First Derivative Operators

1. Sobel Operator

• Purpose: Approximates the 2D gradient of an image. It computes the gradient separately in the horizontal ($x$) and vertical ($y$) directions. It incorporates a slight smoothing effect by giving more weight to the center row/column when calculating the derivative in the orthogonal direction.

• Kernels/Masks (3x3):

  • Gradient in the x-direction ($G_x$): Detects vertical edges. $$G_x=\left[\begin{array}{ccc}-1& 0& +1\\ -2& 0& +2\\ -1& 0& +1\end{array}\right]$$
  • Gradient in the y-direction ($G_y$): Detects horizontal edges. $$G_y=\left[\begin{array}{ccc}-1& -2& -1\\ 0& 0& 0\\ +1& +2& +1\end{array}\right]$$ (Note: Sometimes the sign convention is flipped, or the $G_y$ mask is shown as the transpose of $G_x$ with appropriate sign changes. The core idea of differentiation and smoothing remains.)

• How it works:

  1. Convolve the input image $I$ with the $G_x$ kernel to get the gradient approximation in the x-direction: $I_x=G_x\ast I$.
  2. Convolve the input image $I$ with the $G_y$ kernel to get the gradient approximation in the y-direction: $I_y=G_y\ast I$.
  3. Gradient Magnitude: Typically calculated as $M=\sqrt{I_x^2+I_y^2}$ or approximated as $M\approx |I_x|+|I_y|$ for computational efficiency.
  4. Gradient Direction: Calculated as $\theta =ta{n}^{-1}(I_y,I_x)$. ($ta{n}^{-1}$ handles all quadrants correctly).

• Properties:

  • Uses a $3\times 3$ neighborhood.
  • The weighting factor of 2 for the center row/column provides some averaging/smoothing perpendicular to the derivative direction, making it slightly less sensitive to noise compared to simpler operators like Prewitts or Roberts.
  • Widely used due to its balance between performance and computational cost.

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2. Prewitts Operator

• Purpose: Similar to Sobel, it approximates the 2D gradient using first differences, but without the emphasis on center pixels (uniform weighting).

• Kernels/Masks (3x3):

  • Gradient in the x-direction ($G_x$): Detects vertical edges. $$G_x=\left[\begin{array}{ccc}-1& 0& +1\\ -1& 0& +1\\ -1& 0& +1\end{array}\right]$$
  • Gradient in the y-direction ($G_y$): Detects horizontal edges. $$G_y=\left[\begin{array}{ccc}-1& -1& -1\\ 0& 0& 0\\ +1& +1& +1\end{array}\right]$$

• How it works: The process is identical to the Sobel operator: convolve with $G_x$ and $G_y$, then calculate magnitude and direction.

  1. $I_x=G_x\ast I$
  2. $I_y=G_y\ast I$
  3. $M=\sqrt{I_x^2+I_y^2}$ or $M\approx |I_x|+|I_y|$
  4. $\theta =\text{atan2}(I_y,I_x)$

• Properties:

  • Uses a $3\times 3$ neighborhood.
  • Simpler than Sobel as it uses weights of only -1, 0, +1.
  • The lack of center weighting makes it slightly more sensitive to noise than Sobel.
  • Easy to implement.

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3. Roberts (Cross) Operator

• Purpose: One of the earliest edge detectors, it approximates the gradient using differences between diagonally adjacent pixels.

• Kernels/Masks (2x2):

  • Gradient approximation 1 ($G_x$ - often associated with one diagonal): $$G_x=\left[\begin{array}{cc}+1& 0\\ 0& -1\end{array}\right]$$
  • Gradient approximation 2 ($G_y$ - often associated with the other diagonal): $$G_y=\left[\begin{array}{cc}0& +1\\ -1& 0\end{array}\right]$$

• How it works:

  1. Convolve the input image $I$ with $G_x$: $I_x=G_x\ast I$. Note that the origin of the $2\times 2$ kernel is usually taken as the top-left pixel.
  2. Convolve the input image $I$ with $G_y$: $I_y=G_y\ast I$.
  3. Gradient Magnitude: $M=\sqrt{I_x^2+I_y^2}$ or $M\approx |I_x|+|I_y|$.
  4. Gradient Direction: $\theta =\text{atan2}(I_y,I_x)$. (Interpretation might relate to diagonal directions).

• Properties:

  • Uses a minimal $2\times 2$ neighborhood.
  • Computationally very fast due to the small kernel size and simple coefficients.
  • Highly sensitive to noise because it considers only a very small neighborhood and involves no smoothing.
  • Provides a poor approximation of the gradient compared to $3\times 3$ operators.
  • Historically significant but less commonly used in modern applications compared to Sobel or Prewitts, especially when noise is a concern.