Distance transform
Prerequisites
Before starting this lesson, you should be familiar with:
Learning Objectives
After completing this lesson, learners should be able to:
Understand how to use distance transform to quantify morphology of objects
Understand how to use distance transform to quantify distance between objects
Understand approximate nature of distance transform
Motivation
We use distance transform to quantify how a structure of interest is away from object boundaries or other structures. The distance transform is also use to characterize the morphology of an object in 2D and 3D, find its center, dimensions, etc.. Finally distance transform can be used as a pre-processing step to improve the segmentation results and split touching objects.
Concept map
Figure
Chamfer distance (modified from MorpholibJ Manual)
Several methods (metrics) exist for computing distance maps. The MorphoLibJ library implements distance transforms based on chamfer distances. Chamfer distances approximate Euclidean distances with integer weights, and are simpler to compute than exact Euclidean distance (Borgefors, 1984, 1986). As chamfer weights are only an approximation of the real Euclidean distance, some differences are expected compared to the actual Euclidean distance map.
Several choices for chamfer weights are illustrated in above Figure (showing the not normalized distance).
- The “Chessboard” distance, also named Chebyshev distance, gives the number of moves that an imaginary Chess-king needs to reach a specific pixel. A king can move one pixel in each direction.
- The “City-block” distance, also named Manhattan metric, weights diagonal moves differently.
- The “Borgefors” weights were claimed to be best approximation when considering the 3-by-3 neighborhood.
- The “Chess knight” distance also takes into account the pixels located at a distance from (±1, ±2) in any direction. It is usually the best choice, as it considers a larger neighborhood.
To remove the scaling effect due to weights > 1, it is necessary to perform a normalization step. In MorphoLibJ this is performed by dividing the resulting image by the first weight (option Normalize weights).
Activities
Distance transform basics
- Open label mask xy_8bit_binary__dist_trafo_a.tif.
- Perform a distance transform.
- Invert the binary input image.
- Perform another distance transform on the new binary image.
- Observe that the datatype (in particular 8-bit) of the distance transform image limits the distances.
- Discuss different metrics.
- Add a calibration to the inverted image and recompute the distance transform.
- Observe whether the image calibration is considered for the distances.
Distance measurements
- Open label mask: xy_8bit_labels__dist_trafo_b.tif.
- Create an inverted binary mask for one of the labels (e.g. label 1)
- Compute the distance transform
- Measure intensity (= distances to nearest other objects) of objects in the distance map
- Show that the measured values correspond to the distance of label 2 and 3 to label 1.
Region selection by distance gating
- Open reference object image: xy_8bit_binary__single_object.tif.
- Invert the binary
- Compute the distance transform
- Perform an intensity gating on the distance map, e.g. by thresholding, to create a mask with all pixels of a certain distance, e.g. between 100-120, to the reference object.
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Exercises
Measure distance to center of cell
We would like to measure the distance within a binary mask to a specific point in the cell. This is called Geodesic distance.
- Use xy_8bit_glialcell.tif and the corresponding binary image xy_8bit_binary__glialcell.tif
- Compute the geodesic distance map of the binary image with respect to a reference point close to the soma of the cell (approx x_pixel = 88, y_pixel = 74)
Measure thickness of glial branches
The goal is to combine skeletonization and distance map computation to measure skeleton branch length and thickness. For this exercise you need the binary image xy_8bit_binary__glialcell.tif and the skeletonized and normalized version xy_8bit_binary__glialcell_skeleton.tif
- Compute the distance transform of the binary
- Multiply the skeleton image with the distance map
- Perform an analyze of the skeleton
Assessment
Discuss with your neighbor
- Knowing the image calibration, how could we convert a 2D distance map to physical values?
- Knowing the image calibration, how could we convert a 3D distance map to physical values? Is it important if the image is isotropic sampled or not?
Follow-up material
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