refac: optimize WelfordStatistics update and finalization (#325)

### Description

This PR replaces the current inefficient implementation of Welford
statustics update and finalization with a faster numpy based
implementation.

- **What**: Refactored methods in `running_stats.py`.
- **Why**: Performance.
- **How**: Replacing for loops and list comprehensions with numpy
vectorized ops.

#### Notes

No tests are there at the moment. However results provided by refactored
code match the ones using the old version.

---

**Please ensure your PR meets the following requirements:**

- [x] Code builds and passes tests locally, including doctests
- [ ] New tests have been added (for bug fixes/features)
- [x] Pre-commit passes
- [ ] PR to the documentation exists (for bug fixes / features)

src/careamics/dataset/dataset_utils/running_stats.py

@@ -34,36 +34,35 @@ def update_iterative_stats(
     Parameters
     ----------
     count : NDArray
-        Number of elements in the array.
+        Number of elements in the array. Shape: (C,).
     mean : NDArray
-        Mean of the array.
+        Mean of the array. Shape: (C,).
     m2 : NDArray
-        Variance of the array.
+        Variance of the array. Shape: (C,).
     new_values : NDArray
-        New values to add to the mean and variance.
+        New values to add to the mean and variance. Shape: (C, 1, 1, Z, Y, X).
 
     Returns
     -------
     tuple[NDArray, NDArray, NDArray]
         Updated count, mean, and variance.
     """
-    count += np.array([np.prod(channel.shape) for channel in new_values])
-    # newvalues - oldMean
-    delta = [
-        np.subtract(v.flatten(), [m] * len(v.flatten()))
-        for v, m in zip(new_values, mean)
-    ]
+    num_channels = len(new_values)
 
-    mean += np.array([np.sum(d / c) for d, c in zip(delta, count)])
-    # newvalues - newMeant
-    delta2 = [
-        np.subtract(v.flatten(), [m] * len(v.flatten()))
-        for v, m in zip(new_values, mean)
-    ]
+    # --- update channel-wise counts ---
+    count += np.ones_like(count) * np.prod(new_values.shape[1:])
 
-    m2 += np.array([np.sum(d * d2) for d, d2 in zip(delta, delta2)])
+    # --- update channel-wise mean ---
+    # compute (new_values - old_mean) -> shape: (C, Z*Y*X)
+    delta = new_values.reshape(num_channels, -1) - mean.reshape(num_channels, 1)
+    mean += np.sum(delta / count.reshape(num_channels, 1), axis=1)
 
-    return (count, mean, m2)
+    # --- update channel-wise SoS ---
+    # compute (new_values - new_mean) -> shape: (C, Z*Y*X)
+    delta2 = new_values.reshape(num_channels, -1) - mean.reshape(num_channels, 1)
+    m2 += np.sum(delta * delta2, axis=1)
+
+    return count, mean, m2
 
 
 def finalize_iterative_stats(
@@ -74,18 +73,18 @@ def finalize_iterative_stats(
     Parameters
     ----------
     count : NDArray
-        Number of elements in the array.
+        Number of elements in the array. Shape: (C,).
     mean : NDArray
-        Mean of the array.
+        Mean of the array. Shape: (C,).
     m2 : NDArray
-        Variance of the array.
+        Variance of the array. Shape: (C,).
 
     Returns
     -------
     tuple[NDArray, NDArray]
-        Final mean and standard deviation.
+        Final channel-wise mean and standard deviation.
     """
-    std = np.array([np.sqrt(m / c) for m, c in zip(m2, count)])
+    std = np.sqrt(m2 / count)
     if any(c < 2 for c in count):
         return np.full(mean.shape, np.nan), np.full(std.shape, np.nan)
     else: