FA-91451 / Digital signal filters / Open access

Median filter takes the middle sample without sorting · case 01

The output is the centre sample of the window, i.e. no filtering at all.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The window slice is used in time order rather than sorted.

VERIFIED REPAIR

Sort the window before selecting the middle.

Unsuccessful approach: The attempted repair sorts a de-duplicated set, which changes the median whenever values repeat.

Case contract

Input [K, samples]; K odd ("bad-kernel" otherwise). Output i is the median of samples[i-K//2 .. i+K//2] clipped to the valid range (shrinking windows at the edges); an even count uses the mean of the two middle values. Return exact fraction strings.

Why this case matters

Median filters remove impulse noise without blurring edges; window or tie handling slips shift features.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    k, xs = x
    if k < 1 or k % 2 == 0:
        return 'bad-kernel'
    h = k // 2
    out = []
    for i in range(len(xs)):
        win = list(xs[max(0, i - h):i + h + 1])
        m = len(win)
        if m % 2:
            out.append(str(Fraction(win[m // 2])))
        else:
            out.append(str(Fraction(win[m // 2 - 1] + win[m // 2], 2)))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: unsorted window', [3, [5, 1, 9, 2]], ['3', '5', '2', '11/2']], ['regression: edge even median', [3, [4, 1, 7]], ['5/2', '4', '4']], ['regression: duplicates', [3, [2, 2, 5, 2]], ['2', '2', '2', '7/2']], ['control: negative odd sum', [3, [-3, 0, 5]], ['-3/2', '0', '5/2']], ['control: even kernel', [2, [1, 2, 3]], 'bad-kernel'], ['control: even kernel four', [4, [1, 5, 2, 8]], 'bad-kernel'], ['control: random median 1', [1, [2, 3, 4, -2]], ['2', '3', '4', '-2']]], [['regression: random median 2', [3, [5, 5, 3, -5, 8, -1]], ['5', '5', '3', '3', '-1', '7/2']], ['regression: random median 6', [3, [3, 9, 2, 4, 6, 3, -4]], ['6', '3', '4', '4', '4', '3', '-1/2']], ['repair check: random median 4', [3, [0, -4, -4, 3]], ['-2', '-4', '-4', '-1/2']], ['control: random median 3', [1, [2, -3, 7, 8, -2, 3, 8, 1]], ['2', '-3', '7', '8', '-2', '3', '8', '1']], ['control: random median 7', [1, [7, 6, 6, -2, 6, 0, 8, -4]], ['7', '6', '6', '-2', '6', '0', '8', '-4']], ['control: random median 8', [1, [2, 8, -4, 9, 7, 7]], ['2', '8', '-4', '9', '7', '7']], ['control: