title: Stoney Royd Cemetery

abstract: Stoney Royd Cemetery is a cemetery in Halifax, West Yorkshire, England. Stoney Royd House was a brick house built for Christopher Rawson (1777-1849), the third son of John Rawson of Bolton "a little before 1764". It was demolished in the second half of the 20th century. One of its original gate lodges remains as part of the cemetery. In 1860 Halifax Corporation bought the site to turn it into a cemetery. Edward Milner won a competition for its design. It opened in 1861, and the northern section was consecrated on 11 September 1862. The southern section was for Nonconformist burials. Two chapels were built: one each for Church of England and Nonconformist funerals. The Church of England one was a cruciform Gothic Revival building with a tower, broach spire and polygonal apse. In 1973 the Department of the Environment made it a Grade II listed building. In 2007 its roof collapsed in a storm, and Calderdale Metropolitan Borough Council obtained listed building consent to demolish it. The tower and spire survive. The cemetery contains monuments for 76 war dead, under the care of the Commonwealth War Graves Commission.

text:
# Stoney Royd Cemetery

**Stoney (or Stony) Royd Cemetery** is a cemetery in Halifax, West Yorkshire, England.

Stoney Royd House was a brick house built for Christopher Rawson (1777-1849), the third son of John Rawson of Bolton "a little before 1764". It was demolished in the second half of the 20th century. One of its original gate lodges remains as part of the cemetery.

In 1860 Halifax Corporation bought the site to turn it into a cemetery. Edward Milner won a competition for its design. It opened in 1861, and the northern section was consecrated on 11 September 1862. The southern section was for Nonconformist burials.

Two chapels were built: one each for Church of England and Nonconformist funerals. The Church of England one was a cruciform Gothic Revival building with a tower, broach spire and polygonal apse. In 1973 the Department of the Environment made it a Grade II listed building. In 2007 its roof collapsed in a storm, and Calderdale Metropolitan Borough Council obtained listed building consent to demolish it. The tower and spire survive.

The cemetery contains monuments for 76 war dead, under the care of the Commonwealth War Graves Commission.

53°42′50″N 1°51′2″W / 53.71389°N 1.85056°W / 53.71389; -1.85056


title: Stormarn (gau)

abstract: Stormarn was a gau which, alongside Holstein and Dithmarschen, was one of the three Northern Albingian Saxon gaus. The Gau of Stormarn lay in the southeast of Schleswig-Holstein. As well as the present county of Stormarn it also covered the territory of present-day Pinneberg, part of Steinburg, parts of Segeberg and the present land of the Hanseatic city of Hamburg north of the Elbe. The "Stormarni" were recorded in documents from the 11th century, like the Hamburg Church History by Adam of Bremen. The document refers to the Stormarni as one of the Saxon tribes.

text:
# Stormarn (gau)

**Stormarn** (German pronunciation: \[ˈʃtɔʁmaʁn\] ) was a *gau* which, alongside Holstein and Dithmarschen, was one of the three Northern Albingian Saxon gaus.

The Gau of Stormarn lay in the southeast of Schleswig-Holstein. As well as the present county of Stormarn it also covered the territory of present-day Pinneberg, part of Steinburg, parts of Segeberg and the present land of the Hanseatic city of Hamburg north of the Elbe. The "Stormarni" were recorded in documents from the 11th century, like the Hamburg Church History (*Hamburgische Kirchengeschichte*) by Adam of Bremen. The document refers to the Stormarni as one of the Saxon tribes.


title: Storm Dudley

abstract: From 14 to 19 February 2022, a European windstorm struck the United Kingdom, Germany, Poland, The Netherlands, the Czech Republic, and Lithuania. The storm was named Storm Dudley by the UK Met Office on 14 February 2022. The storm caused 225,000 people to lose power, killed nine people, and spawned 24 tornadoes across Europe. The storm occurred just before the stronger and more impactful Storm Eunice, which caused over 3.1 million power outages, 17 fatalities, and over €1.83 billion in damage.

text:
# Storm Dudley

From 14 to 19 February 2022, a European windstorm struck the United Kingdom, Germany, Poland, The Netherlands, the Czech Republic, and Lithuania. The storm was named **Storm Dudley** by the UK Met Office on 14 February 2022. The storm caused 225,000 people to lose power, killed nine people, and spawned 24 tornadoes across Europe. The storm occurred just before the stronger and more impactful Storm Eunice, which caused over 3.1 million power outages, 17 fatalities, and over €1.83 billion in damage.

## Meteorological history

Storm Dudley, along with Storm Eunice and Storm Franklin, formed by a sting jet, a narrow band of winds that can create narrow and intense winds in a small area. It was also partly due to a large polar vortex in the Arctic that formed these storms. The storm trio isn't particularly rare for the area, as it has happened before. The storm also brought a very rare formation of mammatus clouds, and brought wind gusts of 81 miles per hour (130 km/h) in Capel Cruig, Wales, 74 miles per hour (119 km/h) in Yorkshire and 71 miles per hour (114 km/h) in Drumalbin, Scotland.

## Preparations and impacts

### United Kingdom

The United Kingdom saw one fatality due to Storm Dudley. Dudley was expected to bring winds of 60 to 70 miles per hour (97 to 113 km/h), which caused travel restrictions. The Environment Agency issued 38 flood alerts for the UK. In Yorkshire, a falling power line electrocuted 13 dogs. During the cleanup of Storm Dudley, Storm Eunice struck, further compounding the damage. PwC estimated the insurance losses for the storm to be between £40 million and £75 million.

### Germany

Storm Dudley causes three deaths in Germany, and spawned winds up to 152 kilometres per hour (94 mph) in the country. The Elbe River flowed 2 meters (6.6 ft) higher than normal, and a ferry had its windows shattered. The storm also spawned one tornado in Germany, in Brandenburg, which was rated as an F1/T3 tornado.

### Poland

The storm produced a very intense squall line of severe thunderstorms that moved through Poland, producing damaging straight-line winds and numerous embedded tornadoes. A man in Western Poland near Gorzów Wielkopolski died when a tree fell on his car. Meanwhile, a tornado in Kraków caused a 20 meters (66 ft) long crane to collapse, killing two people and injuring two others. Another tornado in Sierosław injured one, and an F2 tornado in Wójcice injured two. Overall, 23 tornadoes were confirmed in Poland.

### Lithuania

Storm Dudley killed two people in Lithuania due to strong winds.

## Tornado outbreak

Confirmed tornadoes by Fujita rating
| FU | F0 | F1 | F2 | F3 | F4 | F5 | Total |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 4 | 0 | 7 | 13 | 0 | 0 | 0 | 24 |

