.
b) Bilinear (Tent or Bartlett) Filter
– A separable filter that gives more weight to central pixels. – The 2D kernel is the outer product of two linear vectors.
1/16 × [1 2 1 2 4 2 1 2 1]
Example: Gentle smoothing with better edge preservation than box filter.
c) Gaussian Filter
– Smooths the image using a kernel derived from the Gaussian function. – Provides isotropic, smooth blurring, and is widely used in preprocessing.
1/256 × [1 4 6 4 1 4 16 24 16 4 6 24 36 24 6 4 16 24 16 4 1 4 6 4 1]
Example: Used for denoising, blurring before edge detection.
d) Sobel Filter (Edge Detection)
– Used to detect vertical or horizontal edges. – Combines smoothing and differentiation.
Horizontal Sobel: [ -1 0 1 -2 0 2 -1 0 1 ]
Example: Enhances vertical edges in an image.
e) Laplacian Filter (Second Derivative)
– Captures areas where intensity changes rapidly. – Used to find edges and corners.
[ 1 -2 1 -2 4 -2 1 -2 1 ]
Example: Corner detection, edge enhancement.
4. Separable Filters
Many 2D filters can be expressed as the product of two 1D filters: one horizontal and one vertical. This reduces the computational cost from O(K²) to O(2K).
Example: Gaussian and Tent filters are separable.
5. Applications of Linear Filters
- Blurring and denoising
- Edge detection (Sobel, Laplacian)
- Image sharpening (Unsharp masking)
- Feature detection (corners, interest points)