Quantitative Analysis¶

Description¶

This notebook extracts quantitative measures from the final label maps:

• Volume (voxel count) per labeled region
• Center of Mass (CoM) in voxel coordinates
• Equivalent diameter assuming a perfect sphere

Outputs¶

• Tabulated metrics per label (volume, CoM, equivalent diameter)
• Optional CSV export for downstream analysis

Part of Pipeline¶

• Previous Step: Watershed Segmentation
• Next Step: N/A

In [1]:

# Import required libraries
import numpy as np

# Integrate figures into notebook
%matplotlib notebook
%matplotlib inline

# Import required libraries import numpy as np # Integrate figures into notebook %matplotlib notebook %matplotlib inline

Load the labeled, i.e. segmented, image from the previous step¶

In [2]:

# Load binary image
labeled_image = np.load(file='../src/data/labeled_image.npy')

# Load binary image labeled_image = np.load(file='../src/data/labeled_image.npy')

Step 1 — Collect unique labels¶

We collect unique label IDs from the 3D label maps and exclude background (0).

In [3]:

# Custom
labels = np.unique(labeled_image)
print("Labels:", labels.tolist())

# Custom labels = np.unique(labeled_image) print("Labels:", labels.tolist())

Labels: [0, 1, 2]

Step 2 — Index grids¶

We create index arrays (z_idx, y_idx, x_idx) matching the label map shape so we can compute CoM by averaging index positions where each label occurs

In [4]:

# Index grids (z, y, x) for the custom map
z_idx_c, y_idx_c, x_idx_c = np.indices(labeled_image.shape)

# Index grids for the library map
z_idx_l, y_idx_l, x_idx_l = np.indices(labeled_image.shape)

print("Shape check (custom):", labeled_image.shape)
print("Shape check (library):", labeled_image.shape)

# Index grids (z, y, x) for the custom map z_idx_c, y_idx_c, x_idx_c = np.indices(labeled_image.shape) # Index grids for the library map z_idx_l, y_idx_l, x_idx_l = np.indices(labeled_image.shape) print("Shape check (custom):", labeled_image.shape) print("Shape check (library):", labeled_image.shape)

Shape check (custom): (16, 16, 16)
Shape check (library): (16, 16, 16)

Step 3 — Volume and CoM for custom segmentation¶

For each label:

• mask = (labels_3d == lab)
• voxel_count = mask.sum()
• com_z = z_idx[mask].sum() / voxel_count, likewise for com_y, com_x.

In [5]:

# Prepare result arrays
volume = np.zeros(labels.shape, dtype=int)
CoM = np.zeros((labels.size, 3), dtype=float)  # (z, y, x)

# Loop over labels (custom)
for i, lab in enumerate(labels):
    mask = (labeled_image == lab)
    count = int(mask.sum())
    volume[i] = count

    if count > 0:
        com_z = z_idx_c[mask].sum() / count
        com_y = y_idx_c[mask].sum() / count
        com_x = x_idx_c[mask].sum() / count
        CoM[i] = (com_z, com_y, com_x)
    else:
        CoM[i] = (np.nan, np.nan, np.nan)

print("Custom — volumes:", volume.tolist())
print("Custom — CoM (z,y,x) sample:", CoM[:min(3, len(CoM))])

# Prepare result arrays volume = np.zeros(labels.shape, dtype=int) CoM = np.zeros((labels.size, 3), dtype=float) # (z, y, x) # Loop over labels (custom) for i, lab in enumerate(labels): mask = (labeled_image == lab) count = int(mask.sum()) volume[i] = count if count > 0: com_z = z_idx_c[mask].sum() / count com_y = y_idx_c[mask].sum() / count com_x = x_idx_c[mask].sum() / count CoM[i] = (com_z, com_y, com_x) else: CoM[i] = (np.nan, np.nan, np.nan) print("Custom — volumes:", volume.tolist()) print("Custom — CoM (z,y,x) sample:", CoM[:min(3, len(CoM))])

Custom — volumes: [3725, 253, 118]
Custom — CoM (z,y,x) sample: [[7.51758389 7.62255034 7.72214765]
 [6.99209486 4.96047431 3.96047431]
 [8.03389831 9.07627119 8.07627119]]

Step 5 — Equivalent diameter¶

We derive the equivalent diameter assuming a sphere with the same volume: $d_\text{eq} = \left(\frac{6\,V}{\pi}\right)^{1/3}$ (Units: voxels)

In [6]:

equiv_diameter     = ((6.0 * volume     / np.pi) ** (1.0 / 3.0))

equiv_diameter = ((6.0 * volume / np.pi) ** (1.0 / 3.0))

Step 6 — Tables¶

We print label, volume, CoM (z, y, x) (rounded for display), and eq. diameter.

In [7]:

print("Label | Volume (voxels) | COM (z, y, x)           | Eq. Diameter (voxels)")
print("----------------------------------------------------------------------")
for i, lab in enumerate(labels):
    z, y, x = CoM[i]
    com_str = f"({z:.0f}, {y:.0f}, {x:.0f})"
    print(f"{lab:5d} | {volume[i]:15d} | {com_str:25s} | {equiv_diameter[i]:.2f}")

print("Label | Volume (voxels) | COM (z, y, x) | Eq. Diameter (voxels)") print("----------------------------------------------------------------------") for i, lab in enumerate(labels): z, y, x = CoM[i] com_str = f"({z:.0f}, {y:.0f}, {x:.0f})" print(f"{lab:5d} | {volume[i]:15d} | {com_str:25s} | {equiv_diameter[i]:.2f}")

Label | Volume (voxels) | COM (z, y, x)           | Eq. Diameter (voxels)
----------------------------------------------------------------------
    0 |            3725 | (8, 8, 8)                 | 19.23
    1 |             253 | (7, 5, 4)                 | 7.85
    2 |             118 | (8, 9, 8)                 | 6.09

Exercise 1: Label Count Consistency

• Task: Compare the number of labels (excluding background) to the number of spheres you generated in Step 01.
• Questions: If counts differ, why? (e.g., overlapping spheres, merged watershed basins, missing seeds)

Exercise 2: Sphere Sanity: Diameter vs Known Radius

• Task: For spherical segments, check whether Eq. Diameter ≈ 2 × known radius (voxel units).
• Questions: Where do you see deviations? Consider discretization, partial overlap, boundary leakage.