| List of confirmed tornadoes – Thursday, 17 February 2022 | | | | | | | |
| --- | --- | --- | --- | --- | --- | --- | --- |
| F# | T# | Location | District/
County | Coord. | Time (UTC) | Path length | Comments/Damage |
| **Germany** | | | | | | | |
| **F1** | **T3** | Casekow | Brandenburg | 53°13′N 14°12′E / 53.21°N 14.20°E / 53.21; 14.20 (Brandenburg (17 Feb, F1)) | 00:46 | 1.1 km, max. width 70m | This high-end F1 tornado damaged multiple buildings in and around Casekow. Homes in town sustained considerable roof damage, farmhouses were badly damaged outside of town, and horse stables were destroyed. Roof and solar panels were lifted and thrown, with parts shredded to pieces. Heavy damage occurred in a small wooded area, with many trees debranched and snapped. |
| **Poland** | | | | | | | |
| **F1** | N/A | Skwierzyna | Lubuskie | 52°34′N 15°31′E / 52.57°N 15.52°E / 52.57; 15.52 (Lubuskie (17 Feb, F1)) | 02:00 (+/- 5 min.) | Unknown | Roofs of homes were damaged and windows were blown out. Large tree branches were snapped, and pieces of wood were thrown and impaled into a building's insulation layer. |
| **F2** | N/A | Izdebno | Wielkopolskie | 52°40′N 16°10′E / 52.67°N 16.17°E / 52.67; 16.17 (Wielkopolskie (17 Feb, F2)) | 02:24 | 1.1 km | Significant damage to a local forest was reported, and drone photos showed that a large swath of trees was flattened in a convergent pattern. |
| **F2** | N/A | Sucharzewo | Wielkopolskie | 52°09′N 16°50′E / 52.15°N 16.83°E / 52.15; 16.83 (Wielkopolskie (17 Feb, F2)) | 03:03 | Unknown | Significant damage to houses and outbuildings occurred, with some structures largely destroyed. A brick cowshed collapsed, killing a bull inside. Swaths of trees were mowed down in forested areas, and some debarking was noted. |
| **FU** | N/A | Krzykosy | Wielkopolskie | 52°15′N 18°50′E / 52.25°N 18.83°E / 52.25; 18.83 (Wielkopolskie (17 Feb, FU)) | 03:11 | Unknown | Power transmission lines and trees were downed, and residences sustained roof damage. At least one building was unroofed entirely. |
| **F2** | N/A | Młodzikowice | Wielkopolskie | 52°07′N 17°14′E / 52.12°N 17.24°E / 52.12; 17.24 (Wielkopolskie (17 Feb, F2)) | 03:18 | 0.9 km | A masonry outbuilding and a garage were largely destroyed, along with the roof of a cowshed with 50 cows living inside the building. Large trees were snapped in a convergent pattern in forested areas. |
| **F2** | **T4** | Grzywna Szlachecka, Kuczwały | Kujawsko-Pomorskie | 53°08′N 18°38′E / 53.14°N 18.64°E / 53.14; 18.64 (Kujawsko-Pomorskie (17 Feb, F2)) | 03:22 | 600m | Masonry outbuildings were destroyed, and vehicles were also damaged. Homes sustained roof, insulation, and window damage, and some buildings had damage to walls. |
| **F2** | **T4** | Nowe Miasto nad Wartą | Wielkopolskie | 52°05′N 17°24′E / 52.09°N 17.40°E / 52.09; 17.40 (Wielkopolskie (17 Feb, F2)) | 03:25 | 1.7 km, max. width 240m | A swath of large trees were snapped and uprooted in a convergent pattern as this tornado moved through forested areas. |
| **F2** | N/A | Lgów | Wielkopolskie | 52°06′N 17°32′E / 52.10°N 17.54°E / 52.10; 17.54 (Wielkopolskie (17 Feb, F2)) | 03:32 | Unknown | Severe damage occurred in Lgów, where homes had their roofs torn off, outbuildings were destroyed, and debris was scattered throughout the community. Cars had windows blown out and were damaged by flying debris, and numerous large trees were snapped in a forest outside of town. |
| **F2** | N/A | Kosewo | Wielkopolskie | 52°24′N 17°57′E / 52.40°N 17.95°E / 52.40; 17.95 (Wielkopolskie (17 Feb, F2)) | 03:34 | 0.1 km max. width 100 m | A very brief but strong tornado downed trees in a convergent pattern and tossed boats at Lake Powidzkie. A residential building was heavily damaged, and sustained damage to its windows and insulation layer. |
| **F2** | **T5** | Dobrzyca, Sośnica | Wielkopolskie | 52°24′N 17°57′E / 52.40°N 17.95°E / 52.40; 17.95 (Wielkopolskie (17 Feb, F2)) | 03:42 | Unknown | Significant structural damage occurred in Dobrzyca, where numerous homes had their roofs destroyed or torn off, and some sustained loss of their upper-floor exterior walls. Power lines were downed, streets were strewn with debris, and vehicles were also damaged. Large trees were snapped in and around town. |
| **F2** | N/A | Stara Ciświca | Wielkopolskie | 52°01′N 17°59′E / 52.01°N 17.98°E / 52.01; 17.98 (Wielkopolskie (17 Feb, F2)) | 03:50 (+/- 15 min.) | Unknown | Homes and a church had roofing torn off. Several outbuildings were destroyed, and others had walls partially collapsed. Some ducks were killed at a duck farm in the area, and many trees were uprooted and snapped. |
| **F1** | **T3** | Żurawin | Wielkopolskie | 52°05′N 18°11′E / 52.08°N 18.18°E / 52.08; 18.18 (Wielkopolskie (17 Feb, F1)) | 03:55 (+/- 15 min.) | Unknown | A house had its metal roof blown off, with wooden beams and metal sheets blown at least 150 metres away from the structure. A few trees were also damaged. |
| **F2** | N/A | Tykadłów, Żelazków, Zborów, Podzborów | Wielkopolskie | 51°51′N 18°11′E / 51.85°N 18.18°E / 51.85; 18.18 (Wielkopolskie (17 Feb, F2)) | 04:00 (+/- 5 min.) | Unknown | Extensive damage occurred as this tornado impacted multiple villages, where homes and other structures had their roofs destroyed, and some had walls collapsed. Cars and buildings had their windows blown out, and many trees were snapped or uprooted. A soccer field was damaged as well. |
| **F1** | **T3** | Laski | Wielkopolskie | 52°00′N 18°34′E / 52.00°N 18.57°E / 52.00; 18.57 (Wielkopolskie (17 Feb, F1)) | 04:09 | Unknown | Trucks and trailers were overturned, and roofs were almost entirely torn off from residential buildings, with debris left deposited in nearby trees. |
| **F1** | **T3** | Łęka | Wielkopolskie | 52°08′N 18°38′E / 52.13°N 18.64°E / 52.13; 18.64 (Wielkopolskie (17 Feb, F1)) | 04:10 (+/- 5 min.) | Unknown | Roofs were damaged or destroyed in the village as a result of this high-end F1 tornado. In one case, a wooden beam was found speared into a building. Trees in a nearby forest were downed in a convergent pattern. |
| **F1** | **T3** | Lipicze Górne, Ziemięcin | Łódzkie | 52°08′N 18°33′E / 52.13°N 18.55°E / 52.13; 18.55 (Łódzkie (17 Feb, F1)) | 04:12 | Unknown | Roofs of outbuildings were destroyed, and at least two had walls partly collapsed. Parts of sheet metal and wooden beams from a building were blown off and scattered 100–150 metres away. Trees were downed in a local forest as well. |
| **F2** | N/A | Słomków Kościelny | Wielkopolskie | 52°00′N 18°40′E / 52.00°N 18.66°E / 52.00; 18.66 (Wielkopolskie (17 Feb, F2)) | 04:13 | Unknown | A summer house was completely destroyed, and a nearby car was overturned and heavily damaged. |
| **F2** | N/A | Wójcice, Smaszków | Łódzkie | 51°39′N 18°28′E / 51.65°N 18.47°E / 51.65; 18.47 (Łódzkie (17 Feb, F2)) | 04:15 (+/- 5 min.) | Unknown | Numerous houses, outbuildings and trees sustained severe damage in both villages. A greenhouse was damaged, large masonry outbuildings were destroyed, and homes had their roofs torn off. Power transmission and communication lines were downed, and trees were snapped or uprooted. A 65-year-old woman sustained head injuries from a falling roof truss, and her 16-year-old grandson was injured by flying glass. |
| **F1** | **T3** | Aleksandria | Łódzkie | 51°37′N 18°29′E / 51.61°N 18.48°E / 51.61; 18.48 (Łódzkie (17 Feb, F1)) | 04:20 (+/- 5 min.) | Unknown | Several buildings in close proximity to each other were considerably damaged. Trucks and trailers were overturned, and walls were partly collapsed. |
| **F2** | N/A | Różyce Grochowe, Sulimy, Mrożewice | Łódzkie | 51°58′N 19°11′E / 51.97°N 19.19°E / 51.97; 19.19 (Łódzkie (17 Feb, F2)) | 04:30 (+/- 5 min.) | 3 km | Structures in multiple villages sustained major roof damage, with debris scattered over 400 metres away. Large masonry outbuildings were almost fully destroyed and an area of forest was flattened, with many large trees snapped or uprooted. |
| **FU** | N/A | Sierosław | Łódzkie | 51°30′N 19°37′E / 51.50°N 19.62°E / 51.50; 19.62 (Łódzkie (17 Feb, F2)) | 05:05 (+/- 5 min.) | Unknown | Almost every building in Sierosław suffered some degree of damage, some of which had their roofs torn off, including a few that sustained damage to their brick exterior walls. Farms in the area were significantly damaged, and metal truss electrical transmission towers were blown over. Two trucks were overturned outside of town, and one of the drivers sustained injuries. |
| **FU** | N/A | Kraków | Małopolskie | 50°01′N 20°03′E / 50.02°N 20.05°E / 50.02; 20.05 (Małopolskie (February, FU)) | 8:05 (+/- 5 min.) | Unknown | **2 deaths** – A large construction crane was toppled to the ground, killing two construction workers. The roof of a supermarket was heavily damaged, with debris from the building strewn throughout the area. Some cars were damaged by flying roof debris, and two people were injured. |

## Highest wind gust per country

| Country | Gust | Location | Date |
| --- | --- | --- | --- |
| Belgium | 112 km/h | Zeebrugge | 16 February |
| Czech Republic | 160 km/h | Boskovice | 17 February |
| Denmark | 130 km/h | Frøslevlejren | 17 February |
| France | 164 km/h | Calais | 16 February |
| Germany | 153 km/h | Großer Arber | 17 February |
| Guernsey | 156 km/h | Saint Anne | 16 February |
| Ireland | 148 km/h | Waterford | 16 February |
| Jersey | 154 km/h | Saint John | 16 February |
| Luxembourg | 102 km/h | Lac de la Haute-Sûre | 17 February |
| Netherlands | 116 km/h | Lauwersoog | 17 February |
| Poland | 146 km/h | Kikół | 17 February |
| United Kingdom | 166 km/h | North Walsham | 16 February |

## InfoBox

Storm Dudley
|  | |
| --- | --- |
|  | |
|  | |
|  | |
| Type | Tornado outbreak, European windstorm |
| Duration | 14 – 19 February 2022 |
|  | |
|  | |
|  | |
| Highest gust | 166 km/h (103 mph), North Walsham. |
|  | |
|  | |
|  | |
| Tornadoes
confirmed | 24 |
| Max. rating1 | F2 tornado |
| Duration of
tornado outbreak2 | 8 hours, 14 minutes |
|  | |
|  | |
|  | |
| Fatalities | 9 deaths, 5 injuries |
| Damage | $863 million (2022 USD), €794 million (2022 Euros) |
| Power outages | 225,000 |
| Areas affected | United Kingdom, Germany, Poland, The Netherlands, Czech Republic, Lithuania |
| 1*Most severe tornado damage; see Fujita scale*
2*Time from first tornado to last tornado* | |


title: Story of My Life (Lesley Roy song)

abstract: "Story of My Life" is a song by Irish singer-songwriter Lesley Roy. The song would have represented Ireland in the Eurovision Song Contest 2020.

text:
# Story of My Life (Lesley Roy song)

"**Story of My Life**" is a song by Irish singer-songwriter Lesley Roy. The song would have represented Ireland in the Eurovision Song Contest 2020.

## Background

The Irish broadcaster RTÉ invited established songwriters and artists to submit songs from which they, in conjunction with 2FM, made a shortlist and chose their entry for the Eurovision Song Contest 2020.

## Eurovision Song Contest

The song was meant to represent Ireland in the Eurovision Song Contest 2020, after Lesley Roy was internally selected by the national broadcaster Raidió Teilifís Éireann (RTÉ). On 28 January 2020, a special allocation draw was held which placed each country into one of the two semi-finals, as well as which half of the show they would perform in. Ireland was placed into the first semi-final, to be held on 12 May 2020, and was scheduled to perform in the first half of the show. However, on March 18, 2020, the EBU announced the cancellation of Eurovision 2020 as a direct result of the Coronavirus pandemic in Europe.

## Music video

A music video to accompany the release of "Story of My Life" was first released onto YouTube on 5 March 2020. The music video features Roy wearing pink-colored attire. The video was directed by Kate Dolan.