random median 9', [1, [2, 5, 7, 9]], ['2', '5', '7', '9']]], [['regression: random median 12', [3, [7, 1, -5, -5, -5, 2, 5, -5]], ['4', '1', '-5', '-5', '-5', '2', '2', '0']], ['regression: random median 13', [3, [1, -4, -1, 3, 9, 4, 6, 2]], ['-3/2', '-1', '-1', '3', '4', '6', '4', '4']], ['repair check: random median 5', [3, [2, -2, -2, -2, -4, -4]], ['0', '-2', '-2', '-2', '-4', '-4']], ['control: random median 10', [1, [9, 3, -5, 8]], ['9', '3', '-5', '8']], ['control: random median 14', [1, [2, 8, 6, 5, -4, -3, -1]], ['2', '8', '6', '5', '-4', '-3', '-1']], ['control: random median 15', [1, [8, 2]], ['8', '2']], ['control: random median 17', [1, [-5, -2, -4, 5, 9, 2, -4]], ['-5', '-2', '-4', '5', '9', '2', '-4']]], [['regression: random median 18', [5, [-4, 9, 8, -3, -2]], ['8', '5/2', '-2', '3', '-2']], ['regression: random median 19', [3, [7, 8, -4]], ['15/2', '7', '2']], ['regression: random median 12', [3, [7, 1, -5, -5, -5, 2, 5, -5]], ['4', '1', '-5', '-5', '-5', '2', '2', '0']], ['control: random median 23', [1, [4, 1, -1, 3, -1]], ['4', '1', '-1', '3', '-1']], ['control: random median 25', [1, [-5, 6]], ['-5', '6']], ['control: random median 26', [3, [-4, 5]], ['1/2', '1/2']], ['control: random median 27', [1, [-2, 5, -4, 7, 1, -3]], ['-2', '5', '-4', '7', '1', '-3']]], [['regression: random median 22', [5, [3, -1, 4, 8, 9]], ['3', '7/2', '4', '6', '8']], ['regression: random median 24', [3, [8, -2, -5, 4, 2]], ['3', '-2', '-2', '2', '3']], ['repair check: random median 20', [3, [-3, -5, -5]], ['-4', '-5', '-5']], ['control: random median 28', [1, [1, -1, 2, 9, 2, 3, 6]], ['1', '-1', '2', '9', '2', '3', '6']], ['control: random median 29', [5, [9, 8, 5]], ['8', '8', '8']], ['control: random median 34', [3, [-1, 5]], ['2', '2']], ['control: random median 35', [1, [-3, 0, 9]], ['-3', '0', '9']]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: unsorted window	`['3', '1', '9', '11/2']`	`['3', '5', '2', '11/2']`	Failed
regression: edge even median	`['5/2', '1', '4']`	`['5/2', '4', '4']`	Failed
regression: duplicates	`['2', '2', '5', '7/2']`	`['2', '2', '2', '7/2']`	Failed
control: negative odd sum	`['-3/2', '0', '5/2']`	`['-3/2', '0', '5/2']`	Passed
control: even kernel	`bad-kernel`	`bad-kernel`	Passed
control: even kernel four	`bad-kernel`	`bad-kernel`	Passed
control: random median 1	`['2', '3', '4', '-2']`	`['2', '3', '4', '-2']`	Passed

