[Paper Review] An Efficient Statistical Method for Image Noise Level Estimation

Posted by Euisuk's Dev Log on July 25, 2025

[Paper Review] An Efficient Statistical Method for Image Noise Level Estimation

์›๋ณธ ๊ฒŒ์‹œ๊ธ€: https://velog.io/@euisuk-chung/Paper-Review-An-Efficient-Statistical-Method-for-Image-Noise-Level-Estimation

๋งํฌ : https://www.cv-foundation.org/openaccess/content_iccv_2015/papers/Chen_An_Efficient_Statistical_ICCV_2015_paper.pdf

(์ฐธ๊ณ ) ์ด ๋…ผ๋ฌธ์€ 2015๋…„์— ๋ฐœํ‘œ๋œ ๊ฒƒ์œผ๋กœ, ํ˜„์žฌ ๊ธฐ์ค€์—์„œ๋Š” ์ตœ์‹  ๊ธฐ์ˆ ์ด๋ผ๊ณ  ๋ณด๊ธฐ ์–ด๋ ต์Šต๋‹ˆ๋‹ค. ๋ณธ๋ฌธ์—์„œ ์–ธ๊ธ‰๋œ โ€œ์ตœ๊ทผโ€ ๋˜๋Š” โ€œ์ตœ์‹ โ€์ด๋ผ๋Š” ํ‘œํ˜„์€ ๋ชจ๋‘ ๋‹น์‹œ ๊ธฐ์ค€์ด๋ฉฐ, ํ˜„์žฌ์˜ ๊ธฐ์ˆ  ํ๋ฆ„๊ณผ๋Š” ๋‹ค๋ฅผ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค.


  1. ์„œ๋ก 

์ด ๋…ผ๋ฌธ์€ ๋‹จ์ผ ์˜์ƒ์—์„œ ๊ฐ€๋ฒ•์  ์˜ํ‰๊ท  ๊ฐ€์šฐ์‹œ์•ˆ ๋…ธ์ด์ฆˆ(additive zero-mean Gaussian noise)์˜ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์ถ”์ •ํ•˜๋Š” ๋ฌธ์ œ๋ฅผ ๋‹ค๋ฃน๋‹ˆ๋‹ค. ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์€ ์˜์ƒ ์žก์Œ ์ œ๊ฑฐ(image denoising), ๊ด‘ํ•™ ํ๋ฆ„(optical flow), ์˜์ƒ ๋ถ„ํ• (image segmentation), ์ดˆํ•ด์ƒ๋„(super resolution) ๋“ฑ ๋‹ค์–‘ํ•œ ์ปดํ“จํ„ฐ ๋น„์ „ ์•Œ๊ณ ๋ฆฌ์ฆ˜์— ํ•„์ˆ˜์ ์ธ ํŒŒ๋ผ๋ฏธํ„ฐ์ž…๋‹ˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์‹ค์ œ ์ƒํ™ฉ์—์„œ๋Š” ์˜์ƒ์˜ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์•Œ ์ˆ˜ ์—†๋Š” ๊ฒฝ์šฐ๊ฐ€ ๋งŽ์œผ๋ฉฐ, ํŠนํžˆ ํ…์Šค์ฒ˜๊ฐ€ ํ’๋ถ€ํ•œ ์˜์ƒ์˜ ๊ฒฝ์šฐ ์ •ํ™•ํ•œ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ์ถ”์ •์€ ์—ฌ์ „ํžˆ ์–ด๋ ค์šด ๊ณผ์ œ๋กœ ๋‚จ์•„ ์žˆ์Šต๋‹ˆ๋‹ค.

๊ธฐ์กด์˜ ๋…ธ์ด์ฆˆ ์ถ”์ • ๋ฐฉ๋ฒ•๋“ค([2, 17, 13, 20, 24])์€ ์ฒ˜๋ฆฌํ•  ์˜์ƒ์— ์ถฉ๋ถ„ํ•œ ์–‘์˜ ํ‰ํƒ„ ์˜์—ญ(flat areas)์ด ํฌํ•จ๋˜์–ด ์žˆ๋‹ค๋Š” ๊ฐ€์ •์„ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•ฉ๋‹ˆ๋‹ค. ํ•˜์ง€๋งŒ ์ด๋Š” ์ž์—ฐ ์˜์ƒ ์ฒ˜๋ฆฌ์—์„œ๋Š” ํ•ญ์ƒ ์ ์šฉ๋˜๋Š” ๊ฐ€์ •์ด ์•„๋‹™๋‹ˆ๋‹ค. ์ตœ๊ทผ ์ œ์•ˆ๋œ ์ตœ์‹  ์•Œ๊ณ ๋ฆฌ์ฆ˜๋“ค([19, 23])์€ ์ด๋Ÿฌํ•œ ๊ฐ€์ •์ด ํ•„์š” ์—†๋‹ค๊ณ  ์ฃผ์žฅํ•˜์ง€๋งŒ, ๋‹ค์Œ๊ณผ ๊ฐ™์€ ๋‘ ๊ฐ€์ง€ ์•ฝ์ ์„ ๊ฐ€์ง€๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค.

  1. ์ฒซ์งธ, [19]์—์„œ ์–ธ๊ธ‰ํ–ˆ๋“ฏ์ด, ์ €๋žญํฌ(low-rank) ํŒจ์น˜ ์„ ํƒ์˜ ์ˆ˜๋ ด์„ฑ ๋ฐ ์„ฑ๋Šฅ์ด ์ด๋ก ์ ์œผ๋กœ ๋ณด์žฅ๋˜์ง€ ์•Š์œผ๋ฉฐ ๊ฒฝํ—˜์ ์œผ๋กœ๋„ ๋†’์€ ์ •ํ™•๋„๋ฅผ ๋ณด์ด์ง€ ์•Š์Šต๋‹ˆ๋‹ค.
  2. ๋‘˜์งธ, ๋ณธ ๋…ผ๋ฌธ์˜ 2.2์ ˆ์—์„œ ์ด๋ก ์ ์œผ๋กœ ์„ค๋ช…ํ•˜๋“ฏ์ด, ์ด๋“ค ๋ฐฉ๋ฒ•์€ ์„ ํƒ๋œ ์ €๋žญํฌ ํŒจ์น˜์˜ ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ์˜ ๊ฐ€์žฅ ์ž‘์€ ๊ณ ์œ ๊ฐ’(eigenvalue)์„ ๋…ธ์ด์ฆˆ ์ถ”์ • ๊ฒฐ๊ณผ๋กœ ์‚ฌ์šฉํ•˜๊ธฐ ๋•Œ๋ฌธ์— ์ฒ˜๋ฆฌ๋œ ์˜์ƒ์˜ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ๊ณผ์†Œํ‰๊ฐ€(underestimate)ํ•˜๋Š” ๊ฒฝํ–ฅ์ด ์žˆ์Šต๋‹ˆ๋‹ค.

์ด๋Ÿฌํ•œ ๋ฌธ์ œ๋“ค์„ ํ•ด๊ฒฐํ•˜๊ธฐ ์œ„ํ•ด, ๋ณธ ๋…ผ๋ฌธ์€ ์ƒˆ๋กœ์šด ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ์ถ”์ • ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์•ˆํ•ฉ๋‹ˆ๋‹ค. ์ด ์—ฐ๊ตฌ๋Š” ๋…ธ์ด์ฆˆ๊ฐ€ ์—†๋Š” ์˜์ƒ์—์„œ ์ถ”์ถœ๋œ ํŒจ์น˜๋“ค์ด ์ฃผ๋ณ€ ๊ณต๊ฐ„(ambient space)์— ๊ท ์ผํ•˜๊ฒŒ ๋ถ„ํฌํ•˜๊ธฐ๋ณด๋‹ค ์ข…์ข… ์ €์ฐจ์› ๋ถ€๋ถ„๊ณต๊ฐ„(low-dimensional subspace)์— ์กด์žฌํ•œ๋‹ค๋Š” ๊ด€์ฐฐ์— ๊ธฐ๋ฐ˜ํ•ฉ๋‹ˆ๋‹ค.

์ด๋Ÿฌํ•œ ํŠน์„ฑ์€ ๋ถ€๋ถ„๊ณต๊ฐ„ ํด๋Ÿฌ์Šคํ„ฐ๋ง(subspace clustering) ๋ฐฉ๋ฒ•([8, 29])์—์„œ ๋„๋ฆฌ ํ™œ์šฉ๋˜์—ˆ์Šต๋‹ˆ๋‹ค. ์ด ์ €์ฐจ์› ๋ถ€๋ถ„๊ณต๊ฐ„์€ ์ฃผ์„ฑ๋ถ„ ๋ถ„์„(Principal Component Analysis, PCA)์„ ํ†ตํ•ด ํ•™์Šต๋  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์˜ 2.1์ ˆ์—์„œ ๋ถ„์„ํ•˜๋“ฏ์ด, ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ(noise variance)์€ ์ž‰์—ฌ ์ฐจ์›(redundant dimensions)์˜ ๊ณ ์œ ๊ฐ’์œผ๋กœ๋ถ€ํ„ฐ ์ถ”์ •๋  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. ์ด๋Ÿฌํ•œ ๋ฐฉ์‹์œผ๋กœ, ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ์ถ”์ • ๋ฌธ์ œ๋Š” PCA๋ฅผ ์œ„ํ•œ ์ž‰์—ฌ ์ฐจ์›์„ ์„ ํƒํ•˜๋Š” ๋ฌธ์ œ๋กœ ์žฌ๊ตฌ์„ฑ๋ฉ๋‹ˆ๋‹ค.