## Charts

| Chart (2020) | Peak
position |
| --- | --- |
| Irish Homegrown Top 20 (OCC) | 12 |
|

## InfoBox

| "Story of My Life" | |
| --- | --- |
|  | |
| Single by Lesley Roy | |
| Released | 13 March 2020 |
| Genre | Dance-pop |
| Length | 3:05 |
| Label | Universal Denmark |
| Songwriter(s) |  |
| Lesley Roy singles chronology | |
| "Free the Flame (Ember Moon)"
(2016) "**Story of My Life**"
(2020) "Gold"
(2020) | |
|  | |
| Eurovision Song Contest 2020 entry | |
| Country | Ireland |
| Artist(s) | Lesley Roy |
| Language | English |
| Composer(s) |  |
| Lyricist(s) | * Lesley Roy * Robert Marvin * Catt Gravitt * Tom Shapiro |
| Finals performance | |
| Semi-final result | *Contest cancelled* |
| Entry chronology | |
| **◄ "22" (2019)** | |
| **"Maps" (2021) ►** | |
|  | |


title: Storyteller (Raghav album)

abstract: Storyteller is the debut album from Canadian R&B singer Raghav. Combining elements of his South-Asian background with reggae and US style R&B, Raghav has defined his own style as showcased in the UK top ten hits with "Can't Get Enough", "So Confused" and "Angel Eyes". The album reached No. 36 on the UK Albums Chart.

text:
# Storyteller (Raghav album)

***Storyteller*** is the debut album from Canadian R&B singer Raghav. Combining elements of his South-Asian background with reggae and US style R&B, Raghav has defined his own style as showcased in the UK top ten hits with "Can't Get Enough", "So Confused" and "Angel Eyes". The album reached No. 36 on the UK Albums Chart.

## Track listing

| # | Title | Featured guest(s) | Time |
| --- | --- | --- | --- |
| 1 | "Lets Work It Out" | Jahaziel | 4:08 |
| 2 | "No I" |  | 2:26 |
| 3 | "No No" |  | 4:57 |
| 4 | "So Confused" | 2Play feat. Raghav and Jucxi | 3:15 |
| 5 | "Bad Bad Bad" |  | 3:54 |
| 6 | "C'mon" |  | 3:33 |
| 7 | "Weakness" |  | 2:44 |
| 8 | "Winter in My Mind" |  | 4:22 |
| 9 | "Ain't Nobody" |  | 4:08 |
| 10 | "Can't Get Enough" | Iceberg Slimm | 3:23 |
| 11 | "Angel Eyes" | Jucxi & Frankey Maxx | 4:13 |
| 12 | "Sooner or Later" | Kardinal Offishall | 3:55 |
| 13 | "Baby I'm Amazed" |  | 5:51 |
| 14 | "Another Bad Goodbye" |  | 4:07 |
| 15 | "Chor Diya (The Ultimate Sacrifice)" |  | 4:18 |
| 16 | "Teri Baaton Mein Aisa Uljha Jiya" |  | 4:07 |
|

## InfoBox

| Storyteller | |
| --- | --- |
|  | |
| Studio album by Raghav | |
| Released | September 6, 2004 (UK) |
| Recorded | 2003–04 |
| Genre |  |
| Label | V2/A&R |
| Producer |  |
| Raghav chronology | |
| ***Storyteller***
(2004) *Identity*
(2009) | |


title: Stovepipe Johnson

abstract: Adam Rankin "Stovepipe" Johnson was an antebellum Western frontiersman and later an officer in the Confederate States Army during the American Civil War. Johnson obtained fame leading the Newburgh Raid using a force of only about 35 men. Johnson and his men confiscated supplies and ammunition without firing a shot by deceiving Newburgh's defenders into thinking Confederate cannons surrounded the town. In reality, the "cannons" were an assemblage of a stove pipe, a charred log, and wagon wheels, forever giving Johnson the nickname of Adam "Stovepipe" Johnson. Permanently blinded during an 1864 skirmish, in 1887, Johnson founded the town of Marble Falls, Texas, which became known as "the blind man's town."

text:
# Stovepipe Johnson

**Adam Rankin "Stovepipe" Johnson** (February 6, 1834  October 20, 1922) was an antebellum Western frontiersman and later an officer in the Confederate States Army during the American Civil War. Johnson obtained fame leading the Newburgh Raid using a force of only about 35 men. Johnson and his men confiscated supplies and ammunition without firing a shot by deceiving Newburgh's defenders into thinking Confederate cannons surrounded the town. In reality, the "cannons" were an assemblage of a stove pipe, a charred log, and wagon wheels, forever giving Johnson the nickname of Adam "Stovepipe" Johnson. Permanently blinded during an 1864 skirmish, in 1887, Johnson founded the town of Marble Falls, Texas, which became known as "the blind man's town."

## Early life

Johnson was born in Henderson, Kentucky, a son of Thomas J. and Juliet (Rankin) Johnson. Johnson was educated in the local schools and worked in a drugstore from 12 to 20. In 1854, he moved to Hamilton Valley in Burnet County, Texas, and worked as a surveyor on the West Texas frontier. He was a noted Indian fighter and provided supplies and animals for the Butterfield Overland Mail stations. On January 1, 1861, he married Josephine Eastland of Austin, with whom he had nine children.

## American Civil War

When the American Civil War began, Johnson returned to Kentucky. He joined Nathan B. Forrest's cavalry battalion as a scout, fighting at the Battle of Sacramento. Johnson escaped capture with Forrest after Fort Donelson, when the Confederate commanders decided to surrender their post to the Union besiegers. Johnson later received a promotion to colonel for his exploits with his 10th Kentucky Partisan Rangers, a regiment he raised that often operated deep behind U.S. Army lines in Kentucky. Johnson's men harassed Union supply lines and attacked isolated garrisons. In July 1862, in his Newburgh Raid, Johnson captured the town of Newburgh, Indiana, bluffing its sizable Union militia force into surrendering with only twelve of his men and a stovepipe mounted and a burnt black log on the running gears of an abandoned wagon to form a Quaker cannon. His capture of the first Northern city to fall to the Confederates made the news even in Europe, and Johnson's men thereafter nicknamed him "Stovepipe".

In 1863, Johnson assumed command of a brigade in the cavalry division of Brig. Gen. John Hunt Morgan. He reluctantly participated in Morgan's Raid, though he was only supposed to raid on the Kentucky side of the river. Following the Confederate disaster at the Battle of Buffington Island, Johnson led nearly 350 of his men across the rain-swollen Ohio River to safety. The remainder of Morgan's division was trapped on the Ohio side of the river and eventually forced to surrender.

Johnson was appointed brigadier general on September 6, 1864, to rank from June 1, 1864, though his appointment was never confirmed by the Confederate Congress. On August 21, 1864, he was blinded by an accidental shot from one of his men during a skirmish at Grubb's Crossroads, near Princeton, Kentucky. Abandoned because of his injuries, U.S. soldiers captured him, and Johnson became a prisoner of war in Fort Warren. Johnson was exchanged near the war's end and, despite his blindness, attempted to return to active duty before the Confederacy finally surrendered.

## Postbellum

Johnson returned to Texas after being exchanged and paroled in 1865. Although blind, he founded a town, established a company, and worked to harness the water power of the Colorado River. One of his grandsons was Rankin Johnson Sr., a former Major League pitcher for the Boston Red Sox and St. Louis Cardinals.

He died in Burnet, Texas in 1922 at the age of 88, and is interred at the Texas State Cemetery in Austin, Texas. He rests beside his wife Josephine and near his grandson, Judge George Christian Sr., and a great-grandson, George Christian Jr.

## Further reading

## InfoBox

| Adam Rankin Johnson | |
| --- | --- |
|  | |
| Nickname(s) | "Stovepipe" |
| Born | (1834-02-06)February 6, 1834
Henderson, Kentucky, United States |
| Died | October 20, 1922(1922-10-20) (aged 88)
Burnet, Texas, United States |
| Place of burial | Texas State Cemetery
Austin, Texas |
| Allegiance | United States of America
Confederate States of America |
| Service/branch | Confederate States Army |
| Years of service | 1861–1865 |
| Rank | Colonel
Brigadier General (appointment not confirmed) |
| Unit | 3rd Tennessee Cavalry (Forrest's) |
| Commands | 10th Kentucky Partisan Rangers
Johnson's Cavalry Brigade |
| Battles/wars | American Civil War |


title: Stovepipe No. 1

abstract: Stovepipe No. 1, born probably Samuel Chambers Jones was an American blues musician and songster, active in the Cincinnati area of the United States. He made his first recordings in 1924.

text:
# Stovepipe No. 1

**Stovepipe No. 1**, (August 7, 1890 - Unknown) born probably **Samuel Chambers Jones** was an American blues musician and songster, active in the Cincinnati area of the United States. He made his first recordings in 1924.

## Biography

Jones was born in Paducah, Kentucky on August 7, 1890. By the early 1910s, he had relocated to Cincinnati, Ohio, where he was known as a street singer and one-man band who played in many different styles and for black and white audiences. Jones became known for playing a stovepipe in the same manner as a jug, as well as wearing a stovepipe hat; contributing to his later nickname.

In 1924, Jones made his recording debut for Gennett Records. While playing in the streets of Cincinnati, Jones had adopted the nickname "Daddy Stovepipe", in reference to his stovepipe playing. Jones wished to be billed as "Daddy Stovepipe" on his records, however, Chicago based blues musician Johnny Watson had coincidentally recorded under that name six days prior. Jones then chose to adopt the name **Stovepipe No. 1**, perhaps as a way to brand himself as the 'original'. Jones also recorded under the name "Stovepipe Jones", "Stovepipe Jazz Band", and "Sam Jones". None of Jones' Gennett recordings were issued.

A few months later, Jones recorded for Columbia Records in New York City. He recorded 20 sides in these sessions, only six of which were issued, on both the Columbia and Harmony labels. The issued songs show a variety of music styles, from gospel to folk music.

Jones returned to the studio in 1927, being accompanied by blues guitarist and singer David Crockett. Crockett was a more sophisticated guitar player than Jones, who became a regular recording partner throughout the remainder of his recording career. In these sessions, they recorded 6 sides, four of them being issued by Okeh Records. This was Jones' last session where he was given leading credit.

Jones may have also recorded with Bob Coleman's Cincinnati Jug Band during the 1920s

In 1930, Jones and Crockett, along with a group of unidentified musicians, entered the studio under the name "King David's Jug Band". The band recorded 6 sides for Okeh, all of which were issued. This was the last time Jones recorded.