SHA-256 / 28321666975f230bd1cda5814c5492299506eb70bef7c0bd77b3286efbfa16e5

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    k, xs = x
    if k < 1 or k % 2 == 0:
        return 'bad-kernel'
    h = k // 2
    out = []
    for i in range(len(xs)):
        win = sorted(set(xs[max(0, i - h):i + h + 1]))
        m = len(win)
        if m % 2:
            out.append(str(Fraction(win[m // 2])))
        else:
            out.append(str(Fraction(win[m // 2 - 1] + win[m // 2], 2)))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: unsorted window', [3, [5, 1, 9, 2]], ['3', '5', '2', '11/2']], ['regression: edge even median', [3, [4, 1, 7]], ['5/2', '4', '4']], ['regression: duplicates', [3, [2, 2, 5, 2]], ['2', '2', '2', '7/2']], ['control: negative odd sum', [3, [-3, 0, 5]], ['-3/2', '0', '5/2']], ['control: even kernel', [2, [1, 2, 3]], 'bad-kernel'], ['control: even kernel four', [4, [1, 5, 2, 8]], 'bad-kernel'], ['control: random median 1', [1, [2, 3, 4, -2]], ['2', '3', '4', '-2']]], [['regression: random median 2', [3, [5, 5, 3, -5, 8, -1]], ['5', '5', '3', '3', '-1', '7/2']], ['regression: random median 6', [3, [3, 9, 2, 4, 6, 3, -4]], ['6', '3', '4', '4', '4', '3', '-1/2']], ['repair check: random median 4', [3, [0, -4, -4, 3]], ['-2', '-4', '-4', '-1/2']], ['control: random median 3', [1, [2, -3, 7, 8, -2, 3, 8, 1]], ['2', '-3', '7', '8', '-2', '3', '8', '1']], ['control: random median 7', [1, [7, 6, 6, -2, 6, 0, 8, -4]], ['7', '6', '6', '-2', '6', '0', '8', '-4']], ['control: random median 8', [1, [2, 8, -4, 9, 7, 7]], ['2', '8', '-4', '9', '7', '7']], ['control: random median 9', [1, [2, 5, 7, 9]], ['2', '5', '7', '9']]], [['regression: random median 12', [3, [7, 1, -5, -5, -5, 2, 5, -5]], ['4', '1', '-5', '-5', '-5', '2', '2', '0']], ['regression: random median 13', [3, [1, -4, -1, 3, 9, 4, 6, 2]], ['-3/2', '-1', '-1', '3', '4', '6', '4', '4']], ['repair check: random median 5', [3, [2, -2, -2, -2, -4, -4]], ['0', '-2', '-2', '-2', '-4', '-4']], ['control: random median 10', [1, [9, 3, -5, 8]], ['9', '3', '-5', '8']], ['control: random median 14', [1, [2, 8, 6, 5, -4, -3, -1]], ['2', '8', '6', '5', '-4', '-3', '-1']], ['control: random median 15', [1, [8, 2]], ['8', '2']], ['control: random median 17', [1, [-5, -2, -4, 5, 9, 2, -4]], ['-5', '-2', '-4', '5', '9', '2', '-4']]], [['regression: random median 18', [5, [-4, 9, 8, -3, -2]], ['8', '5/2', '-2', '3', '-2']], ['regression: random median 19', [3, [7, 8, -4]], ['15/2', '7', '2']], ['regression: random median 12', [3, [7, 1, -5, -5, -5, 2, 5, -5]], ['4', '1', '-5', '-5', '-5', '2', '2', '0']], ['control: random median 23', [1, [4, 1, -1, 3, -1]], ['4', '1', '-1', '3', '-1']], ['control: random median 25', [1, [-5, 6]], ['-5', '6']], ['control: random median 26', [3, [-4, 5]], ['1/2', '1/2']], ['control: random median 27', [1, [-2, 5, -4, 7, 1, -3]], ['-2', '5', '-4', '7', '1', '-3']]], [['regression: random median 22', [5, [3, -1, 4, 8, 9]], ['3', '7/2', '4', '6', '8']], ['regression: random median 24', [3, [8, -2, -5, 4, 2]], ['3', '-2', '-2', '2', '3']], ['repair check: random median 20', [3, [-3, -5, -5]], ['-4', '-5', '-5']], ['control: random median 28', [1, [1, -1, 2, 9, 2, 3, 6]], ['1', '-1', '2', '9', '2', '3', '6']], ['control: random median 29', [5, [9, 8, 5]], ['8', '8', '8']], ['control: random median 34', [3, [-1, 5]], ['2', '2']], ['control: random median 35', [1, [-3, 0, 9]], ['-3', '0', '9']]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: unsorted window	`['3', '5', '2', '11/2']`	`['3', '5', '2', '11/2']`	Passed
regression: edge even median	`['5/2', '4', '4']`	`['5/2', '4', '4']`	Passed
regression: duplicates	`['2', '7/2', '7/2', '7/2']`	`['2', '2', '2', '7/2']`	Failed
control: negative odd sum	`['-3/2', '0', '5/2']`	`['-3/2', '0', '5/2']`	Passed
control: even kernel	`bad-kernel`	`bad-kernel`	Passed
control: even kernel four	`bad-kernel`	`bad-kernel`	Passed
control: random median 1	`['2', '3', '4', '-2']`	`['2', '3', '4', '-2']`	Passed

