Distance Maps

Distance Maps compute the Euclidean distance transform of a binary or coded image:

1. Input — A coded image defines the objects. All non-zero pixels are "object"; zero pixels are "background."
2. Distance computation — For each background pixel, compute the Euclidean distance to the nearest object pixel. For each object pixel, optionally compute the distance to the nearest background pixel (boundary distance).
3. Output — A grayscale image where pixel intensity encodes distance. Background pixels near objects are dim (small distance); pixels far from objects are bright (large distance). Object pixels can encode their distance from the nearest boundary.

The Euclidean distance transform can be computed efficiently using a two-pass algorithm that processes rows then columns, achieving O(N) time for an N-pixel image despite the apparent O(N²) nature of the all-pairs distance problem.

Distance metrics (Gonzalez & Woods): Three common distance metrics for image analysis: Euclidean (D_e = √(Δx² + Δy²)) — the true straight-line distance, producing circular iso-distance contours; City block / Manhattan (D_4 = |Δx| + |Δy|) — the path along grid axes, producing diamond-shaped contours; Chessboard (D_8 = max(|Δx|, |Δy|)) — allows diagonal steps at unit cost, producing square contours. For biological spatial analysis, Euclidean distance is standard because cell-to-cell distances in tissue don't follow grid constraints.

The inside distance transform and shape: The inside distance transform of a convex object has a single peak at its center, with the peak value equal to the inscribed circle radius. For touching objects, the valley between their peaks indicates the contact region. This is why the negated inside distance transform serves as an ideal watershed input — each object's center is a local minimum (catchment basin), and the contact region is a ridge (watershed boundary). Solomon & Breckon note that the Euclidean distance transform "calculates distance from each background point to nearest foreground point" and when negated and used as a watershed surface, it performs shape-based segmentation.

Proximity zones and buffer analysis: Thresholding a distance map at a specific value creates a buffer zone — the set of all pixels within that distance of the objects. This is equivalent to morphological dilation by that distance but computed continuously rather than for a single integer value. A distance map thus encodes all possible dilation distances simultaneously, enabling flexible proximity analysis without rerunning the dilation at each distance.

Nearest-neighbor statistics: For two sets of objects (e.g., tumor cells and immune cells), the distance from each tumor cell's centroid to the nearest immune cell can be read directly from the immune cell distance map. The distribution of these nearest-neighbor distances characterizes the spatial relationship between the two populations: a narrow distribution centered on small distances indicates close spatial association; a wide distribution centered on large distances indicates spatial separation.

Analyzing immune cell infiltration as a function of distance from the tumor boundary:

1. Create a tumor mask from CK+ cells (binary: tumor vs. non-tumor)
2. Compute Outside Distance Map from the tumor mask → every pixel now encodes its distance from the nearest tumor cell
3. Bin immune cells by their distance value: 0-20 µm, 20-50 µm, 50-100 µm, 100+ µm from tumor
4. Compute immune cell density in each distance bin

Result: An immune infiltration profile showing how immune cell density changes with distance from the tumor boundary — high at the margin (immune-inflamed), low everywhere (immune-desert), or high only at distance (immune-excluded).