It is not known when Jones died. He is remembered as being alive in the 1960s.

## Recordings

### 1924

Recorded May 16, 1924 in Richmond, Indiana for Gennett Records:

* Six Street Blues - Unissued
* Them Pitiful Blues - Unissued
* Dixie Barn Dance - Unissued
* Spanish Rag - Unissued
* Hummin' Blues - Unissued
* In Dey Go - Unissued

Recorded August 18, 1924 in New York City for Columbia Records:

* Stovepipe Blues - Unissued
* Spanish Rag - Unissued
* Sixth Street Blues - Unissued
* Loveland Blues - Unissued

Recorded August 19, 1924 in New York City for Columbia Records:

* Lord Don't You Know I Don't Have No Friend Like You - Columbia 210-D
* When The Saints Go Marching Through - Unissued
* I've Got Salvation In My Heart - Columbia 210-D
* Soon One Morning Death Came Creeping In The Room - Unissued
* I'm Going To Wait On The Lord - Unissued
* Bye And Bye When The Morning Came - Unissued
* Pitiful Blues - Unissued
* Sundown Blues - Unissued
* Dan Tucker - Unissued

Recorded August 20, 1924 in New York City for Columbia Records:

* John Henry - Unissued
* Lonesome John - Columbia 15011-D
* Cripple Creed and Sourwood Mountain - Columbia 201-D
* Turkey In The Straw - Columbia 201-D
* Arkansas Traveler - Unissued
* Fisher's Hornpipe - Unissued
* Fisher's Hornpipe - Columbia 15011-D

### 1927

Recorded April 25, 1927 for Okeh Records:

* Court Street Blues - Okeh 8514
* Sundown Blues - Unissued

Recorded April 26, 1927 for Okeh Records:

* Untitled Song - unissued
* A Woman Gets Tired Of The Same Man All The Time - Okeh 8514
* A Chicken Can Waltz The Gravy Around - Okeh 8543
* Bed Slats - Okeh 8543

### 1930

With **King David's Jug Band**

Recorded December 11, 1930 for Okeh Records:

* What's That Tastes Like Gravy - Okeh 8913
* Rising Sun Blues - Okeh 8913
* Sweet Potato Blues - Okeh 8901
* Tear It Down - Okeh 8961
* I Can Deal Worry - Okeh 8901
* Georgia Bo Bo - Okeh 8961

## InfoBox

| Stovepipe No. 1 | |
| --- | --- |
| Birth name | probably Samuel Chambers Jones |
| Born | (1890-08-07)August 7, 1890<br>Paducah, Kentucky, U.S. |
| Died | Unknown |
| Genres |  |
| Occupation | Musician |
| Instruments | * Guitar * Vocals * Harmonica * Stovepipe |
| Years active | 1920s - 1960s |


title: Strike Under

abstract: Strike Under was an influential Chicago punk rock band of the early 1980s. It was started by Steve Bjorklund (singer-guitarist) after the demise of his previous band, The Rabbits. The principal musicians, besides Bjorklund, were his brother Chris (guitarist), Pierre Kezdy (bassist), and Bob Furem (drummer).

text:
# Strike Under

**Strike Under** was an influential Chicago punk rock band of the early 1980s. It was started by Steve Bjorklund (singer-guitarist) after the demise of his previous band, The Rabbits. The principal musicians, besides Bjorklund, were his brother Chris (guitarist), Pierre Kezdy (bassist), and Bob Furem (drummer).

## Background

Strike Under was founded in 1979 or 1980. Although American punk is generally considered to have started four or five years earlier, punk and new wave music arrived in Chicago far later than on either coast, so that Strike Under is correctly still considered an "early" Chicago punk band. They can also be seen as one of the first and most important bands of the "second wave" of Chicago punk, as it turned toward a more hardcore scene and sound, and as an archetype of Chicago hardcore.

In 1981, Strike Under released a 5-song 12" EP record, *Immediate Action*, notable as the first release by the new label Wax Trax Records (the record is commonly known as WAX 001, although the actual catalog number is Wax Trax 103015X). The album cover resembled a book of matches (the band's name derives from the phrase commonly seen on cheap matchbooks). Songs included "Sunday Night Disorientation", "Context", "Closing In", "Elephant's Graveyard" (which later appeared on the Wax Trax *Best of* boxed set), and the title track. *Immediate Action* received mixed reviews; however, as one of the first actual recordings by a local punk outfit in the Windy City, it looms large in the memory of denizens of the early Chicago punk scene, and has been called an "integral document to the history of punk rock in Chicago".

Strike Under, besides releasing its own record, played out at clubs such as O'Banion's, and also appeared on the 1981 *Busted at Oz* LP, which is usually viewed as an historic document of the second wave of Chicago punk, recorded at the club Oz before it closed down.

Strike Under broke up in late 1981. Steve Bjorklund briefly formed another Chicago punk band, Terminal Beach, then ultimately relocated to Minneapolis to pursue playing and recording with various punk and industrial music bands, including Breaking Circus. The other three members formed Trial By Fire, which played throughout Chicago in 1982. After Trial By Fire broke up, Pierre Kezdy went on to play with Naked Raygun and Pegboy. Chris Bjorklund played first with Bloodsport and then with The Effigies during the mid and late 1980s, the latter being another leading 1980s Chicago hardcore band who happened to be fronted by Pierre Kezdy's brother, John Kezdy.

Strike Under was profiled in the 2007 documentary *You Weren't There*. The film featured interviews with both Bjorklund brothers and Pierre Kezdy.

## Discography

### Releases

* *Immediate Action* 12" EP (Wax Trax!, 1981/2020)
* *09-05-08* 7" single (Wax Trax!, 2020) - "One Eye", "Damaged Goods"

### Compilation appearances

* "Elephant's Graveyard" on the *Black Box - Wax Trax! Records: The First 13 Years* CD box set (Wax Trax!/TVT 1994)
* "Elephant's Graveyard" on the *You Weren't There: A History Of Chicago Punk 1977-1983* soundtrack LP (Factory 25 Records 2009)
* "Fucking Uniforms" & "Anarchy Song" on the *Busted At Oz* compilation LP (Autumn Records 1981/Permanent Records 2011)

## InfoBox

| Strike Under | |
| --- | --- |
|  | |
| Background information | |
| Origin | Chicago, Illinois, US |
| Genres | Punk rock |
| Years active | 1980–1981 |
| Labels | Wax Trax! |
|  | |
| Members | Steve Bjorklund, Chris Bjorklund, Pierre Kezdy, Bob Furem |
|  | |
| Website | Myspace page |


title: Strike Revilla

abstract: Strike Bautista Revilla is a Filipino politician and incumbent Representative of Cavite's 2nd District in the Philippine House of Representatives. He is a member of the National Unity Party. He previously served as mayor of Bacoor and was a member of the board of directors of the Philippine Charity Sweepstakes Office (PCSO) from August 2005 to March 2007.

text:
# Strike Revilla

**Strike Bautista Revilla** (Tagalog pronunciation: \[rɛˈbiljɐ\]; born **Edwin Mortel Bautista**; March 2, 1970) is a Filipino politician and incumbent Representative of Cavite's 2nd District in the Philippine House of Representatives. He is a member of the National Unity Party. He previously served as mayor of Bacoor and was a member of the board of directors of the Philippine Charity Sweepstakes Office (PCSO) from August 2005 to March 2007.

## Background

Born as Edwin Mortel Bautista, he is the 61st child of Jose Acuña Bautista (Ramon Bautista Revilla, Sr.) and the fifth child of Azucena Guzman Mortel Bautista. He is the younger brother of incumbent Senator Ramon "Bong" Revilla Jr. He finished his elementary education at Jesus Good Shepherd School in 1983 and attended high school at San Dimas High School in Los Angeles County, California from 1983 to 1987. He finished Commerce major in Management at Colegio de San Juan de Letran in Manila, where he also got his master's degree in Business Administration.

Prior to entering politics, Revilla managed various family businesses, including the Angelus Eternal Garden memorial park, which has branches in Bacoor and Imus, Cavite.

## Political career

At 25, Revilla was first elected as councilor of Bacoor in 1995. In 1998, he was elected to the Sangguniang Panlalawigan of Cavite where he served for two terms until 2004. He was appointed by President Gloria Macapagal Arroyo to the board of directors of the PCSO in 2005.

## InfoBox

| The HonourableStrike Revilla | |
| --- | --- |
| Official portrait, 2019 | |
|  | |
| Mayor of Bacoor | |
| **Incumbent** | |
| **Assumed office**
June 30, 2022 | |
| Preceded by | Lani Mercado |
| **In office**
June 30, 2007  June 30, 2016 | |
| Preceded by | Jessie B. Castillo |
| Succeeded by | Lani Mercado |
| Deputy Speaker of the House of Representatives of the Philippines | |
| **In office**
December 14, 2020  June 30, 2022 | |
| Member of the Philippine House of Representatives from Cavite's 2nd district | |
| **In office**
June 30, 2016  June 30, 2022 | |
| Preceded by | Lani Mercado |
| Succeeded by | Lani Mercado |
| Member of the Cavite Provincial Board from the 1st district | |
| **In office**
June 30, 1998  June 30, 2004 | |
| Member of the Bacoor Municipal Council | |
| **In office**
June 30, 1995  June 30, 1998 | |
|  | |
| Personal details | |
| Born | Edwin Mortel Bautista
(1970-03-02) March 2, 1970
Bacoor, Cavite, Philippines |
| Political party | Nacionalista (2021–present) |
| Other political
affiliations | NUP (2018–2021)
PDP–Laban (2016–2018)
Lakas–CMD (until 2016) |
| Relatives | Ramon Revilla, Sr. (father)
Bong Revilla (brother) |
| Alma mater | Colegio de San Juan de Letran (BS, MBA) |
| Occupation | Businessman |
|  | |


abstract: In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

text:
# Structure theorem for finitely generated modules over a principal ideal domain

In mathematics, in the field of abstract algebra, the **structure theorem for finitely generated modules over a principal ideal domain** is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

## Statement

When a vector space over a field *F* has a finite generating set, then one may extract from it a basis consisting of a finite number *n* of vectors, and the space is therefore isomorphic to *F**n*. The corresponding statement with *F* generalized to a principal ideal domain *R* is no longer true, since a basis for a finitely generated module over *R* might not exist. However such a module is still isomorphic to a quotient of some module *Rn* with *n* finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis of *Rn* to the generators of the module, and take the quotient by its kernel.) By changing the choice of generating set, one can in fact describe the module as the quotient of some *Rn* by a particularly simple submodule, and this is the structure theorem.

