Chapter 3: Image Processing Quiz

Chapter 3: Image Processing

• Contents:
  • Pixel Processing
  • LTI Systems (Linear Time Invariant) and Convolution
  • Linear Image Filters
  • Non-Linear Image Filters

3.1 Pixel Processing

• Transforming a pixel's brightness value depends only on its own value, not the values of other pixels.
• It's a mapping of one brightness/color value to another.
• g(x, y) = T(f(x, y)) where:
  • F(x,y) is the original pixel value
  • g(x,y) is the transformed pixel value
  • T is the transformation function
• Examples of transformations:
  • f-128
  • f+128
  • 255-f
  • f/2
  • f*2
  • 0.3f R + 0.6f G + 0.1*f B

3.2 LTI Systems (Linear Time Invariant) and Convolution

• Linearity: The output is scaled by the same amount as the input. Its response to a weighted sum of inputs is equal to the weighted sum of responses to each individual input.

• Time Invariance: The output is delayed the same as the input. Delaying the input to the system delays the output by the same amount of time.

• Unit Impulse Response (h[n]):

  • The system output (response) to a unit impulse input δ[n].
  • Key for determining the system output for any input (x[n]) through superposition.
  • Convolving the input signal (x[n]) with the impulse response (h[n])

• Convolution:

  • A method for determining the output of an LTI system.
  • The result (y[n]) of convolving the input (x[n]) with the impulse response (h[n]) is represented as y[n] = x[n] * h[n].
  • Involves calculating scaled and shifted versions of h[n]for each input x[k], then adding these results for the output.
  • δ[n] is the Kronecker delta function

• Review of DT Sampling (Shifting) Property:

  • The signal x[n] is essentially a weighted sum of shifted unit impulses: x[n] = Σₖ x[k]δ[n - k].

3.3 Linear Image Filters

• Convolution with Discrete Images:
  • h[i,j] (impulse response) is the mask, kernel, or filter.
  • The output of the filter at location [i,j] is calculated through summing the product of corresponding pixels. (g[i,j] = Σₘ=⁻ᵏⁿ=⁻ᵏ f(i +m, j +n)h(m,n))
  • The filter kernel is overlaid on the image, shifted over different positions, multiplied, and the values summed.
  • Boundary problems often require padding or reflection to handle border regions accurately.
  • Specific examples of kernels are provided, such as a "Box Filter".
  • Gaussian kernel is separable

3.4 Non-Linear Image Filters

• Smoothing:

  • Aims to reduce noise (salt and pepper noise).
  • Applying a linear filter (Gaussian) can reduce noise but also blurs edges and details within an image.

• Median Filtering:

  • A non-linear approach for noise reduction.
  • The median of pixel values within a specified neighborhood (window size) is calculated as the new pixel value.
  • It's effective at removing isolated noise points without as much blurring.

• Bilateral Filtering:

  • Another non-linear method for preserving edges while smoothing an image.
  • Combines the spatial and intensity information related to pixel neighborhood data for calculating the new pixel value.

Test your understanding of key concepts in image processing, including pixel processing and linear time invariant (LTI) systems. This quiz covers important topics such as convolution and various image filters. Enhance your knowledge of how pixel transformations affect image quality and representation.