(์ฐธ๊ณ ) ์ž‰์—ฌ ์ฐจ์›์˜ ์˜๋ฏธ: ๋…ผ๋ฌธ์—์„œ๋Š” ๊นจ๋—ํ•œ ์ด๋ฏธ์ง€ ํŒจ์น˜๋“ค์ด ์‚ฌ์‹ค์ƒ โ€œ์ €์ฐจ์› ๋ถ€๋ถ„ ๊ณต๊ฐ„(low-dimensional subspace)โ€์— ๋†“์—ฌ ์žˆ๋‹ค๊ณ  ๊ฐ€์ •ํ•ฉ๋‹ˆ๋‹ค. ์ด๋Š” ์ด๋ฏธ์ง€ ํŒจ์น˜์— ํฌํ•จ๋œ ์‹ค์ œ ์ •๋ณด(์˜ˆ: ์ด๋ฏธ์ง€์˜ ๊ตฌ์กฐ, ํ…์Šค์ฒ˜ ๋“ฑ)๋Š” ์ „์ฒด ํŒจ์น˜์˜ ํ”ฝ์…€ ์ˆ˜(๊ณ ์ฐจ์›)๋ณด๋‹ค ํ›จ์”ฌ ์ ์€ ์ˆ˜์˜ ์ฐจ์›์œผ๋กœ๋„ ์ถฉ๋ถ„ํžˆ ์„ค๋ช…๋  ์ˆ˜ ์žˆ๋‹ค๋Š” ๋œป์ž…๋‹ˆ๋‹ค.

์ด ๋ฌธ์ œ๋Š” ํ†ต๊ณ„ํ•™ ๋ฐ ์‹ ํ˜ธ ์ฒ˜๋ฆฌ ๋ถ„์•ผ์—์„œ ๋ชจ๋ธ ์„ ํƒ(model selection) ๋ฌธ์ œ๋กœ ์—ฐ๊ตฌ๋˜์–ด ์™”์Šต๋‹ˆ๋‹ค([10, 15, 21, 28]). ๊ทธ๋Ÿฌ๋‚˜ ์ด๋“ค ๋ฐฉ๋ฒ•์€ ๊ด€์ธก๋œ ์‹ ํ˜ธ๋ฅผ ํ‘œํ˜„ํ•˜๊ธฐ ์œ„ํ•ด ๋” ์ ์€ ์ž ์žฌ ๊ตฌ์„ฑ์š”์†Œ(latent components)๋ฅผ ์‚ฌ์šฉํ•˜๋Š” ๋ฐ ์ค‘์ ์„ ๋‘๊ธฐ ๋•Œ๋ฌธ์—, ์‹ ํ˜ธ ๊ตฌ์„ฑ์š”์†Œ๋ฅผ ๋…ธ์ด์ฆˆ๋กœ ๊ฐ„์ฃผํ•˜์—ฌ ์˜์ƒ ๋…ธ์ด์ฆˆ๋ฅผ ๊ณผ๋Œ€ํ‰๊ฐ€(overestimate)ํ•˜๋Š” ๊ฒฐ๊ณผ๋ฅผ ์ดˆ๋ž˜ํ•ฉ๋‹ˆ๋‹ค.

๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ์ž‰์—ฌ ์ฐจ์›์˜ ๊ณ ์œ ๊ฐ’์ด ๋™์ผํ•œ ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅด๋Š” ํ™•๋ฅ  ๋ณ€์ˆ˜๋ผ๋Š” ํ†ต๊ณ„์  ์†์„ฑ์„ ํ™œ์šฉํ•˜์—ฌ ์ด ๋ฌธ์ œ๋ฅผ ํ•ด๊ฒฐํ•˜๋Š” ํšจ๊ณผ์ ์ธ ๋ฐฉ๋ฒ•์„ ์ œ์•ˆํ•ฉ๋‹ˆ๋‹ค. ์ด๋Š” 2.2์ ˆ์—์„œ ์ž…์ฆ๋˜์—ˆ์œผ๋ฉฐ, ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์€ ์ฃผ์„ฑ๋ถ„(principal dimensions)์˜ ์ˆ˜๊ฐ€ ํŠน์ • ์ž„๊ณ„๊ฐ’๋ณด๋‹ค ์ž‘์„ ๋•Œ ์ •ํ™•ํ•œ ๋…ธ์ด์ฆˆ ์ถ”์ •์„ ๋‹ฌ์„ฑํ•  ๊ฒƒ์œผ๋กœ 2.3์ ˆ์—์„œ ์ž…์ฆ๋ฉ๋‹ˆ๋‹ค.

๋ณธ ๋…ผ๋ฌธ์˜ ์ฃผ์š” ๊ธฐ์—ฌ๋Š” ๋‹ค์Œ๊ณผ ๊ฐ™์ด ์š”์•ฝํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค:

  • ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ฯƒ2\sigma^2ฯƒ2์™€ ํŒจ์น˜์˜ ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ ๊ณ ์œ ๊ฐ’ ์‚ฌ์ด์˜ ํ†ต๊ณ„์  ๊ด€๊ณ„๋ฅผ ์ตœ์ดˆ๋กœ ์ถ”์ •ํ•ฉ๋‹ˆ๋‹ค.
  • ๊ณ ์œ ๊ฐ’์œผ๋กœ๋ถ€ํ„ฐ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ฯƒ2\sigma^2ฯƒ2์„ ์ถ”์ •ํ•˜๋Š” ๋น„๋ชจ์ˆ˜(nonparametric) ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์•ˆํ•˜๋ฉฐ, ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ์„ฑ๋Šฅ์€ ์ด๋ก ์ ์œผ๋กœ ๋ณด์žฅ๋ฉ๋‹ˆ๋‹ค. ๊ฒฝํ—˜์ ์œผ๋กœ ๋ณธ ๋ฐฉ๋ฒ•์€ ๊ฐ€์žฅ ๊ฒฌ๊ณ ํ•˜๋ฉฐ ๋Œ€๋ถ€๋ถ„์˜ ๊ฒฝ์šฐ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ์ถ”์ •์—์„œ ์ตœ๊ณ ์˜ ์„ฑ๋Šฅ์„ ๋‹ฌ์„ฑํ•ฉ๋‹ˆ๋‹ค. ๋˜ํ•œ, ๋ณธ ๋ฐฉ๋ฒ•์€ ๊ฐ€์žฅ ์ ์€ ์‹คํ–‰ ์‹œ๊ฐ„์„ ์†Œ๋ชจํ•˜๋ฉฐ [19, 23]๋ณด๋‹ค ๊ฑฐ์˜ 8๋ฐฐ ๋น ๋ฆ…๋‹ˆ๋‹ค.
  • ๋””๋…ธ์ด์ง• ์•Œ๊ณ ๋ฆฌ์ฆ˜ BM3D๊ฐ€ ๋ณธ ์•Œ๊ณ ๋ฆฌ์ฆ˜์œผ๋กœ ์ถ”์ •๋œ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ์‚ฌ์šฉํ•  ๋•Œ ์ตœ์ ์˜ ์„ฑ๋Šฅ์„ ๋‹ฌ์„ฑํ•จ์„ ์ถ”๊ฐ€๋กœ ์ž…์ฆํ•ฉ๋‹ˆ๋‹ค.
  1. ๊ด€๋ จ ์—ฐ๊ตฌ

๊ธฐ์กด์˜ ๋งŽ์€ ๋…ธ์ด์ฆˆ ์ถ”์ • ๋ฐฉ๋ฒ•๋“ค([2, 17, 13, 20, 24])์€ ์˜์ƒ ๋‚ด์— ์ถฉ๋ถ„ํ•œ ํ‰ํƒ„ ์˜์—ญ์ด ์กด์žฌํ•œ๋‹ค๋Š” ๊ฐ€์ •์— ์˜์กดํ•ฉ๋‹ˆ๋‹ค. ์ด ๊ฐ€์ •์€ ์ž์—ฐ ์˜์ƒ์—์„œ ํ•ญ์ƒ ์œ ํšจํ•˜์ง€ ์•Š์•„ ํ…์Šค์ฒ˜๊ฐ€ ํ’๋ถ€ํ•œ ์˜์ƒ์—์„œ ์ •ํ™•ํ•œ ์ถ”์ •์„ ์–ด๋ ต๊ฒŒ ํ•ฉ๋‹ˆ๋‹ค.

