为何方形二维中值滤波器不可分离？非方形的是否可分离？

Great question—let's break this down clearly, since the difference between linear filters (like mean) and nonlinear ones (like median) is the core of this issue.

First, quick recap: a separable filter means you can split a 2D filtering operation into two sequential 1D passes (e.g., apply a row filter first, then a column filter) and get exactly the same result as running the full 2D filter directly.

Mean filters are separable because they're linear #

You already get this, but let's formalize it briefly: mean filtering is a linear operation (it follows the superposition principle: mean(a + b) = mean(a) + mean(b) and mean(k*a) = k*mean(a)). A 3x3 mean kernel is just the outer product of a 1x3 row kernel and a 3x1 column kernel. Linearity lets us split the operation without losing any information—row-then-column mean is mathematically identical to the full 2D mean.

Median filters are non-separable because they're nonlinear #

Median filtering is an order-statistic operation: it depends on sorting every element in the mask and picking the middle value. This is fundamentally nonlinear, and nonlinear operations don't work with separation. Let's prove this with a concrete example:

Take this 3x3 matrix:

1  3  5
2  4  6
7  8  9

• Direct 3x3 median: Sort all 9 elements: [1,2,3,4,5,6,7,8,9] → median is 5.
• Separable approach (row median then column median):
  1. Row medians: 3 (first row), 4 (second row), 8 (third row).
  2. Column median of these three values: 4.

The results (5 vs 4) don't match. Why? Because when we take row medians first, we throw away critical information about the global order of all elements. The row median only cares about order within each row, not how elements from different rows compare. The full 2D median needs the complete sorted list of all 9 elements, which can't be reconstructed from just row (or column) medians.

This applies to square median masks of any size—2x2, 5x5, etc.—the nonlinearity of the median operation means splitting into 1D passes will always lose ordering information that changes the final result.

Are non-square median filters separable? #

Short answer: Almost never. The same nonlinearity problem applies here too. Let's use a 2x3 mask example to show why:

Consider this matrix:

1  2  9
3  8  4

• Direct 2x3 median: Sort all 6 elements: [1,2,3,4,8,9] → median is the average of the 3rd and 4th values: (3+4)/2 = 3.5.
• Separable approach (row median then column median):
  1. Row medians: 2 (first row), 4 (second row).
  2. Column median of these two values: 3.

Again, the results don't align. Even non-square masks rely on the global order of all elements in the mask, which can't be captured by sequential 1D median operations. There's no way to split the median calculation into row/column passes without discarding information that affects the final median value.

内容的提问来源于stack exchange，提问作者playitright