The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.

### Invariant factor decomposition

For every finitely generated module *M* over a principal ideal domain *R*, there is a unique decreasing sequence of proper ideals $(d\_{1})\supseteq (d\_{2})\supseteq \cdots \supseteq (d\_{n})$ such that *M* is isomorphic to the sum of cyclic modules:

$$M\cong \bigoplus \_{i}R/(d\_{i})=R/(d\_{1})\oplus R/(d\_{2})\oplus \cdots \oplus R/(d\_{n}).$$

The generators $d\_{i}$ of the ideals are unique up to multiplication by a unit, and are called invariant factors of *M*. Since the ideals should be proper, these factors must not themselves be invertible (this avoids trivial factors in the sum), and the inclusion of the ideals means one has divisibility $d\_{1}\\,|\\,d\_{2}\\,|\\,\cdots \\,|\\,d\_{n}$. The free part is visible in the part of the decomposition corresponding to factors $d\_{i}=0$. Such factors, if any, occur at the end of the sequence.

While the direct sum is uniquely determined by *M*, the isomorphism giving the decomposition itself is *not unique* in general. For instance if *R* is actually a field, then all occurring ideals must be zero, and one obtains the decomposition of a finite dimensional vector space into a direct sum of one-dimensional subspaces; the number of such factors is fixed, namely the dimension of the space, but there is a lot of freedom for choosing the subspaces themselves (if dim *M* \> 1).

The nonzero $d\_{i}$ elements, together with the number of $d\_{i}$ which are zero, form a complete set of invariants for the module. Explicitly, this means that any two modules sharing the same set of invariants are necessarily isomorphic.

Some prefer to write the free part of *M* separately:

$$R^{f}\oplus \bigoplus \_{i}R/(d\_{i})=R^{f}\oplus R/(d\_{1})\oplus R/(d\_{2})\oplus \cdots \oplus R/(d\_{n-f})\\;,$$

where the visible $d\_{i}$ are nonzero, and *f* is the number of $d\_{i}$'s in the original sequence which are 0.

### Primary decomposition

Every finitely generated module *M* over a principal ideal domain *R* is isomorphic to one of the form

$$\bigoplus \_{i}R/(q\_{i})\\;,$$

where $(q\_{i})\neq R$ and the $(q\_{i})$ are primary ideals. The $q\_{i}$ are unique (up to multiplication by units).

The elements $q\_{i}$ are called the *elementary divisors* of *M*. In a PID, nonzero primary ideals are powers of primes, and so $(q\_{i})=(p\_{i}^{r\_{i})=(p\_{i})^{r\_{i}}}$. When $q\_{i}=0$, the resulting indecomposable module is $R$ itself, and this is inside the part of *M* that is a free module.

The summands $R/(q\_{i})$ are indecomposable, so the primary decomposition is a decomposition into indecomposable modules, and thus every finitely generated module over a PID is a completely decomposable module. Since PID's are Noetherian rings, this can be seen as a manifestation of the Lasker-Noether theorem.

As before, it is possible to write the free part (where $q\_{i}=0$) separately and express *M* as

$$R^{f}\oplus (\bigoplus \_{i}R/(q\_{i}))\\;,$$

where the visible $q\_{i}$ are nonzero.

## Proofs

One proof proceeds as follows:

* Every finitely generated module over a PID is also finitely presented because a PID is Noetherian, an even stronger condition than coherence.
* Take a presentation, which is a map $R^{r}\to R^{g}$ (relations to generators), and put it in Smith normal form.

This yields the invariant factor decomposition, and the diagonal entries of Smith normal form are the invariant factors.

Another outline of a proof:

* Denote by *tM* the torsion submodule of *M*. Torsion module can be embedded as a submodule of *M* and this gives short exact sequence:

$$0\rightarrow tM\rightarrow M\rightarrow M/tM\rightarrow 0$$

Where the map is a projection. *M*/*tM* is a finitely generated torsion free module, and such a module over a commutative PID is a free module of finite rank, so it is isomorphic to: $R^{n}$ for a positive integer *n*. Since every free module is projective module, then exists right inverse of the projection map (it suffices to lift each of the generators of *M/tM* into *M*). By splitting lemma (left split) M splits into: $M=tM\oplus F$.

* For a prime element *p* in *R* we can then speak of $N\_{p}=\\{m\in tM\mid \exists i,mp^{i}=0\\}$. This is a submodule of *tM*, and it turns out that each *N**p* is a direct sum of cyclic modules, and that *tM* is a direct sum of *N**p* for a finite number of distinct primes *p*.
* Putting the previous two steps together, *M* is decomposed into cyclic modules of the indicated types.

## Corollaries

This includes the classification of finite-dimensional vector spaces as a special case, where $R=K$. Since fields have no non-trivial ideals, every finitely generated vector space is free.

Taking $R=\mathbb {Z} $ yields the fundamental theorem of finitely generated abelian groups.

Let *T* be a linear operator on a finite-dimensional vector space *V* over *K*. Taking $R=K\[T\]$, the algebra of polynomials with coefficients in *K* evaluated at *T*, yields structure information about *T*. *V* can be viewed as a finitely generated module over $K\[T\]$. The last invariant factor is the minimal polynomial, and the product of invariant factors is the characteristic polynomial. Combined with a standard matrix form for $K\[T\]/p(T)$, this yields various canonical forms:

## Uniqueness

While the invariants (rank, invariant factors, and elementary divisors) are unique, the isomorphism between *M* and its canonical form is not unique, and does not even preserve the direct sum decomposition. This follows because there are non-trivial automorphisms of these modules which do not preserve the summands.

However, one has a canonical torsion submodule *T*, and similar canonical submodules corresponding to each (distinct) invariant factor, which yield a canonical sequence:

$$0\<\cdots \<T\<M.$$

Compare composition series in Jordan–Hölder theorem.

For instance, if $M\approx \mathbf {Z} \oplus \mathbf {Z} /2$, and $(1,{\bar {0}),(0,{\bar {1})}}$ is one basis, then $(1,{\bar {1}),(0,{\bar {1})}}$ is another basis, and the change of basis matrix $\begin{bmatrix}1&0\\1&1\end{bmatrix}$ does not preserve the summand $\mathbf {Z} $. However, it does preserve the $\mathbf {Z} /2$ summand, as this is the torsion submodule (equivalently here, the 2-torsion elements).

## Generalizations

### Groups

The Jordan–Hölder theorem is a more general result for finite groups (or modules over an arbitrary ring). In this generality, one obtains a composition series, rather than a direct sum.

The Krull–Schmidt theorem and related results give conditions under which a module has something like a primary decomposition, a decomposition as a direct sum of indecomposable modules in which the summands are unique up to order.

### Primary decomposition

The primary decomposition generalizes to finitely generated modules over commutative Noetherian rings, and this result is called the Lasker–Noether theorem.

### Indecomposable modules

By contrast, unique decomposition into *indecomposable* submodules does not generalize as far, and the failure is measured by the ideal class group, which vanishes for PIDs.

For rings that are not principal ideal domains, unique decomposition need not even hold for modules over a ring generated by two elements. For the ring *R* = **Z**\[√−5\], both the module *R* and its submodule *M* generated by 2 and 1 + √−5 are indecomposable. While *R* is not isomorphic to *M*, *R*  *R* is isomorphic to *M*  *M*; thus the images of the *M* summands give indecomposable submodules *L*1, *L*2 \< *R*  *R* which give a different decomposition of *R*  *R*. The failure of uniquely factorizing *R*  *R* into a direct sum of indecomposable modules is directly related (via the ideal class group) to the failure of the unique factorization of elements of *R* into irreducible elements of *R*.

However, over a Dedekind domain the ideal class group is the only obstruction, and the structure theorem generalizes to finitely generated modules over a Dedekind domain with minor modifications. There is still a unique torsion part, with a torsionfree complement (unique up to isomorphism), but a torsionfree module over a Dedekind domain is no longer necessarily free. Torsionfree modules over a Dedekind domain are determined (up to isomorphism) by rank and Steinitz class (which takes value in the ideal class group), and the decomposition into a direct sum of copies of *R* (rank one free modules) is replaced by a direct sum into rank one projective modules: the individual summands are not uniquely determined, but the Steinitz class (of the sum) is.

### Non-finitely generated modules

Similarly for modules that are not finitely generated, one cannot expect such a nice decomposition: even the number of factors may vary. There are **Z**-submodules of **Q**4 which are simultaneously direct sums of two indecomposable modules and direct sums of three indecomposable modules, showing the analogue of the primary decomposition cannot hold for infinitely generated modules, even over the integers, **Z**.

Another issue that arises with non-finitely generated modules is that there are torsion-free modules which are not free. For instance, consider the ring **Z** of integers. Then **Q** is a torsion-free **Z**-module which is not free. Another classical example of such a module is the Baer–Specker group, the group of all sequences of integers under termwise addition. In general, the question of which infinitely generated torsion-free abelian groups are free depends on which large cardinals exist. A consequence is that any structure theorem for infinitely generated modules depends on a choice of set theory axioms and may be invalid under a different choice.

* Dummit, David S.; Foote, Richard M. (2004), *Abstract algebra* (3rd ed.), New York: Wiley, ISBN 978-0-471-43334-7, MR 2286236
* Hungerford, Thomas W. (1980), *Algebra*, New York: Springer, pp. 218–226, Section IV.6: Modules over a Principal Ideal Domain, ISBN 978-0-387-90518-1
* Jacobson, Nathan (1985), *Basic algebra. I* (2 ed.), New York: W. H. Freeman and Company, pp. xviii+499, ISBN 0-7167-1480-9, MR 0780184
* Lam, T. Y. (1999), *Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, Springer-Verlag, ISBN 978-0-387-98428-5


title: Structure tensor

abstract: In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant to the observing coordinates. The structure tensor is often used in image processing and computer vision.

text:
# Structure tensor

In mathematics, the **structure tensor**, also referred to as the **second-moment matrix**, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant to the observing coordinates. The structure tensor is often used in image processing and computer vision.