(๊ด€๋ จ์—ฐ๊ตฌ) ๋…ธ์ด์ฆˆ ์ถ”์ • ๋ฐฉ๋ฒ•๋“ค

  • [2]: Segmentation of images based on intensity gradient information

    • ์ €์ž: R. Bracho, A. C. Sanderson
    • ์„ค๋ช…: 1985๋…„์— ๋ฐœํ‘œ๋œ ์ด ๋…ผ๋ฌธ์€ ์ด๋ฏธ์ง€์˜ ๊ฐ•๋„ ๊ธฐ์šธ๊ธฐ ์ •๋ณด(intensity gradient information)๋ฅผ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•œ ์ด๋ฏธ์ง€ ๋ถ„ํ• (Segmentation)์— ๋Œ€ํ•ด ๋‹ค๋ฃน๋‹ˆ๋‹ค.
  • [17]: Refined filtering of image noise using local statistics

    • ์ €์ž: Jong-Sen Lee
    • ์„ค๋ช…: 1981๋…„ ๋…ผ๋ฌธ์œผ๋กœ, ์ง€์—ญ ํ†ต๊ณ„๊ฐ’(local statistics)๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ์ด๋ฏธ์ง€ ๋…ธ์ด์ฆˆ๋ฅผ ์ •์ œํ•˜๋Š” ํ•„ํ„ฐ๋ง ๋ฐฉ๋ฒ•์— ์ดˆ์ ์„ ๋งž์ถฅ๋‹ˆ๋‹ค.
  • [13]: Fast Noise Variance Estimation

    • ์ €์ž: John Immerkรฆr
    • ์„ค๋ช…: 1996๋…„ ๋…ผ๋ฌธ์œผ๋กœ, ๋น ๋ฅด๊ณ  ํšจ์œจ์ ์ธ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ์ถ”์ • ๋ฐฉ๋ฒ•์„ ์ œ์•ˆํ•ฉ๋‹ˆ๋‹ค.
  • [20]: A fast parallel algorithm for blind estimation of noise variance

    • ์ €์ž: P. Meer, J. Jolion, A. Rosenfeld
    • ์„ค๋ช…: 1990๋…„ ๋…ผ๋ฌธ์œผ๋กœ, โ€˜๋ธ”๋ผ์ธ๋“œโ€™ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ์ถ”์ •์„ ์œ„ํ•œ ๋น ๋ฅด๊ณ  ๋ณ‘๋ ฌ์ ์ธ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์•ˆํ•ฉ๋‹ˆ๋‹ค.
  • [24]: Estimation of image noise variance

    • ์ €์ž: K. Rank, M. Lendl, R. Unbehauen
    • ์„ค๋ช…: 1999๋…„ ๋…ผ๋ฌธ์œผ๋กœ, ์ด๋ฏธ์ง€ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ์ถ”์ •์— ๋Œ€ํ•œ ์—ฐ๊ตฌ์ž…๋‹ˆ๋‹ค.

์ตœ๊ทผ์˜ ์ตœ์ฒจ๋‹จ ๋ฐฉ๋ฒ•์ธ [19]์™€ [23]์€ ํ‰ํƒ„ ์˜์—ญ์ด ์—†๋Š” ์˜์ƒ์—์„œ๋„ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์ •ํ™•ํ•˜๊ฒŒ ์ถ”์ •ํ•  ์ˆ˜ ์žˆ๋‹ค๊ณ  ์ฃผ์žฅํ•ฉ๋‹ˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ด๋“ค์€ ์ €๋žญํฌ ํŒจ์น˜์˜ ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ์—์„œ ๊ฐ€์žฅ ์ž‘์€ ๊ณ ์œ ๊ฐ’ ฮปr\lambda_rฮปrโ€‹์„ ๋…ธ์ด์ฆˆ ์ถ”์ • ๊ฒฐ๊ณผ๋กœ ์‚ฌ์šฉํ•ฉ๋‹ˆ๋‹ค.

๋ณธ ๋…ผ๋ฌธ์˜ 2.2์ ˆ์—์„œ ์ด๋ก ์ ์œผ๋กœ ๋ถ„์„ํ–ˆ๋“ฏ์ด, ์ด ์ ‘๊ทผ ๋ฐฉ์‹์€ ์ฒ˜๋ฆฌ๋œ ์˜์ƒ์˜ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์ผ๊ด€๋˜๊ฒŒ ๊ณผ์†Œํ‰๊ฐ€ํ•˜๋Š” ๋ฌธ์ œ๋ฅผ ์•ผ๊ธฐํ•ฉ๋‹ˆ๋‹ค. ์ด๋Š” ์ž‰์—ฌ ์ฐจ์›(redundant space)์˜ ๊ณ ์œ ๊ฐ’์ด ๊ฐ€์šฐ์‹œ์•ˆ ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅธ๋‹ค๋Š” ์ ๊ณผ, ์ •๋ ฌ๋œ ๊ณ ์œ ๊ฐ’์˜ ๊ธฐ๋Œ€๊ฐ’์ด ํŠน์ • ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅธ๋‹ค๋Š” ์ (Theorem 1, Blom์˜ ์ •๋ฆฌ)์„ ํ†ตํ•ด ์„ค๋ช…๋ฉ๋‹ˆ๋‹ค. ํŠนํžˆ, ์ž‰์—ฌ ์ฐจ์›(redundant space)์˜ ์ˆ˜๊ฐ€ 1๋ณด๋‹ค ํด ๋•Œ ๊ฐ€์žฅ ์ž‘์€ ๊ณ ์œ ๊ฐ’์˜ ๊ธฐ๋Œ€๊ฐ’์€ ์‹ค์ œ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ๋ณด๋‹ค ์ž‘์•„์ง‘๋‹ˆ๋‹ค.

(๊ด€๋ จ์—ฐ๊ตฌ) ๋…ธ์ด์ฆˆ ์ถ”์ • ์ตœ์‹  ๋ฐฉ๋ฒ•๋“ค

  • [19]: Single-Image Noise Level Estimation for Blind Denoising

    • ์ €์ž: Xinhao Liu, Masayuki Tanaka, M. Okutomi
    • ์„ค๋ช…: 2013๋…„ ๋ฐœํ‘œ๋œ ์ด ๋…ผ๋ฌธ์€ ๋‹จ์ผ ์ด๋ฏธ์ง€๋กœ๋ถ€ํ„ฐ โ€˜๋ธ”๋ผ์ธ๋“œโ€™ ๋””๋…ธ์ด์ง•์„ ์œ„ํ•œ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์ถ”์ •ํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ๋‹ค๋ฃน๋‹ˆ๋‹ค.
  • [23]: Image Noise Level Estimation by Principal Component Analysis

    • ์ €์ž: Stanislav Pyatykh, J. Hesser, Lei Zheng
    • ์„ค๋ช…: ์—ญ์‹œ 2013๋…„์— ๋ฐœํ‘œ๋œ ์ด ๋…ผ๋ฌธ์€ ์ฃผ์„ฑ๋ถ„ ๋ถ„์„(PCA)์„ ํ™œ์šฉํ•˜์—ฌ ์ด๋ฏธ์ง€ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์ถ”์ •ํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์•ˆํ•ฉ๋‹ˆ๋‹ค.

๋˜ํ•œ, ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ์ถ”์ •์„ ์ฐจ์› ์„ ํƒ ๋ฌธ์ œ๋กœ ์žฌํ•ด์„ํ•˜์—ฌ PCA์˜ ์ž‰์—ฌ ์ฐจ์›(redundant space)์„ ์„ ํƒํ•˜๋Š” ์ ‘๊ทผ ๋ฐฉ์‹์€ ํ†ต๊ณ„ ๋ฐ ์‹ ํ˜ธ ์ฒ˜๋ฆฌ ๋ถ„์•ผ์˜ ๋ชจ๋ธ ์„ ํƒ ๋ฌธ์ œ([10, 15, 21, 28])์™€ ์œ ์‚ฌํ•ฉ๋‹ˆ๋‹ค. ๊ทธ๋Ÿฌ๋‚˜ ์ด๋Ÿฌํ•œ ๊ธฐ์กด ๋ฐฉ๋ฒ•๋“ค์€ ๊ด€์ธก๋œ ์‹ ํ˜ธ๋ฅผ ํ‘œํ˜„ํ•˜๊ธฐ ์œ„ํ•ด ๋” ์ ์€ ์ž ์žฌ ๊ตฌ์„ฑ์š”์†Œ๋ฅผ ์‚ฌ์šฉํ•˜๋Š” ๋ฐ ์ดˆ์ ์„ ๋งž์ถ”๊ธฐ ๋•Œ๋ฌธ์—, ์‹ ํ˜ธ ๊ตฌ์„ฑ์š”์†Œ(signal components)๋ฅผ ๋…ธ์ด์ฆˆ๋กœ ๊ฐ„์ฃผํ•˜์—ฌ ์˜์ƒ ๋…ธ์ด์ฆˆ๋ฅผ ๊ณผ๋Œ€ํ‰๊ฐ€ํ•˜๋Š” ๊ฒฝํ–ฅ์ด ์žˆ์Šต๋‹ˆ๋‹ค.

