Frei–Chen operator

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The Frei–Chen operator, sometimes called Frei and Chen operator, is used in image processing for edge detection. It was proposed by Werner Frei and Chung-Ching Chen, researchers at USC's Image Processing Institute, in 1977. The idea is to use a set of orthogonal basis vectors related to distinctive image features, which enable the algorithm to extract boundary elements effectively.

Formulation

The operator uses nine 3×3 kernels which are convolved with the original image to calculate the gradient.

We define the nine kernels ${\displaystyle {W1,...,W9}}$ as:

$${\displaystyle W1={\begin{bmatrix}1&{\sqrt {2}}&1\\0&0&0\\-1&-{\sqrt {2}}&-1\end{bmatrix}}\quad W2={\begin{bmatrix}1&0&-1\\{\sqrt {2}}&0&-{\sqrt {2}}\\1&0&-1\end{bmatrix}}}$$

$${\displaystyle W3={\begin{bmatrix}0&-1&{\sqrt {2}}\\1&0&-1\\-{\sqrt {2}}&1&0\end{bmatrix}}\quad W4={\begin{bmatrix}{\sqrt {2}}&-1&0\\-1&0&1\\0&1&-{\sqrt {2}}\end{bmatrix}}}$$

$${\displaystyle W5={\begin{bmatrix}0&1&0\\-1&0&-1\\0&1&0\end{bmatrix}}\quad W6={\begin{bmatrix}-1&0&1\\0&0&0\\1&0&-1\end{bmatrix}}}$$

$${\displaystyle W7={\begin{bmatrix}1&-2&1\\-2&4&-2\\1&-2&1\end{bmatrix}}\quad W8={\begin{bmatrix}-2&1&-2\\1&4&1\\-2&1&-2\end{bmatrix}}}$$

$${\displaystyle W9={\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}}}$$

• The pair ${\displaystyle (W1,W2)}$ is the isotropic average gradient
• The pair ${\displaystyle (W3,W4)}$ is used to detect ripples
• The pair ${\displaystyle (W5,W6)}$ is used to detect lines
• The pair ${\displaystyle (W7,W8)}$ is the discrete laplacian operator
• ${\displaystyle W9}$, the averaging operator, added to complete the basis
• ${\displaystyle W1,...W4}$ are used for edges subspace, ${\displaystyle W5,...W8}$ used for lines subspace and ${\displaystyle W9}$ is used to compute averages

Let ${\displaystyle B}$ be the image sub-area, and ${\displaystyle \theta }$ be the angle (in ${\displaystyle n^{2}}$space), ${\displaystyle e}$ is the number of orthogonal edge basis vectors ${\displaystyle W_{1},...W_{e}}$ spanning the edge subspace.

$${\displaystyle \theta =\arccos \left[\textstyle \sum _{i=1}^{e}\displaystyle (B*W_{i})^{2}/\textstyle \sum _{i=1}^{9}\displaystyle (B*W_{i})^{2}\right]^{\frac {1}{2}}}$$

The larger ${\displaystyle \theta }$, the poorer the fit between B and an element of the edge subspace.

The strategy is to classify image sub-area as containing and edge element only if ${\displaystyle \theta }$ is small which is done by thresholding.

Simple description

The image is convolved with each of the kernel. Thus, 9 results are obtained.

Vectors ${\displaystyle W_{1},...W_{4}}$ are used for edge subspace identification. Hence numerator in the formula will be ${\displaystyle \textstyle \sum _{i=1}^{4}\displaystyle (B*W_{i})^{2}}$. Similarly, for line subspace identification, the numerator will be ${\displaystyle \textstyle \sum _{i=5}^{8}\displaystyle (B*W_{i})^{2}}$.

Using formula, we compute ${\displaystyle \theta }$, if it is above a certain threshold ${\displaystyle r}$, we say that an edge is detected in the image sub-area.

Example comparisons

Here, the Frei–Chen operator, along with three different gradient operators is used to detect edges in the test image.

• 
  Grayscale test image of brick wall and bike rack
• 
  Edge detection using the Frei–Chen operator
• 
  Edge detection using the Roberts cross operator
• 
  Edge detection using the Prewitt operator
• 
  Edge detection using the Sobel operator

See also

• Digital image processing

References