## The 2D structure tensor

### Continuous version

For a function $I$ of two variables *p* = (*x*, *y*), the structure tensor is the 2×2 matrix

$S\_{w}(p)={\begin{bmatrix}\int w(r)(I\_{x}(p-r))^{2}\\,dr&\int w(r)I\_{x}(p-r)I\_{y}(p-r)\\,dr\\\\[10pt\]\int w(r)I\_{x}(p-r)I\_{y}(p-r)\\,dr&\int w(r)(I\_{y}(p-r))^{2}\\,dr\end{bmatrix}}$ where $I\_{x}$ and $I\_{y}$ are the partial derivatives of $I$ with respect to *x* and *y*; the integrals range over the plane $\mathbb {R} ^{2}$; and *w* is some fixed "window function" (such as a Gaussian blur), a distribution on two variables. Note that the matrix $S\_{w}$ is itself a function of *p* = (*x*, *y*).

The formula above can be written also as $\textstyle S\_{w}(p)=\int w(r)S\_{0}(p-r)\\,dr$, where $S\_{0}$ is the matrix-valued function defined by $S\_{0}(p)={\begin{bmatrix}(I\_{x}(p))^{2}&I\_{x}(p)I\_{y}(p)\\\\[10pt\]I\_{x}(p)I\_{y}(p)&(I\_{y}(p))^{2}\end{bmatrix}}$

If the gradient $\nabla I=(I\_{x},I\_{y})^{\text{T}}$ of $I$ is viewed as a 2×1 (single-column) matrix, where $(\cdot )^{\text{T}}$ denotes transpose operation, turning a row vector to a column vector, the matrix $S\_{0}$ can be written as the matrix product $(\nabla I)(\nabla I)^{\text{T}}$ or tensor or outer product $\nabla I\otimes \nabla I$. Note however that the structure tensor $S\_{w}(p)$ cannot be factored in this way in general except if $w$ is a Dirac delta function.

### Discrete version

In image processing and other similar applications, the function $I$ is usually given as a discrete array of samples $I\[p\]$, where *p* is a pair of integer indices. The 2D structure tensor at a given pixel is usually taken to be the discrete sum

$S\_{w}\[p\]={\begin{bmatrix}\sum \_{r}w\[r\](I\_{x}\[p-r\])^{2}&\sum \_{r}w\[r\]I\_{x}\[p-r\]I\_{y}\[p-r\]\\\\[10pt\]\sum \_{r}w\[r\]I\_{x}\[p-r\]I\_{y}\[p-r\]&\sum \_{r}w\[r\](I\_{y}\[p-r\])^{2}\end{bmatrix}}$

Here the summation index *r* ranges over a finite set of index pairs (the "window", typically $\\{-m\ldots +m\\}\times \\{-m\ldots +m\\}$ for some *m*), and *w*\[*r*\] is a fixed "window weight" that depends on *r*, such that the sum of all weights is 1. The values $I\_{x}\[p\],I\_{y}\[p\]$ are the partial derivatives sampled at pixel *p*; which, for instance, may be estimated from by $I$ by finite difference formulas.

The formula of the structure tensor can be written also as $\textstyle S\_{w}\[p\]=\sum \_{r}w\[r\]S\_{0}\[p-r\]$, where $S\_{0}$ is the matrix-valued array such that $S\_{0}\[p\]={\begin{bmatrix}(I\_{x}\[p\])^{2}&I\_{x}\[p\]I\_{y}\[p\]\\\\[10pt\]I\_{x}\[p\]I\_{y}\[p\]&(I\_{y}\[p\])^{2}\end{bmatrix}}$

### Interpretation

The importance of the 2D structure tensor $S\_{w}$ stems from the fact eigenvalues $\lambda \_{1},\lambda \_{2}$ (which can be ordered so that $\lambda \_{1}\geq \lambda \_{2}\geq 0$) and the corresponding eigenvectors $e\_{1},e\_{2}$ summarize the distribution of the gradient $\nabla I=(I\_{x},I\_{y})$ of $I$ within the window defined by $w$ centered at $p$.

Namely, if $\lambda \_{1}\>\lambda \_{2}$, then $e\_{1}$ (or $-e\_{1}$) is the direction that is maximally aligned with the gradient within the window.

In particular, if $\lambda \_{1}\>0,\lambda \_{2}=0$ then the gradient is always a multiple of $e\_{1}$ (positive, negative or zero); this is the case if and only if $I$ within the window varies along the direction $e\_{1}$ but is constant along $e\_{2}$. This condition of eigenvalues is also called linear symmetry condition because then the iso-curves of $I$ consist in parallel lines, i.e there exists a one dimensional function $g$ which can generate the two dimensional function $I$ as $I(x,y)=g(d^{\text{T}p)}$ for some constant vector $d=(d\_{x},d\_{y})^{T}$ and the coordinates $p=(x,y)^{T}$.

If $\lambda \_{1}=\lambda \_{2}$, on the other hand, the gradient in the window has no predominant direction; which happens, for instance, when the image has rotational symmetry within that window. This condition of eigenvalues is also called balanced body, or directional equilibrium condition because it holds when all gradient directions in the window are equally frequent/probable.

Furthermore, the condition $\lambda \_{1}=\lambda \_{2}=0$ happens if and only if the function $I$ is constant ( $\nabla I=(0,0)$) within $W$.

More generally, the value of $\lambda \_{k}$, for *k*=1 or *k*=2, is the $w$-weighted average, in the neighborhood of *p*, of the square of the directional derivative of $I$ along $e\_{k}$. The relative discrepancy between the two eigenvalues of $S\_{w}$ is an indicator of the degree of anisotropy of the gradient in the window, namely how strongly is it biased towards a particular direction (and its opposite). This attribute can be quantified by the **coherence**, defined as

$c\_{w}=\left({\frac {\lambda \_{1}-\lambda \_{2}{\lambda \_{1}+\lambda \_{2}\right)^{2}}}}$

if $\lambda \_{2}\>0$. This quantity is 1 when the gradient is totally aligned, and 0 when it has no preferred direction. The formula is undefined, even in the limit, when the image is constant in the window ( $\lambda \_{1}=\lambda \_{2}=0$). Some authors define it as 0 in that case.

Note that the average of the gradient $\nabla I$ inside the window is **not** a good indicator of anisotropy. Aligned but oppositely oriented gradient vectors would cancel out in this average, whereas in the structure tensor they are properly added together. This is a reason for why $(\nabla I)(\nabla I)^{\text{T}}$ is used in the averaging of the structure tensor to optimize the direction instead of $\nabla I$.

By expanding the effective radius of the window function $w$ (that is, increasing its variance), one can make the structure tensor more robust in the face of noise, at the cost of diminished spatial resolution. The formal basis for this property is described in more detail below, where it is shown that a multi-scale formulation of the structure tensor, referred to as the multi-scale structure tensor, constitutes a *true multi-scale representation of directional data under variations of the spatial extent of the window function*.

### Complex version

The interpretation and implementation of the 2D structure tensor becomes particularly accessible using complex numbers. The structure tensor consists in 3 real numbers

$S\_{w}(p)={\begin{bmatrix}\mu \_{20}&\mu \_{11}\\\\[10pt\]\mu \_{11}&\mu \_{02}\end{bmatrix}}$

where $\textstyle \mu \_{20}=\int (w(r)(I\_{x}(p-r))^{2}\\,dr$ , $\textstyle \mu \_{02}=\int (w(r)(I\_{y}(p-r))^{2}\\,dr$ and $\textstyle \mu \_{11}=\int w(r)I\_{x}(p-r)I\_{y}(p-r)\\,dr$ in which integrals can be replaced by summations for discrete representation. Using Parseval's identity it is clear that the three real numbers are the second order moments of the power spectrum of $I$. The following second order complex moment of the power spectrum of $I$ can then be written as

$\kappa \_{20}=\mu \_{20}-\mu \_{02}+i2\mu \_{11}=\int w(r)(I\_{x}(p-r)+iI\_{y}(p-r))^{2}\\,dr=(\lambda \_{1}-\lambda \_{2})\exp(i2\phi )$

where $i={\sqrt {-1}}$ and $\phi $ is the direction angle of the most significant eigenvector of the structure tensor $\phi =\angle {e\_{1}}$ whereas $\lambda \_{1}$ and $\lambda \_{2}$ are the most and the least significant eigenvalues. From, this it follows that $\kappa \_{20}$ contains both a certainty $|\kappa \_{20}|=\lambda \_{1}-\lambda \_{2}$ and the optimal direction in double angle representation since it is a complex number consisting of two real numbers. It follows also that if the gradient is represented as a complex number, and is remapped by squaring (i.e. the argument angles of the complex gradient is doubled), then averaging acts as an optimizer in the mapped domain, since it directly delivers both the optimal direction (in double angle representation) and the associated certainty. The complex number represents thus how much linear structure (linear symmetry) there is in image $I$, and the complex number is obtained directly by averaging the gradient in its (complex) double angle representation without computing the eigenvalues and the eigenvectors explicitly.

Likewise the following second order complex moment of the power spectrum of $I$, which happens to be always real because $I$ is real,

$\kappa \_{11}=\mu \_{20}+\mu \_{02}=\int w(r)|I\_{x}(p-r)+iI\_{y}(p-r)|^{2}\\,dr=\lambda \_{1}+\lambda \_{2}$

can be obtained, with $\lambda \_{1}$ and $\lambda \_{2}$ being the eigenvalues as before. Notice that this time the magnitude of the complex gradient is squared (which is always real).