(๊ด€๋ จ์—ฐ๊ตฌ) ํ†ต๊ณ„ ๋ฐ ์‹ ํ˜ธ ์ฒ˜๋ฆฌ ๋ถ„์•ผ์˜ ๋ชจ๋ธ ์„ ํƒ ๋ฌธ์ œ

  • [10]: The Optimal Hard Threshold for Singular Values is 4/34/\sqrt {3}4/3โ€‹

    • ์ €์ž: M. Gavish, D. L. Donoho
    • ์„ค๋ช…: 2014๋…„ ๋ฐœํ‘œ๋œ ์ด ๋…ผ๋ฌธ์€ ํŠน์ด๊ฐ’(singular values)์— ๋Œ€ํ•œ ์ตœ์ ์˜ ํ•˜๋“œ ์ž„๊ณ„๊ฐ’(hard threshold)์„ ๋‹ค๋ฃน๋‹ˆ๋‹ค.
  • [15]: Non-Parametric Detection of the Number of Signals: Hypothesis Testing and Random Matrix Theory

    • ์ €์ž: S. Kritchman, B. Nadler
    • ์„ค๋ช…: 2009๋…„ ๋…ผ๋ฌธ์œผ๋กœ, ๊ฐ€์„ค ๊ฒ€์ •(Hypothesis Testing)๊ณผ ๋ฌด์ž‘์œ„ ํ–‰๋ ฌ ์ด๋ก (Random Matrix Theory)์„ ์‚ฌ์šฉํ•˜์—ฌ ์‹ ํ˜ธ์˜ ๊ฐœ์ˆ˜๋ฅผ ๋น„๋ชจ์ˆ˜์ ์œผ๋กœ ํƒ์ง€ํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค.
  • [21]: Perfect Dimensionality Recovery by Variational Bayesian PCA

    • ์ €์ž: S. Nakajima, R. Tomioka, M. Sugiyama, S. D. Babacan
    • ์„ค๋ช…: 2012๋…„ ๋…ผ๋ฌธ์œผ๋กœ, Variational Bayesian PCA๋ฅผ ํ†ตํ•ด ์ฐจ์›(dimensionality)์„ ์™„๋ฒฝํ•˜๊ฒŒ ๋ณต๊ตฌํ•˜๋Š” ๋ฐฉ๋ฒ•์„ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค.
  • [28]: Dimension Estimation in Noisy PCA with SURE and Random Matrix Theory

    • ์ €์ž: M. O. Ulfarsson, V. Solo
    • ์„ค๋ช…: 2008๋…„ ๋…ผ๋ฌธ์œผ๋กœ, ๋…ธ์ด์ฆˆ๊ฐ€ ์žˆ๋Š” PCA์—์„œ SURE(Steinโ€™s Unbiased Risk Estimator)์™€ ๋ฌด์ž‘์œ„ ํ–‰๋ ฌ ์ด๋ก ์„ ์‚ฌ์šฉํ•˜์—ฌ ์ฐจ์› ์ถ”์ •์„ ๋‹ค๋ฃน๋‹ˆ๋‹ค.

๋ณธ ๋…ผ๋ฌธ์€ ์ด๋Ÿฌํ•œ ๊ธฐ์กด ๋ฐฉ๋ฒ•๋“ค์˜ ํ•œ๊ณ„๋ฅผ ๊ทน๋ณตํ•˜๊ธฐ ์œ„ํ•ด, ์ž‰์—ฌ ์ฐจ์›(redundant space)์˜ ๊ณ ์œ ๊ฐ’์ด ํŠน์ • ํ†ต๊ณ„์  ๋ถ„ํฌ(๊ฐ€์šฐ์‹œ์•ˆ ๋ถ„ํฌ)๋ฅผ ๋”ฐ๋ฅธ๋‹ค๋Š” ์ ์„ ํ™œ์šฉํ•˜์—ฌ ๋…ธ์ด์ฆˆ๋ฅผ ๋ณด๋‹ค ์ •ํ™•ํ•˜๊ฒŒ ์ถ”์ •ํ•˜๊ณ , ๊ฐ€์žฅ ์ž‘์€ ๊ณ ์œ ๊ฐ’ ๋Œ€์‹  ์ž‰์—ฌ ์ฐจ์› ์ „์ฒด์˜ ํ†ต๊ณ„์  ํŠน์„ฑ์„ ์‚ฌ์šฉํ•˜์—ฌ ๊ณผ์†Œํ‰๊ฐ€ ๋ฌธ์ œ๋ฅผ ํ•ด๊ฒฐํ•ฉ๋‹ˆ๋‹ค.


  1. ๋ฐฉ๋ฒ•๋ก 

๋ณธ ๋…ผ๋ฌธ์˜ ๋ฐฉ๋ฒ•๋ก ์€ ์˜์ƒ ํŒจ์น˜(patch)๋ฅผ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•˜๋ฉฐ, ํŒจ์น˜์˜ ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ์˜ ๊ณ ์œ ๊ฐ’ ๋ถ„์„์„ ํ†ตํ•ด ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์ถ”์ •ํ•ฉ๋‹ˆ๋‹ค.

3.1. ํŒจ์น˜ ๋ถ„ํ•ด ๋ฐ ๋…ธ์ด์ฆˆ ๋ชจ๋ธ๋ง

๊ด€์ธก๋œ ์˜์ƒ III๋Š” sss๊ฐœ์˜ ํŒจ์น˜ ์ง‘ํ•ฉ Xs={xt}t=1sโˆˆRrร—s\mathbf{X}_s = {\mathbf{x}_t}_{t=1}^s \in \mathbb{R}^{r \times s}Xsโ€‹={xtโ€‹}t=1sโ€‹โˆˆRrร—s๋กœ ๋ถ„ํ•ด๋  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค.

๊ด€์ธก๋œ ์˜์ƒ III๋Š” sss๊ฐœ์˜ ํŒจ์น˜ ์ง‘ํ•ฉ Xs={xt}t=1s\mathbf{X}_s = {\mathbf{x}_t}_{t=1}^sXsโ€‹={xtโ€‹}t=1sโ€‹๋กœ ๋‚˜๋ˆŒ ์ˆ˜ ์žˆ๋‹ค.

  • ์ด๋ฏธ์ง€๋ฅผ ์ž‘์€ ์กฐ๊ฐ๋“ค(ํŒจ์น˜) ๋กœ ๋‚˜๋ˆ•๋‹ˆ๋‹ค.
  • ๊ฐ ํŒจ์น˜๋Š” dร—dร—cd \times d \times cdร—dร—c ํฌ๊ธฐ (์˜ˆ: 8ร—8ร—3)์ด๊ณ , ์ด๋ฅผ ๋ฒกํ„ฐ xtโˆˆRrร—1\mathbf{x}_t \in \mathbb{R}^{r \times 1}xtโ€‹โˆˆRrร—1๋กœ ํŽด์„œ ์‚ฌ์šฉํ•ฉ๋‹ˆ๋‹ค.
  • ์ „์ฒด ์ด๋ฏธ์ง€๋Š” ์ด๋ ‡๊ฒŒ ๋ฒกํ„ฐํ™”๋œ ํŒจ์น˜๋“ค์˜ ์ง‘ํ•ฉ XsโˆˆRrร—s\mathbf{X}_s \in \mathbb{R}^{r \times s}Xsโ€‹โˆˆRrร—s๋กœ ํ‘œํ˜„๋ฉ๋‹ˆ๋‹ค.

์ž„์˜์˜ ๊ด€์ธก ํŒจ์น˜xt\mathbf{x}_txtโ€‹๋Š” ๋‹ค์Œ๊ณผ ๊ฐ™์ด ๋ถ„ํ•ด๋ฉ๋‹ˆ๋‹ค:

xt=x^t+et(1)\mathbf{x}_t = \hat{\mathbf{x}}_t + \mathbf{e}_t \quad (1)xtโ€‹=x^tโ€‹+etโ€‹(1)

  • xt\mathbf{x}_txtโ€‹: ์‹ค์ œ ๊ด€์ธก๋œ ํŒจ์น˜ (๋…ธ์ด์ฆˆ ํฌํ•จ)
  • x^t\hat{\mathbf{x}}_tx^tโ€‹: ๋…ธ์ด์ฆˆ๊ฐ€ ์—†๋Š” ์ง„์งœ ํŒจ์น˜
  • et\mathbf{e}_tetโ€‹: ๋…ธ์ด์ฆˆ๋งŒ ๋”ฐ๋กœ ๋ถ„๋ฆฌํ•œ ์„ฑ๋ถ„

๊ฐ€์ •

  • ๋…ธ์ด์ฆˆ et\mathbf{e}_tetโ€‹๋Š” ํ‰๊ท ์ด 0์ด๊ณ , ๋ถ„์‚ฐ์ด ฯƒ2\sigma^2ฯƒ2์ธ ๊ฐ€์šฐ์‹œ์•ˆ ๋…ธ์ด์ฆˆ์ž…๋‹ˆ๋‹ค.
  • ์ˆ˜ํ•™์ ์œผ๋กœ๋Š” ๋‹ค์Œ๊ณผ ๊ฐ™์ด ํ‘œํ˜„๋ฉ๋‹ˆ๋‹ค:

    etโˆผNr(0,ฯƒ2I)\mathbf{e}_t \sim \mathcal{N}_r(0, \sigma^2 \mathbf{I})etโ€‹โˆผNrโ€‹(0,ฯƒ2I)

    • ๋…ธ์ด์ฆˆ๋Š” ๋žœ๋คํ•˜์ง€๋งŒ, ์ „์ฒด์ ์œผ๋กœ ๋ณด๋ฉด ์ •๊ทœ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅด๋Š” ๋ฌด์ž‘์œ„ ์žก์Œ์ž…๋‹ˆ๋‹ค.
    • ๊ฐ ํ”ฝ์…€์€ ๋…๋ฆฝ์ ์œผ๋กœ ๋…ธ์ด์ฆˆ๊ฐ€ ์„ž์—ฌ ์žˆ๊ณ , ๊ทธ ์„ธ๊ธฐ๋Š” ฯƒ2\sigma^2ฯƒ2๋กœ ํ‘œํ˜„๋ฉ๋‹ˆ๋‹ค.

