Draft:Image Morphology

Image Morphology

Image Morphology is a set of image processing operations that process images based on shapes. It applies structuring elements to an input image, creating an output image of the same size. In morphological operations, the value of each pixel in the output image is based on a comparison of the corresponding pixel in the input image with its neighbors. By choosing the size and shape of the neighborhood, you can construct a morphological operation that is sensitive to specific shapes in the input image.

Image morphology was developed by Georges Matheron and Jean Serra at the École des Mines de Paris in the 1960s. The field was originally used for geological image analysis but has since expanded to a wide range of applications.

Basic Concepts

Structuring Element

A structuring element is a matrix used to probe or interact with a given image. Common shapes for structuring elements include lines, squares, and disks. The choice of structuring element affects the outcome of morphological operations.

Dilation

Dilation adds pixels to the boundaries of objects in an image. The number of pixels added depends on the size and shape of the structuring element used.

Dilation of an image A by a structuring element B is defined as:

A⊕B={z∣(B)z​∩A=∅}

where (B)z​ is the translation of B by the vector z.

Erosion

Erosion removes pixels on object boundaries. The number of pixels removed depends on the size and shape of the structuring element used.

Erosion of an image A by a structuring element B is defined as:

A⊖B={z∣(B)z​⊆A}

where (B)z​ is the translation of B by the vector z.

Opening

Opening is an erosion operation followed by a dilation. It is used to remove small objects from an image while preserving the shape and size of larger objects in the image.

Opening of an image A by a structuring element B is defined as:

A∘B=(A⊖B)⊕B

Opening is erosion followed by dilation.

Closing

Closing is a dilation operation followed by an erosion. It is used to fill small holes and gaps in an image while preserving the shape and size of the objects.

Closing of an image A by a structuring element B is defined as:

A∙B=(A⊕B)⊖B

Closing is dilation followed by erosion.

Advanced Operations

Hit-or-Miss Transform

The hit-or-miss transform is a basic morphological tool used for shape detection. It can be used to find specific patterns of foreground and background pixels.

Thinning

Thinning reduces the thickness of objects in a binary image to produce a skeletonized version of the objects. This operation is useful in pattern recognition and image analysis.

Thickening

Thickening is the dual operation of thinning and is used to grow objects in a binary image.

Pruning

Pruning is used to remove small spurious branches from the skeletons produced by the thinning process.

Applications

Image morphology has a wide range of applications, including:

- Noise Removal: Morphological filters can be used to remove noise from binary and grayscale images.

- Shape Analysis: Morphological operations can extract relevant shapes and features from an image for further analysis.

- Image Segmentation: Morphological techniques are used to separate different regions in an image.

- Object Detection: Morphology can be used to detect specific shapes within an image.

References

1. Serra, J. (1982). "Image Analysis and Mathematical Morphology." Academic Press.

2. Haralick, R.M., Sternberg, S.R., & Zhuang, X. (1987). "Image Analysis Using Mathematical Morphology." IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(4), 532-550.