However, decomposing the structure tensor in its eigenvectors yields its tensor components as

$S\_{w}(p)=\lambda \_{1}e\_{1}e\_{1}^{\text{T}+\lambda \_{2}e\_{2}e\_{2}^{\text{T}=(\lambda \_{1}-\lambda \_{2})e\_{1}e\_{1}^{\text{T}+\lambda \_{2}(e\_{1}e\_{1}^{\text{T}+e\_{2}e\_{2}^{\text{T})=(\lambda \_{1}-\lambda \_{2})e\_{1}e\_{1}^{\text{T}+\lambda \_{2}E}}}}}}$

where $E$ is the identity matrix in 2D because the two eigenvectors are always orthogonal (and sum to unity). The first term in the last expression of the decomposition, $(\lambda \_{1}-\lambda \_{2})e\_{1}e\_{1}^{\text{T}}$, represents the linear symmetry component of the structure tensor containing all directional information (as a rank-1 matrix), whereas the second term represents the balanced body component of the tensor, which lacks any directional information (containing an identity matrix $E$). To know how much directional information there is in $I$ is then the same as checking how large $\lambda \_{1}-\lambda \_{2}$ is compared to $\lambda \_{2}$.

Evidently, $\kappa \_{20}$ is the complex equivalent of the first term in the tensor decomposition, whereas $\tfrac {1}{2}(|\kappa \_{20}|-\kappa \_{11})=\lambda \_{2}$is the equivalent of the second term. Thus the two scalars, comprising three real numbers,

$\begin{aligned}\kappa \_{20}&=&(\lambda \_{1}-\lambda \_{2})\exp(i2\phi )&=w\*(h\*I)^{2}\\\kappa \_{11}&=&\lambda \_{1}+\lambda \_{2}&=w\*|h\*I|^{2}\\\end{aligned}$ where $h(x,y)=(x+iy)\exp(-(x^{2}+y^{2})/(2\sigma ^{2}))$ is the (complex) gradient filter, and $\*$ is convolution, constitute a complex representation of the 2D Structure Tensor. As discussed here and elsewhere $w$ defines the local image which is usually a Gaussian (with a certain variance defining the outer scale), and $\sigma $ is the (inner scale) parameter determining the effective frequency range in which the orientation $2\phi $ is to be estimated.

The elegance of the complex representation stems from that the two components of the structure tensor can be obtained as averages and independently. In turn, this means that $\kappa \_{20}$ and $\kappa \_{11}$ can be used in a scale space representation to describe the evidence for presence of unique orientation and the evidence for the alternative hypothesis, the presence of multiple balanced orientations, without computing the eigenvectors and eigenvalues. A functional, such as squaring the complex numbers have to this date not been shown to exist for structure tensors with dimensions higher than two. In Bigun 91, it has been put forward with due argument that this is because complex numbers are commutative algebras whereas quaternions, the possible candidate to construct such a functional by, constitute a non-commutative algebra.

The complex representation of the structure tensor is frequently used in fingerprint analysis to obtain direction maps containing certainties which in turn are used to enhance them, to find the locations of the global (cores and deltas) and local (minutia) singularities, as well as automatically evaluate the quality of the fingerprints.

## The 3D structure tensor

### Definition

The structure tensor can be defined also for a function $I$ of three variables *p*=(*x*,*y*,*z*) in an entirely analogous way. Namely, in the continuous version we have $\textstyle S\_{w}(p)=\int w(r)S\_{0}(p-r)\\,dr$, where $S\_{0}(p)={\begin{bmatrix}(I\_{x}(p))^{2}&I\_{x}(p)I\_{y}(p)&I\_{x}(p)I\_{z}(p)\\\\[10pt\]I\_{x}(p)I\_{y}(p)&(I\_{y}(p))^{2}&I\_{y}(p)I\_{z}(p)\\\\[10pt\]I\_{x}(p)I\_{z}(p)&I\_{y}(p)I\_{z}(p)&(I\_{z}(p))^{2}\end{bmatrix}}$ where $I\_{x},I\_{y},I\_{z}$ are the three partial derivatives of $I$, and the integral ranges over $\mathbb {R} ^{3}$.

In the discrete version, $\textstyle S\_{w}\[p\]=\sum \_{r}w\[r\]S\_{0}\[p-r\]$, where $S\_{0}\[p\]={\begin{bmatrix}(I\_{x}\[p\])^{2}&I\_{x}\[p\]I\_{y}\[p\]&I\_{x}\[p\]I\_{z}\[p\]\\\\[10pt\]I\_{x}\[p\]I\_{y}\[p\]&(I\_{y}\[p\])^{2}&I\_{y}\[p\]I\_{z}\[p\]\\\\[10pt\]I\_{x}\[p\]I\_{z}\[p\]&I\_{y}\[p\]I\_{z}\[p\]&(I\_{z}\[p\])^{2}\end{bmatrix}}$ and the sum ranges over a finite set of 3D indices, usually $\\{-m\ldots +m\\}\times \\{-m\ldots +m\\}\times \\{-m\ldots +m\\}$ for some m.

### Interpretation

As in the two-dimensional case, the eigenvalues $\lambda \_{1},\lambda \_{2},\lambda \_{3}$ of $S\_{w}\[p\]$, and the corresponding eigenvectors $\hat {e}\_{1},{\hat {e}\_{2},{\hat {e}\_{3}}}$, summarize the distribution of gradient directions within the neighborhood of *p* defined by the window $w$. This information can be visualized as an ellipsoid whose semi-axes are equal to the eigenvalues and directed along their corresponding eigenvectors.

In particular, if the ellipsoid is stretched along one axis only, like a cigar (that is, if $\lambda \_{1}$ is much larger than both $\lambda \_{2}$ and $\lambda \_{3}$), it means that the gradient in the window is predominantly aligned with the direction $e\_{1}$, so that the isosurfaces of $I$ tend to be flat and perpendicular to that vector. This situation occurs, for instance, when *p* lies on a thin plate-like feature, or on the smooth boundary between two regions with contrasting values.

|  |  |  |
| --- | --- | --- |

If the ellipsoid is flattened in one direction only, like a pancake (that is, if $\lambda \_{3}$ is much smaller than both $\lambda \_{1}$ and $\lambda \_{2}$), it means that the gradient directions are spread out but perpendicular to $e\_{3}$; so that the isosurfaces tend to be like tubes parallel to that vector. This situation occurs, for instance, when *p* lies on a thin line-like feature, or on a sharp corner of the boundary between two regions with contrasting values.

|  |  |  |
| --- | --- | --- |

Finally, if the ellipsoid is roughly spherical (that is, if $\lambda \_{1}\approx \lambda \_{2}\approx \lambda \_{3}$), it means that the gradient directions in the window are more or less evenly distributed, with no marked preference; so that the function $I$ is mostly isotropic in that neighborhood. This happens, for instance, when the function has spherical symmetry in the neighborhood of *p*. In particular, if the ellipsoid degenerates to a point (that is, if the three eigenvalues are zero), it means that $I$ is constant (has zero gradient) within the window.

|  |  |  |
| --- | --- | --- |

## The multi-scale structure tensor

The structure tensor is an important tool in scale space analysis. The **multi-scale structure tensor** (or **multi-scale second moment matrix**) of a function $I$ is in contrast to other one-parameter scale-space features an image descriptor that is defined over *two* scale parameters. One scale parameter, referred to as *local scale* $t$, is needed for determining the amount of pre-smoothing when computing the image gradient $(\nabla I)(x;t)$. Another scale parameter, referred to as *integration scale* $s$, is needed for specifying the spatial extent of the window function $w(\xi ;s)$ that determines the weights for the region in space over which the components of the outer product of the gradient by itself $(\nabla I)(\nabla I)^{\text{T}}$ are accumulated.

More precisely, suppose that $I$ is a real-valued signal defined over $\mathbb {R} ^{k}$. For any local scale $t\>0$, let a multi-scale representation $I(x;t)$ of this signal be given by $I(x;t)=h(x;t)\*I(x)$ where $h(x;t)$ represents a pre-smoothing kernel. Furthermore, let $(\nabla I)(x;t)$ denote the gradient of the scale space representation. Then, the *multi-scale structure tensor/second-moment matrix* is defined by $\mu (x;t,s)=\int \_{\xi \in \mathbb {R} ^{k}(\nabla I)(x-\xi ;t)\\,(\nabla I)^{\text{T}(x-\xi ;t)\\,w(\xi ;s)\\,d\xi }}$ Conceptually, one may ask if it would be sufficient to use any self-similar families of smoothing functions $h(x;t)$ and $w(\xi ;s)$. If one naively would apply, for example, a box filter, however, then non-desirable artifacts could easily occur. If one wants the multi-scale structure tensor to be well-behaved over both increasing local scales $t$ and increasing integration scales $s$, then it can be shown that both the smoothing function and the window function *have to* be Gaussian. The conditions that specify this uniqueness are similar to the scale-space axioms that are used for deriving the uniqueness of the Gaussian kernel for a regular Gaussian scale space of image intensities.

There are different ways of handling the two-parameter scale variations in this family of image descriptors. If we keep the local scale parameter $t$ fixed and apply increasingly broadened versions of the window function by increasing the integration scale parameter $s$ only, then we obtain a *true formal scale space representation of the directional data computed at the given local scale* $t$. If we couple the local scale and integration scale by a *relative integration scale* $r\geq 1$, such that $s=rt$ then for any fixed value of $r$, we obtain a reduced self-similar one-parameter variation, which is frequently used to simplify computational algorithms, for example in corner detection, interest point detection, texture analysis and image matching. By varying the relative integration scale $r\geq 1$ in such a self-similar scale variation, we obtain another alternative way of parameterizing the multi-scale nature of directional data obtained by increasing the integration scale.

A conceptually similar construction can be performed for discrete signals, with the convolution integral replaced by a convolution sum and with the continuous Gaussian kernel $g(x;t)$ replaced by the discrete Gaussian kernel $T(n;t)$: $\mu (x;t,s)=\sum \_{n\in \mathbb {Z} ^{k}(\nabla I)(x-n;t)\\,(\nabla I)^{\text{T}(x-n;t)\\,w(n;s)}}$ When quantizing the scale parameters $t$ and $s$ in an actual implementation, a finite geometric progression $\alpha ^{i}$ is usually used, with *i* ranging from 0 to some maximum scale index *m*. Thus, the discrete scale levels will bear certain similarities to image pyramid, although spatial subsampling may not necessarily be used in order to preserve more accurate data for subsequent processing stages.