๊ฒฐ๊ตญ, ์˜์ƒ์˜ ๋…ธ์ด์ฆˆ ์ˆ˜์ค€์„ ์ถ”์ •ํ•œ๋‹ค๋Š” ๊ฒƒ์€
ํŒจ์น˜ ์ง‘ํ•ฉ Xs\mathbf{X}_sXsโ€‹ ์—์„œ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ฯƒ2\sigma^2ฯƒ2 ๋ฅผ ์ถ”์ •ํ•˜๋Š” ๊ฒƒ๊ณผ ๊ฐ™์Šต๋‹ˆ๋‹ค.

์ฆ‰, ๋งŽ์€ ํŒจ์น˜๋“ค์„ ๋ถ„์„ํ•ด์„œ ๊ทธ ์•ˆ์— ์„ž์—ฌ ์žˆ๋Š” ๋…ธ์ด์ฆˆ์˜ ํ‰๊ท ์ ์ธ ์„ธ๊ธฐ(๋ถ„์‚ฐ) ๋ฅผ ๊ณ„์‚ฐํ•˜๋Š” ๊ฒƒ์ด ์ด ๋…ผ๋ฌธ์˜ ํ•ต์‹ฌ์ž…๋‹ˆ๋‹ค.


3.2. ๊ณ ์œ ๊ฐ’๊ณผ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์˜ ๊ด€๊ณ„

์ด๋ฏธ์ง€์˜ ํŒจ์น˜๋“ค์ด ์ €์ฐจ์› ์„ ํ˜• ๋ถ€๋ถ„๊ณต๊ฐ„์— ๋†“์—ฌ ์žˆ๋‹ค๊ณ  ๊ฐ€์ •ํ•˜๋ฉด, ๊ฐ ๊ด€์ธก ํŒจ์น˜ xt\mathbf{x}_txtโ€‹๋Š” ๋‹ค์Œ๊ณผ ๊ฐ™์ด ํ‘œํ˜„๋ฉ๋‹ˆ๋‹ค:

xt=Ayt+et\mathbf{x}_t = \mathbf{A}\mathbf{y}_t + \mathbf{e}_txtโ€‹=Aytโ€‹+etโ€‹

  • AโˆˆRrร—m\mathbf{A} \in \mathbb{R}^{r \times m}AโˆˆRrร—m: ๋ถ€๋ถ„๊ณต๊ฐ„์„ ์ •์˜ํ•˜๋Š” ์ง๊ต ๊ธฐ์ € ํ–‰๋ ฌ
  • yt\mathbf{y}_tytโ€‹: ๋ถ€๋ถ„๊ณต๊ฐ„ ์ƒ์˜ ์ขŒํ‘œ
  • et\mathbf{e}_tetโ€‹: ํ‰๊ท  0, ๋ถ„์‚ฐ ฯƒ2\sigma^2ฯƒ2์˜ ๊ฐ€์šฐ์‹œ์•ˆ ๋…ธ์ด์ฆˆ

์ด ๋ชจ๋ธ์€ ๊ด€์ธก๋œ ํŒจ์น˜๊ฐ€ ์‹ ํ˜ธ ์„ฑ๋ถ„๊ณผ ๋…ธ์ด์ฆˆ ์„ฑ๋ถ„์˜ ํ•ฉ์œผ๋กœ ๊ตฌ์„ฑ๋œ๋‹ค๋Š” ๊ฒƒ์„ ์˜๋ฏธํ•ฉ๋‹ˆ๋‹ค.

๐Ÿ”„ ํšŒ์ „ ํ–‰๋ ฌ์„ ํ†ตํ•œ ์„ฑ๋ถ„ ๋ถ„๋ฆฌ

๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ ฮฃx\Sigma_{\mathbf{x}}ฮฃxโ€‹์˜ ๊ณ ์œ ๊ฐ’ ๋ถ„ํ•ด๋ฅผ ํ†ตํ•ด ์–ป์€ ํšŒ์ „ ํ–‰๋ ฌ R=[A,U]\mathbf{R} = [\mathbf{A}, \mathbf{U}]R=[A,U]๋ฅผ ์‚ฌ์šฉํ•˜๋ฉด, ํŒจ์น˜ xt\mathbf{x}_txtโ€‹๋ฅผ ๋‹ค์Œ๊ณผ ๊ฐ™์ด ๋ถ„๋ฆฌํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค:

RTxt=[yt+ATetUTet]\mathbf{R}^\mathsf{T} \mathbf{x}_t = \begin{bmatrix} \mathbf{y}_t + \mathbf{A}^\mathsf{T} \mathbf{e}_t \ \mathbf{U}^\mathsf{T} \mathbf{e}_t \end{bmatrix}RTxtโ€‹=[ytโ€‹+ATetโ€‹UTetโ€‹โ€‹]

  • ์ƒ๋‹จ: ์‹ ํ˜ธ ์„ฑ๋ถ„ + ์ผ๋ถ€ ๋…ธ์ด์ฆˆ
  • ํ•˜๋‹จ: ๋…ธ์ด์ฆˆ๋งŒ ํฌํ•จ๋œ ์„ฑ๋ถ„ โ†’ ์ž‰์—ฌ ์ฐจ์›

๐Ÿ“Š ๊ณ ์œ ๊ฐ’๊ณผ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ

์ด๋ฏธ์ง€ ํŒจ์น˜๋“ค์˜ ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ ฮฃx\Sigma_{\mathbf{x}}ฮฃxโ€‹๋ฅผ ๊ณ ์œ ๊ฐ’ ๋ถ„ํ•ดํ•˜๋ฉด, ํšŒ์ „ ํ–‰๋ ฌ R\mathbf{R}R์„ ํ†ตํ•ด ๋‹ค์Œ๊ณผ ๊ฐ™์ด ๋Œ€๊ฐํ™”๋ฉ๋‹ˆ๋‹ค:

RTฮฃxR=diag(ฮป1,ฮป2,โ€ฆ,ฮปr)\mathbf{R}^\mathsf{T} \Sigma_{\mathbf{x}} \mathbf{R} = \text{diag}(\lambda_1, \lambda_2, \ldots, \lambda_r)RTฮฃxโ€‹R=diag(ฮป1โ€‹,ฮป2โ€‹,โ€ฆ,ฮปrโ€‹)

์—ฌ๊ธฐ์„œ ๊ฐ ๊ณ ์œ ๊ฐ’ ฮปi\lambda_iฮปiโ€‹๋Š” ํ•ด๋‹น ์ฐจ์›์˜ ๋ฐ์ดํ„ฐ ๋ถ„์‚ฐ์„ ๋‚˜ํƒ€๋ƒ…๋‹ˆ๋‹ค.

  • ์‹ ํ˜ธ์™€ ๋…ธ์ด์ฆˆ ์ฐจ์›์˜ ๊ตฌ๋ถ„
    • ์ด๋ฏธ์ง€ ํŒจ์น˜๋“ค์€ ์‹ค์ œ๋กœ๋Š” ์ €์ฐจ์› ์„ ํ˜• ๋ถ€๋ถ„๊ณต๊ฐ„์— ์ง‘์ค‘๋˜์–ด ์žˆ๋‹ค๊ณ  ๊ฐ€์ •ํ•ฉ๋‹ˆ๋‹ค.
    • ์ด๋•Œ:
      • ์‹ ํ˜ธ๊ฐ€ ํฌํ•จ๋œ ์ฐจ์›์—์„œ๋Š” ฮปi>ฯƒ2\lambda_i > \sigma^2ฮปiโ€‹>ฯƒ2
        โ†’ ๊ตฌ์กฐ์  ์ •๋ณด๊ฐ€ ์กด์žฌํ•˜๋ฏ€๋กœ ๋ถ„์‚ฐ์ด ํผ
      • ๋…ธ์ด์ฆˆ๋งŒ ํฌํ•จ๋œ ์ž‰์—ฌ ์ฐจ์›์—์„œ๋Š” ฮปiโ‰ˆฯƒ2\lambda_i \approx \sigma^2ฮปiโ€‹โ‰ˆฯƒ2
        โ†’ ๊ตฌ์กฐ๊ฐ€ ์—†๊ณ , ์˜ค์ง ๋…ธ์ด์ฆˆ๋งŒ ์กด์žฌํ•˜๋ฏ€๋กœ ๋ถ„์‚ฐ์ด ์ž‘๊ณ  ์ผ์ •ํ•จ

์ด๋Ÿฌํ•œ ํŠน์„ฑ์€ ํšŒ์ „ ํ–‰๋ ฌ R\mathbf{R}R์„ ํ†ตํ•ด ์‹ ํ˜ธ์™€ ๋…ธ์ด์ฆˆ ์„ฑ๋ถ„์ด ๋ถ„๋ฆฌ๋˜์—ˆ๊ธฐ ๋•Œ๋ฌธ์— ๊ฐ€๋Šฅํ•ด์ง‘๋‹ˆ๋‹ค.

  • ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ์ถ”์ • ๋ฐฉ๋ฒ•

    • ๋…ธ์ด์ฆˆ๋งŒ ํฌํ•จ๋œ ์ฐจ์›๋“ค์˜ ๊ณ ์œ ๊ฐ’์€ ํ†ต๊ณ„์ ์œผ๋กœ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ฯƒ2\sigma^2ฯƒ2 ๋ฅผ ์ค‘์‹ฌ์œผ๋กœ ํ•œ ์ •๊ทœ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฆ…๋‹ˆ๋‹ค.
    • ๋”ฐ๋ผ์„œ, ์ด๋“ค ๊ณ ์œ ๊ฐ’์˜ ํ‰๊ท ์„ ์ทจํ•˜๋ฉด ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ์ถ”์ •ํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค:

    ฯƒ2โ‰ˆ1rโˆ’mโˆ‘i=m+1rฮปi\sigma^2 \approx \frac{1}{r - m} \sum_{i=m+1}^{r} \lambda_iฯƒ2โ‰ˆrโˆ’m1โ€‹i=m+1โˆ‘rโ€‹ฮปiโ€‹

    • rrr: ์ „์ฒด ์ฐจ์› ์ˆ˜ (ํŒจ์น˜ ๋ฒกํ„ฐ์˜ ๊ธธ์ด)
    • mmm: ์‹ ํ˜ธ๊ฐ€ ํฌํ•จ๋œ ์ฐจ์› ์ˆ˜
    • rโˆ’mr - mrโˆ’m: ๋…ธ์ด์ฆˆ๋งŒ ํฌํ•จ๋œ ์ž‰์—ฌ ์ฐจ์› ์ˆ˜

์ด ์‹์€ ์ž‘์€ ๊ณ ์œ ๊ฐ’๋“ค(๋…ธ์ด์ฆˆ ์ฐจ์›) ์˜ ํ‰๊ท ์ด ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์˜ ํ†ต๊ณ„์  ์ถ”์ •์น˜๊ฐ€ ๋œ๋‹ค๋Š” ์›๋ฆฌ์— ๊ธฐ๋ฐ˜ํ•ฉ๋‹ˆ๋‹ค.


3.3. ์œ ํ•œ ๊ฐ€์šฐ์‹œ์•ˆ ๋ณ€์ˆ˜์˜ ๋ถ„์‚ฐ ๋ฐ ๊ณผ์†Œํ‰๊ฐ€ ๋ฌธ์ œ ๋ถ„์„

๊ธฐ์กด ์—ฐ๊ตฌ [19], [23]์—์„œ๋Š” ์ตœ์†Œ ๊ณ ์œ ๊ฐ’์„ ์ด๋ฏธ์ง€ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ฯƒ2\sigma^2ฯƒ2์˜ ์ถ”์ •์น˜๋กœ ์‚ฌ์šฉํ–ˆ์Šต๋‹ˆ๋‹ค. ํ•˜์ง€๋งŒ ์‹คํ—˜ ๊ฒฐ๊ณผ, ์ด ๋ฐฉ์‹์€ ๋…ธ์ด์ฆˆ ์ˆ˜์ค€์„ ์ง€์†์ ์œผ๋กœ ๊ณผ์†Œํ‰๊ฐ€ํ•˜๋Š” ๊ฒฝํ–ฅ์ด ์žˆ์Œ์„ ๋ณด์—ฌ์ค๋‹ˆ๋‹ค.

์ด๋ฅผ ์„ค๋ช…ํ•˜๊ธฐ ์œ„ํ•ด ๋…ผ๋ฌธ์—์„œ๋Š” Lemma 1๊ณผ Theorem 1์„ ์ œ์‹œํ•ฉ๋‹ˆ๋‹ค.

Lemma 1: ๊ณ ์œ ๊ฐ’์˜ ํ†ต๊ณ„์  ๋ถ„ํฌ

  • ๊ฐ ์ž‰์—ฌ ์ฐจ์›์˜ ๊ณ ์œ ๊ฐ’ ฮปi\lambda_iฮปiโ€‹๋Š” ๊ฐ€์šฐ์‹œ์•ˆ ๋…ธ์ด์ฆˆ N(0,ฯƒ2)\mathcal{N}(0, \sigma^2)N(0,ฯƒ2)์—์„œ ์œ ๋„๋œ ํ™•๋ฅ  ๋ณ€์ˆ˜๋“ค์˜ ์ œ๊ณฑ ํ‰๊ท ์œผ๋กœ ๊ณ„์‚ฐ๋ฉ๋‹ˆ๋‹ค.
  • ์ด ๊ฐ’์€ sss๊ฐœ์˜ ์ƒ˜ํ”Œ์„ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•˜๋ฉฐ, sโ†’โˆžs \to \inftysโ†’โˆž์ผ ๋•Œ ๋‹ค์Œ ๋ถ„ํฌ๋กœ ์ˆ˜๋ ดํ•ฉ๋‹ˆ๋‹ค:

ฯƒ^i2โˆผN(ฯƒ2,2ฯƒ4s)\hat{\sigma}^2_i \sim \mathcal{N}(\sigma^2, \frac{2\sigma^4}{s})ฯƒ^i2โ€‹โˆผN(ฯƒ2,s2ฯƒ4โ€‹)

  • ์ฆ‰, ๊ณ ์œ ๊ฐ’์€ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ์ค‘์‹ฌ์œผ๋กœ ํ•œ ์ •๊ทœ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅด๋ฉฐ, ์ด๋Š” Monte Carlo ์‹œ๋ฎฌ๋ ˆ์ด์…˜์œผ๋กœ๋„ ๊ฒ€์ฆ๋ฉ๋‹ˆ๋‹ค.

Theorem 1: ์ตœ์†Œ ๊ณ ์œ ๊ฐ’์˜ ๊ธฐ๋Œ€๊ฐ’

  • nnn๊ฐœ์˜ ๊ณ ์œ ๊ฐ’์ด ์ •๊ทœ๋ถ„ํฌ N(ฯƒ2,ฮฝ2)\mathcal{N}(\sigma^2, \nu^2)N(ฯƒ2,ฮฝ2)๋ฅผ ๋”ฐ๋ฅธ๋‹ค๊ณ  ํ•  ๋•Œ, ์ตœ์†Œ ๊ณ ์œ ๊ฐ’์˜ ๊ธฐ๋Œ€๊ฐ’์€ ๋‹ค์Œ๊ณผ ๊ฐ™์ด ๊ทผ์‚ฌ๋ฉ๋‹ˆ๋‹ค:

E(ฮปmin)โ‰ˆฯƒ2+ฮฝโ‹…ฮฆโˆ’1(1โˆ’ฮฑnโˆ’2ฮฑ+1)E(\lambda_{\text{min}}) \approx \sigma^2 + \nu \cdot \Phi^{-1}\left(\frac{1 - \alpha}{n - 2\alpha + 1}\right)E(ฮปminโ€‹)โ‰ˆฯƒ2+ฮฝโ‹…ฮฆโˆ’1(nโˆ’2ฮฑ+11โˆ’ฮฑโ€‹)

  • ์—ฌ๊ธฐ์„œ ฮฆโˆ’1\Phi^{-1}ฮฆโˆ’1๋Š” ํ‘œ์ค€ ์ •๊ทœ๋ถ„ํฌ์˜ ์—ญ๋ˆ„์ ํ•จ์ˆ˜, ฮฑ=0.375\alpha = 0.375ฮฑ=0.375, ฮฝ=2ฯƒ4s\nu = \sqrt{\frac{2\sigma^4}{s}}ฮฝ=s2ฯƒ4โ€‹โ€‹์ž…๋‹ˆ๋‹ค.
  • ์ด ์‹์— ๋”ฐ๋ฅด๋ฉด, ์ตœ์†Œ ๊ณ ์œ ๊ฐ’์˜ ๊ธฐ๋Œ€๊ฐ’์€ ํ•ญ์ƒ ฯƒ2\sigma^2ฯƒ2๋ณด๋‹ค ์ž‘๊ฒŒ ๋‚˜ํƒ€๋‚˜๋ฉฐ, ์ž‰์—ฌ ์ฐจ์›์˜ ์ˆ˜๊ฐ€ ๋งŽ์„์ˆ˜๋ก ๊ณผ์†Œํ‰๊ฐ€ ์ •๋„๊ฐ€ ์ปค์ง‘๋‹ˆ๋‹ค.