SHA-256 / 1f4bc483476d3f253e5c8adef2987060121ee904210371ad4b2808bc8c1b6b71

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    k, xs = x
    if k < 1 or k % 2 == 0:
        return 'bad-kernel'
    h = k // 2
    out = []
    for i in range(len(xs)):
        win = sorted(xs[max(0, i - h):i + h + 1])
        m = len(win)
        if m % 2:
            out.append(str(Fraction(win[m // 2])))
        else:
            out.append(str(Fraction(win[m // 2 - 1] + win[m // 2], 2)))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: unsorted window', [3, [5, 1, 9, 2]], ['3', '5', '2', '11/2']], ['regression: edge even median', [3, [4, 1, 7]], ['5/2', '4', '4']], ['regression: duplicates', [3, [2, 2, 5, 2]], ['2', '2', '2', '7/2']], ['control: negative odd sum', [3, [-3, 0, 5]], ['-3/2', '0', '5/2']], ['control: even kernel', [2, [1, 2, 3]], 'bad-kernel'], ['control: even kernel four', [4, [1, 5, 2, 8]], 'bad-kernel'], ['control: random median 1', [1, [2, 3, 4, -2]], ['2', '3', '4', '-2']]], [['regression: random median 2', [3, [5, 5, 3, -5, 8, -1]], ['5', '5', '3', '3', '-1', '7/2']], ['regression: random median 6', [3, [3, 9, 2, 4, 6, 3, -4]], ['6', '3', '4', '4', '4', '3', '-1/2']], ['repair check: random median 4', [3, [0, -4, -4, 3]], ['-2', '-4', '-4', '-1/2']], ['control: random median 3', [1, [2, -3, 7, 8, -2, 3, 8, 1]], ['2', '-3', '7', '8', '-2', '3', '8', '1']], ['control: random median 7', [1, [7, 6, 6, -2, 6, 0, 8, -4]], ['7', '6', '6', '-2', '6', '0', '8', '-4']], ['control: random median 8', [1, [2, 8, -4, 9, 7, 7]], ['2', '8', '-4', '9', '7', '7']], ['control: random median 9', [1, [2, 5, 7, 9]], ['2', '5', '7', '9']]], [['regression: random median 12', [3, [7, 1, -5, -5, -5, 2, 5, -5]], ['4', '1', '-5', '-5', '-5', '2', '2', '0']], ['regression: random median 13', [3, [1, -4, -1, 3, 9, 4, 6, 2]], ['-3/2', '-1', '-1', '3', '4', '6', '4', '4']], ['repair check: random median 5', [3, [2, -2, -2, -2, -4, -4]], ['0', '-2', '-2', '-2', '-4', '-4']], ['control: random median 10', [1, [9, 3, -5, 8]], ['9', '3', '-5', '8']], ['control: random median 14', [1, [2, 8, 6, 5, -4, -3, -1]], ['2', '8', '6', '5', '-4', '-3', '-1']], ['control: random median 15', [1, [8, 2]], ['8', '2']], ['control: random median 17', [1, [-5, -2, -4, 5, 9, 2, -4]], ['-5', '-2', '-4', '5', '9', '2', '-4']]], [['regression: random median 18', [5, [-4, 9, 8, -3, -2]], ['8', '5/2', '-2', '3', '-2']], ['regression: random median 19', [3, [7, 8, -4]], ['15/2', '7', '2']], ['regression: random median 12', [3, [7, 1, -5, -5, -5, 2, 5, -5]], ['4', '1', '-5', '-5', '-5', '2', '2', '0']], ['control: random median 23', [1, [4, 1, -1, 3, -1]], ['4', '1', '-1', '3', '-1']], ['control: random median 25', [1, [-5, 6]], ['-5', '6']], ['control: random median 26', [3, [-4, 5]], ['1/2', '1/2']], ['control: random median 27', [1, [-2, 5, -4, 7, 1, -3]], ['-2', '5', '-4', '7', '1', '-3']]], [['regression: random median 22', [5, [3, -1, 4, 8, 9]], ['3', '7/2', '4', '6', '8']], ['regression: random median 24', [3, [8, -2, -5, 4, 2]], ['3', '-2', '-2', '2', '3']], ['repair check: random median 20', [3, [-3, -5, -5]], ['-4', '-5', '-5']], ['control: random median 28', [1, [1, -1, 2, 9, 2, 3, 6]], ['1', '-1', '2', '9', '2', '3', '6']], ['control: random median 29', [5, [9, 8, 5]], ['8', '8', '8']], ['control: random median 34', [3, [-1, 5]], ['2', '2']], ['control: random median 35', [1, [-3, 0, 9]], ['-3', '0', '9']]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: unsorted window	`['3', '5', '2', '11/2']`	`['3', '5', '2', '11/2']`	Passed
regression: edge even median	`['5/2', '4', '4']`	`['5/2', '4', '4']`	Passed
regression: duplicates	`['2', '2', '2', '7/2']`	`['2', '2', '2', '7/2']`	Passed
control: negative odd sum	`['-3/2', '0', '5/2']`	`['-3/2', '0', '5/2']`	Passed
control: even kernel	`bad-kernel`	`bad-kernel`	Passed
control: even kernel four	`bad-kernel`	`bad-kernel`	Passed
control: random median 1	`['2', '3', '4', '-2']`	`['2', '3', '4', '-2']`	Passed

SHA-256 / 4394eabf5edbbfccee1767d53740382bcd00aabb03b53f82a63d85ee2aedbd6d

Verification & scope

A deterministic bounded teaching model with a stipulated toy contract; exact rational arithmetic or fixed-decimal rounding keeps outputs strict JSON. It is not a production DSP library and claims no standards conformance. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:51:36.142691+00:00.

Case digest / ae4f5814f888ba48aaf1511351ca9f9f57c4932cef07e6698edc293d9ac4f8ef