## Applications

The eigenvalues of the structure tensor play a significant role in many image processing algorithms, for problems like corner detection, interest point detection, and feature tracking. The structure tensor also plays a central role in the Lucas-Kanade optical flow algorithm, and in its extensions to estimate affine shape adaptation; where the magnitude of $\lambda \_{2}$ is an indicator of the reliability of the computed result. The tensor has been used for scale space analysis, estimation of local surface orientation from monocular or binocular cues, non-linear fingerprint enhancement, diffusion-based image processing, and several other image processing problems. The structure tensor can be also applied in geology to filter seismic data.

### Processing spatio-temporal video data with the structure tensor

The three-dimensional structure tensor has been used to analyze three-dimensional video data (viewed as a function of *x*, *y*, and time *t*). If one in this context aims at image descriptors that are *invariant* under Galilean transformations, to make it possible to compare image measurements that have been obtained under variations of a priori unknown image velocities $v=(v\_{x},v\_{y})^{\text{T}}$ $\begin{bmatrix}x'\\y'\\t'\end{bmatrix}=G{\begin{bmatrix}x\\y\\t\end{bmatrix}={\begin{bmatrix}x-v\_{x}\\,t\\y-v\_{y}\\,t\\t\end{bmatrix},}}$ it is, however, from a computational viewpoint preferable to parameterize the components in the structure tensor/second-moment matrix $S$ using the notion of *Galilean diagonalization* $S'=R\_{\text{space}^{-{\text{T}\\,G^{-{\text{T}\\,S\\,G^{-1}\\,R\_{\text{space}^{-1}={\begin{bmatrix}\nu \_{1}&\\,&\\,\\\\,&\nu \_{2}&\\,\\\\,&\\,&\nu \_{3}\end{bmatrix}}}}}}}}$ where $G$ denotes a Galilean transformation of spacetime and $R\_{\text{space}}$ a two-dimensional rotation over the spatial domain, compared to the abovementioned use of eigenvalues of a 3-D structure tensor, which corresponds to an eigenvalue decomposition and a (non-physical) three-dimensional rotation of spacetime $S''=R\_{\text{spacetime}^{-{\text{T}\\,S\\,R\_{\text{spacetime}^{-1}={\begin{bmatrix}\lambda \_{1}&&\\\&\lambda \_{2}&\\\&&\lambda \_{3}\end{bmatrix}.}}}}}$ To obtain true Galilean invariance, however, also the shape of the spatio-temporal window function needs to be adapted, corresponding to the transfer of affine shape adaptation from spatial to spatio-temporal image data. In combination with local spatio-temporal histogram descriptors, these concepts together allow for Galilean invariant recognition of spatio-temporal events.

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## Resources


title: Stump the Guesser

abstract: Stump the Guesser is a Canadian short film by Guy Maddin, Evan Johnson and Galen Johnson, which was released in 2020. A silent black-and-white film based on early Soviet cinema tropes, it stars Adam Brooks as a man who works as a guesser at the fair, but whose mindreading tricks suddenly begin to fail him; simultaneously, he meets a long-lost sister he never knew he had and falls in love with her, and sets out to disprove the theory of heredity in hopes of being able to marry her. The film premiered on February 25, 2020 at the 70th Berlin International Film Festival. The film was named to the Toronto International Film Festival's year-end Canada's Top Ten list for short films in 2020.

text:
# Stump the Guesser

***Stump the Guesser*** is a Canadian short film by Guy Maddin, Evan Johnson and Galen Johnson, which was released in 2020. A silent black-and-white film based on early Soviet cinema tropes, it stars Adam Brooks as a man who works as a guesser at the fair, but whose mindreading tricks suddenly begin to fail him; simultaneously, he meets a long-lost sister he never knew he had and falls in love with her, and sets out to disprove the theory of heredity in hopes of being able to marry her.

The film premiered on February 25, 2020 at the 70th Berlin International Film Festival.

The film was named to the Toronto International Film Festival's year-end Canada's Top Ten list for short films in 2020.

## InfoBox

| Stump the Guesser | |
| --- | --- |
| Film poster | |
| Directed by | Guy Maddin
Evan Johnson
Galen Johnson |
| Written by | Guy Maddin
Evan Johnson
Galen Johnson |
| Produced by | Juliette Hagopian |
| Starring | Adam Brooks
Brent Neale
Stephanie Berrington |
| Cinematography | Ryan Simmons |
| Edited by | Evan Johnson
Galen Johnson |
| Production
company | Julijette |
| Release date | * February 25, 2020 (2020-02-25) (Berlinale) |
| Running time | 19 minutes |
| Country | Canada |
| Language | English |


title: Strehl ratio

abstract: The Strehl ratio is a measure of the quality of optical image formation, originally proposed by Karl Strehl, after whom the term is named. Used variously in situations where optical resolution is compromised due to lens aberrations or due to imaging through the turbulent atmosphere, the Strehl ratio has a value between 0 and 1, with a hypothetical, perfectly unaberrated optical system having a Strehl ratio of 1.

text:
# Strehl ratio

The **Strehl ratio** is a measure of the quality of optical image formation, originally proposed by Karl Strehl, after whom the term is named. Used variously in situations where optical resolution is compromised due to lens aberrations or due to imaging through the turbulent atmosphere, the Strehl ratio has a value between 0 and 1, with a hypothetical, perfectly unaberrated optical system having a Strehl ratio of 1.

## Mathematical definition

The Strehl ratio $S$ is frequently defined as the ratio of the peak aberrated image intensity from a point source compared to the maximum attainable intensity using an ideal optical system limited only by diffraction over the system's aperture. It is also often expressed in terms not of the peak intensity but the intensity at the image center (intersection of the optical axis with the focal plane) due to an on-axis source; in most important cases these definitions result in a very similar figure (or identical figure, when the point of peak intensity must be exactly at the center due to symmetry). Using the latter definition, the Strehl ratio $S$ can be computed in terms of the wavefront-error $\delta (x,y)$: the offset of the wavefront due to an on-axis point source, compared to that produced by an ideal focusing system over the aperture A(x,y). Using Fraunhofer diffraction theory, one computes the wave amplitude using the Fourier transform of the aberrated pupil function evaluated at 0,0 (center of the image plane) where the phase factors of the Fourier transform formula are reduced to unity. Since the Strehl ratio refers to intensity, it is found from the squared magnitude of that amplitude:

$$S=|\langle e^{i\phi }\rangle |^{2}=|\langle e^{i2\pi \delta /\lambda }\rangle |^{2}$$

where *i* is the imaginary unit, $\phi =2\pi \delta /\lambda $ is the phase error over the aperture at wavelength λ, and the average of the complex quantity inside the brackets is taken over the aperture A(x,y).

The Strehl ratio can be estimated using only the statistics of the phase deviation $\phi $, according to a formula rediscovered by Mahajan but known long before in antenna theory as the Ruze formula

$$S\approx {e^{-\sigma ^{2}}}$$

where sigma (σ) is the root mean square deviation over the aperture of the wavefront phase: $\sigma ^{2}=\langle (\phi -{\bar {\phi })^{2}\rangle }$.

## The Airy disk

Computer-generated image of the Airy diskGraph of the Airy intensity function vs. normalized radius

Due to diffraction, even a focusing system which is perfect according to geometrical optics will have a limited spatial resolution. In the usual case of a uniform circular aperture, the point spread function (PSF) which describes the image formed from an object with no spatial extent (a "point source"), is given by the Airy disk as illustrated here. For a circular aperture, the peak intensity found at the center of the Airy disk defines the point source image intensity required for a Strehl ratio of unity. An imperfect optical system using the same physical aperture will generally produce a broader PSF in which the peak intensity is reduced according to the factor given by the Strehl ratio. An optical system with only minor imperfections in this sense may be referred to as "diffraction limited" as its PSF closely resembles the Airy disk; a Strehl ratio of greater than .8 is frequently cited as a criterion for the use of that designation.

Note that for a given aperture the size of the Airy disk grows linearly with the wavelength $\lambda $, and consequently the peak intensity falls according to $\lambda ^{-2}$ so that the reference point for unity Strehl ratio is changed. Typically, as wavelength is increased, an imperfect optical system will have a broader PSF with a decreased peak intensity. However the peak intensity of the reference Airy disk would have decreased even more at that longer wavelength, resulting in a *better* Strehl ratio at longer wavelengths (typically) even though the actual image resolution is poorer.

## Usage

The ratio is commonly used to assess the quality of astronomical seeing in the presence of atmospheric turbulence and assess the performance of any adaptive optical correction system. It is also used for the selection of short exposure images in the lucky imaging method.

In industry, the Strehl ratio has become a popular way to summarize the performance of an optical design because it gives the performance of a real system, of finite cost and complexity, relative to a theoretically perfect system, which would be infinitely expensive and complex to build and would still have a finite point spread function. It provides a simple method to decide whether a system with a Strehl ratio of, for example, 0.95 is good enough, or whether twice as much should be spent to try to get a Strehl ratio of perhaps 0.97 or 0.98.

## Limitations

Characterizing the form of the point-spread function by a single number, as the Strehl Ratio does, will be meaningful and sensible only if the point-spread function is little distorted from its ideal (aberration-free) form, which will be true for a well-corrected system that operates close to the diffraction limit. That includes most telescopes and microscopes, but excludes most photographic systems, for example. The Strehl ratio has been linked via the work of André Maréchal to an aberration tolerancing theory which is very useful to designers of well-corrected optical systems, allowing a meaningful link between the aberrations of geometrical optics and the diffraction theory of physical optics. A significant shortcoming of the Strehl ratio as a method of image assessment is that, although it is relatively easy to calculate for an optical design prescription on paper, it is normally difficult to measure for a real optical system, not least because the theoretical maximum peak intensity is not readily available.

* Discussion page R.F. Royce' explanation of Strehl ratio in lay terms
* Strehl meter W.M. Keck Observatory Strehl calculator page
* Definition page Eric Weisstein's World of Physics
* Strehl ratio Telescope Optics Net practical explanation of Strehl ratio for amateur telescope makers