โš ๏ธ ๊ฒฐ๋ก  ๋ฐ ํ•œ๊ณ„

  • ๊ธฐ์กด ๋ฐฉ๋ฒ•([19], [23])์€ ์ตœ์†Œ ๊ณ ์œ ๊ฐ’์„ ๋…ธ์ด์ฆˆ ์ถ”์ •์น˜๋กœ ์‚ฌ์šฉํ–ˆ์ง€๋งŒ, ์ด๋Š” ํ†ต๊ณ„์ ์œผ๋กœ ๊ณผ์†Œํ‰๊ฐ€๋  ์ˆ˜๋ฐ–์— ์—†๋Š” ๊ตฌ์กฐ์ž…๋‹ˆ๋‹ค.
  • ํŠนํžˆ, ๊ณ ์ฐจ์› ์ด๋ฏธ์ง€๋‚˜ ํŒจ์น˜ ์ˆ˜๊ฐ€ ๋งŽ์„์ˆ˜๋ก ์ด ์˜ค์ฐจ๋Š” ๋” ์ปค์ง€๋ฉฐ, ์ •ํ™•ํ•œ ๋…ธ์ด์ฆˆ ์ถ”์ •์„ ์œ„ํ•ด์„œ๋Š” ๊ณ ์œ ๊ฐ’ ์ „์ฒด ๋ถ„ํฌ๋ฅผ ๊ณ ๋ คํ•œ ํ‰๊ท  ๊ธฐ๋ฐ˜ ์ ‘๊ทผ์ด ํ•„์š”ํ•ฉ๋‹ˆ๋‹ค.

3.4. ์ฐจ์› ์„ ํƒ ์•Œ๊ณ ๋ฆฌ์ฆ˜

์•ž์„  ์„น์…˜์—์„œ ์„ค๋ช…ํ•œ ๋ฐ”์™€ ๊ฐ™์ด, ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ์ •ํ™•ํ•˜๊ฒŒ ์ถ”์ •ํ•˜๊ธฐ ์œ„ํ•ด์„œ๋Š” ๋…ธ์ด์ฆˆ๋งŒ ํฌํ•จ๋œ ๊ณ ์œ ๊ฐ’ ์ง‘ํ•ฉ S2\mathbf{S}_2S2โ€‹ ์ „์ฒด๋ฅผ ๊ณ ๋ คํ•ด์•ผ ํ•ฉ๋‹ˆ๋‹ค. ํ•˜์ง€๋งŒ ์‹ค์ œ๋กœ๋Š” ์‹ ํ˜ธ๊ฐ€ ํฌํ•จ๋œ ์ฐจ์› ์ˆ˜ mmm ๋ฅผ ์‚ฌ์ „์— ์•Œ ์ˆ˜ ์—†๊ธฐ ๋•Œ๋ฌธ์—, ์ด๋ฅผ ์ž๋™์œผ๋กœ ์ถ”์ •ํ•˜๋Š” ์•Œ๊ณ ๋ฆฌ์ฆ˜์ด ํ•„์š”ํ•ฉ๋‹ˆ๋‹ค.

๋…ผ๋ฌธ์—์„œ๋Š” ์ด๋ฅผ ํ•ด๊ฒฐํ•˜๊ธฐ ์œ„ํ•ด ์ƒ์œ„ ์ด์ƒ์น˜(upper outliers) ๋ฅผ ๊ธฐ๋ฐ˜์œผ๋กœ ํ•œ ์ฐจ์› ์„ ํƒ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ์ œ์•ˆํ•ฉ๋‹ˆ๋‹ค.

๐Ÿง  ํ•ต์‹ฌ ์•„์ด๋””์–ด

  • ๊ณ ์œ ๊ฐ’ ์ง‘ํ•ฉ S={ฮป1,ฮป2,โ€ฆ,ฮปr}\mathbf{S} = {\lambda_1, \lambda_2, \ldots, \lambda_r}S={ฮป1โ€‹,ฮป2โ€‹,โ€ฆ,ฮปrโ€‹}์—์„œ
    • ์‹ ํ˜ธ๊ฐ€ ํฌํ•จ๋œ ์ฐจ์›์˜ ๊ณ ์œ ๊ฐ’์€ ํ‰๊ท ๋ณด๋‹ค ํฌ๊ณ 
    • ๋…ธ์ด์ฆˆ๋งŒ ํฌํ•จ๋œ ์ฐจ์›์˜ ๊ณ ์œ ๊ฐ’์€ ํ‰๊ท ๊ณผ ์ค‘์•™๊ฐ’์ด ๊ฑฐ์˜ ๊ฐ™์Œ
  • ๋”ฐ๋ผ์„œ, ๊ณ ์œ ๊ฐ’ ์ง‘ํ•ฉ์—์„œ ํ‰๊ท ์ด ์ค‘์•™๊ฐ’๋ณด๋‹ค ํฐ ๊ฒฝ์šฐ, ์ƒ์œ„ ์ด์ƒ์น˜๊ฐ€ ์กด์žฌํ•œ๋‹ค๊ณ  ํŒ๋‹จํ•˜๊ณ 
    • ๊ฐ€์žฅ ํฐ ๊ณ ์œ ๊ฐ’์„ ์ œ๊ฑฐํ•˜๋ฉฐ ๋ฐ˜๋ณต
  • ํ‰๊ท ์ด ์ค‘์•™๊ฐ’๊ณผ ๊ฐ™์•„์ง€๋Š” ์ˆœ๊ฐ„, ์‹ ํ˜ธ ์ฐจ์› ์ œ๊ฑฐ ์™„๋ฃŒ โ†’ ์ž‰์—ฌ ์ฐจ์›๋งŒ ๋‚จ์Œ

๐Ÿ“ Theorem 2: ํ‰๊ท ๊ณผ ์ค‘์•™๊ฐ’์˜ ๊ด€๊ณ„

  • m>0m > 0m>0์ผ ๋•Œ: ๊ณ ์œ ๊ฐ’ ์ง‘ํ•ฉ์˜ ํ‰๊ท ์€ ์ค‘์•™๊ฐ’๋ณด๋‹ค ํผ
  • m=0m = 0m=0 ๋˜๋Š” ์ด์ƒ์น˜๊ฐ€ ์—†์„ ๋•Œ: ํ‰๊ท ๊ณผ ์ค‘์•™๊ฐ’์ด ๊ฐ™์Œ
  • ์ด ๊ด€๊ณ„๋ฅผ ํ†ตํ•ด ์‹ ํ˜ธ ์ฐจ์› ์ˆ˜ mmm ๋ฅผ ์ถ”์ •ํ•  ์ˆ˜ ์žˆ์Œ

โš™๏ธ Algorithm 1: ์˜์ƒ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ์ถ”์ •

  1. ์ž…๋ ฅ ์˜์ƒ III์—์„œ sss๊ฐœ์˜ ํŒจ์น˜๋ฅผ ์ถ”์ถœ
  2. ํ‰๊ท  ๋ฒกํ„ฐ ฮผ\muฮผ ๊ณ„์‚ฐ
  3. ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ ฮฃ\Sigmaฮฃ ๊ณ„์‚ฐ
  4. ๊ณ ์œ ๊ฐ’ ฮป1โ‰ฅฮป2โ‰ฅโ‹ฏโ‰ฅฮปr\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_rฮป1โ€‹โ‰ฅฮป2โ€‹โ‰ฅโ‹ฏโ‰ฅฮปrโ€‹ ์ •๋ ฌ
  5. ๋ฐ˜๋ณต๋ฌธ์„ ํ†ตํ•ด ๊ณ ์œ ๊ฐ’ ์ง‘ํ•ฉ์˜ ํ‰๊ท ๊ณผ ์ค‘์•™๊ฐ’ ๋น„๊ต
  6. ํ‰๊ท  = ์ค‘์•™๊ฐ’์ด ๋˜๋Š” ์ˆœ๊ฐ„์˜ ํ‰๊ท ๊ฐ’์„ ฯ„\tauฯ„๋กœ ์„ค์ •
  7. ๋…ธ์ด์ฆˆ ํ‘œ์ค€ํŽธ์ฐจ ฯƒ=ฯ„\sigma = \sqrt{\tau}ฯƒ=ฯ„โ€‹ ๋ฐ˜ํ™˜

3.5. ๊ตฌํ˜„ ๋ฐ ์‹œ๊ฐ„ ๋ณต์žก๋„

๋ณธ ๋ฐฉ๋ฒ•์˜ ํ•ต์‹ฌ ์•„์ด๋””์–ด๋Š” ์ฃผ์„ฑ๋ถ„ ์ฐจ์› S1\mathbf{S}_1S1โ€‹์˜ ๊ณ ์œ ๊ฐ’์„ S\mathbf{S}S์—์„œ ์ œ๊ฑฐํ•จ์œผ๋กœ์จ ์ •ํ™•ํ•œ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ์–ป๋Š” ๊ฒƒ์ž…๋‹ˆ๋‹ค. ์œ„ Algorithm 1์— ์„ค๋ช…๋œ ๋‹จ๊ณ„์— ๋”ฐ๋ผ ์‹œ๊ฐ„ ๋ณต์žก๋„๋ฅผ ๋ถ„์„ํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค.

  • sss๊ฐœ์˜ rrr-์ฐจ์› ์ƒ˜ํ”Œ์„ ๊ฐ–๋Š” X\mathbf{X}X์ง‘ํ•ฉ์„ ์ƒ์„ฑํ•˜๋Š” ๋ณต์žก๋„๋Š” O(sr)O(sr)O(sr)์ž…๋‹ˆ๋‹ค.
  • ๋ฐ์ดํ„ฐ์…‹ X\mathbf{X}X์˜ ํ‰๊ท  ๋ฒกํ„ฐ ฮผ\mathbf{\mu}ฮผ ๋ฐ ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ ฮฃ\mathbf{\Sigma}ฮฃ๋ฅผ ๊ณ„์‚ฐํ•˜๋Š” ๊ฒƒ์€ ๊ฐ๊ฐ O(sr)O(sr)O(sr) ๋ฐ O(sr2)O(sr^2)O(sr2)์ž…๋‹ˆ๋‹ค.
  • ฮฃ\mathbf{\Sigma}ฮฃ์˜ ๊ณ ์œ  ๋ถ„ํ•ด๋Š” O(r3)O(r^3)O(r3)์ž…๋‹ˆ๋‹ค.
  • 4๋‹จ๊ณ„์˜ ์ •๋ ฌ ๊ณผ์ •์€ O(rlogโกr)O(r \log r)O(rlogr)(ํ˜น์€ O(r2)O(r^2)O(r2)if using selection sort as implied by loop)๋ฅผ ์†Œ๋น„ํ•˜๋ฉฐ, 5-9๋‹จ๊ณ„์˜ ํ™•์ธ ์ ˆ์ฐจ๋Š” ์ตœ์•…์˜ ๊ฒฝ์šฐ rrr๋ฒˆ ๋ฐ˜๋ณตํ•˜๋ฉฐ ๊ฐ ๋ฐ˜๋ณต์—์„œ ํ‰๊ท ๊ณผ ์ค‘์•™๊ฐ’์„ ๊ณ„์‚ฐํ•˜๋ฏ€๋กœ O(rโ‹…r)=O(r2)O(r \cdot r) = O(r^2)O(rโ‹…r)=O(r2)๊ฐ€ ๊ฑธ๋ฆฝ๋‹ˆ๋‹ค.

  • Algorithm 1์˜ r2r^2r2๋Š” ๊ณ ์œ ๊ฐ’ ๊ฐœ์ˆ˜ rrr์— ๋Œ€ํ•ด rrr๋ฒˆ ๋ฐ˜๋ณตํ•˜๋ฉฐ ๋งค ๋ฐ˜๋ณต๋งˆ๋‹ค ๋ถ€๋ถ„ ์ง‘ํ•ฉ์˜ ํ‰๊ท ์„ ๊ณ„์‚ฐํ•˜๊ณ , ์ค‘์•™๊ฐ’์„ ์ฐพ๋Š” ๋ฐ O(r)O(r)O(r)๊ฐ€ ๊ฑธ๋ฆฌ๊ธฐ ๋•Œ๋ฌธ์ž…๋‹ˆ๋‹ค.

๋”ฐ๋ผ์„œ, ๋ณธ ๋ฐฉ๋ฒ•์˜ ์ด ์‹œ๊ฐ„ ๋ณต์žก๋„๋Š” O(sr2+r3)O(sr^2 + r^3)O(sr2+r3)์ด๋ฉฐ, ์ด๋Š” ๋‹คํ•ญ ์‹œ๊ฐ„(polynomial time)์— ํ•ด๊ฒฐ๋  ์ˆ˜ ์žˆ์Œ์„ ์˜๋ฏธํ•ฉ๋‹ˆ๋‹ค.


  1. ์‹คํ—˜

์ œ์•ˆ๋œ ๋…ธ์ด์ฆˆ ์ถ”์ • ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ์„ฑ๋Šฅ์„ ํ‰๊ฐ€ํ•˜๊ธฐ ์œ„ํ•ด ํ•ฉ์„ฑ ๋…ธ์ด์ฆˆ ์˜์ƒ, ์‹ค์ œ ๋…ธ์ด์ฆˆ ์˜์ƒ, ๊ทธ๋ฆฌ๊ณ  ์˜์ƒ ๋ณต์› ์‘์šฉ(BM3D) ์—์„œ์˜ ํšจ๊ณผ๋ฅผ ์‹คํ—˜์ ์œผ๋กœ ๊ฒ€์ฆํ–ˆ์Šต๋‹ˆ๋‹ค. ๋น„๊ต ๋Œ€์ƒ์€ ๊ธฐ์กด์˜ ๋Œ€ํ‘œ์ ์ธ ๋…ธ์ด์ฆˆ ์ถ”์ • ๋ฐฉ๋ฒ• [19], [23]์ด๋ฉฐ, ๋ชจ๋“  ์‹คํ—˜์€ ๋™์ผํ•œ ํ™˜๊ฒฝ(MATLAB R2013a, Intel i7 CPU)์—์„œ ์ˆ˜ํ–‰๋˜์—ˆ์Šต๋‹ˆ๋‹ค.


4.1 ํŒŒ๋ผ๋ฏธํ„ฐ ์„ค์ •

  • ์•Œ๊ณ ๋ฆฌ์ฆ˜์—์„œ ์œ ์ผํ•œ ์‚ฌ์ „ ์„ค์ • ํŒŒ๋ผ๋ฏธํ„ฐ๋Š” ํŒจ์น˜ ํฌ๊ธฐ ( d ) ์ž…๋‹ˆ๋‹ค.
  • ํŒจ์น˜ ํฌ๊ธฐ๊ฐ€ ํด์ˆ˜๋ก ํ…์Šค์ฒ˜ ํ‘œํ˜„์— ์œ ๋ฆฌํ•˜์ง€๋งŒ, ์ƒ˜ํ”Œ ์ˆ˜ ( s )๊ฐ€ ์ค„์–ด๋“ค์–ด ํ†ต๊ณ„์  ์‹ ๋ขฐ๋„๊ฐ€ ๋‚ฎ์•„์ง€๊ณ  ์‹คํ–‰ ์‹œ๊ฐ„์ด ์ฆ๊ฐ€ํ•ฉ๋‹ˆ๋‹ค.
  • ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ํŒจ์น˜ ํฌ๊ธฐ ( d = 8 ) ์„ ๊ธฐ๋ณธ๊ฐ’์œผ๋กœ ์‚ฌ์šฉํ•˜์—ฌ ์„ฑ๋Šฅ๊ณผ ํšจ์œจ์„ฑ์˜ ๊ท ํ˜•์„ ๋งž์ท„์Šต๋‹ˆ๋‹ค.

4.2 ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ ์ถ”์ • ๊ฒฐ๊ณผ

4.2.1 TID2008 ๋ฐ์ดํ„ฐ์…‹

  • ํ•ฉ์„ฑ๋œ ๋ฐฑ์ƒ‰ ๋…ธ์ด์ฆˆ๋ฅผ ์ถ”๊ฐ€ํ•œ ์˜์ƒ์— ๋Œ€ํ•ด ๋…ธ์ด์ฆˆ ์ˆ˜์ค€์„ ์ถ”์ •
  • ํ‰๊ฐ€ ์ง€ํ‘œ:
    • Bias: ์ถ”์ •์น˜์˜ ์ •ํ™•๋„
    • Std: ์ถ”์ •์น˜์˜ ๊ฒฌ๊ณ ์„ฑ
    • โˆšMSE: ์ „์ฒด ์„ฑ๋Šฅ

  • ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์€ ๋Œ€๋ถ€๋ถ„์˜ ๊ฒฝ์šฐ ๊ธฐ์กด ๋ฐฉ๋ฒ•๋ณด๋‹ค ๋” ์ •ํ™•ํ•˜๊ณ  ๊ฒฌ๊ณ ํ•œ ์ถ”์ •์„ ๋ณด์˜€์œผ๋ฉฐ, ์‹คํ–‰ ์‹œ๊ฐ„๋„ ์•ฝ 8๋ฐฐ ๋น ๋ฆ„ (0.5์ดˆ vs 4์ดˆ)

4.2.2 BSDS500 ๋ฐ์ดํ„ฐ์…‹

  • ํ…์Šค์ฒ˜๊ฐ€ ํ’๋ถ€ํ•œ ์˜์ƒ์—์„œ์˜ ์„ฑ๋Šฅ์„ ํ‰๊ฐ€
  • ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์€ ๋ชจ๋“  ์ง€ํ‘œ์—์„œ ๊ธฐ์กด ๋ฐฉ๋ฒ•๋ณด๋‹ค ์šฐ์ˆ˜ํ•œ ์„ฑ๋Šฅ์„ ๋ณด์˜€์œผ๋ฉฐ, Gavish์˜ ๋ฐฉ๋ฒ•๋ณด๋‹ค๋„ ๋” ๋‚ฎ์€ MSE๋ฅผ ๊ธฐ๋กํ•จ

4.2.3 ์‹ค์ œ ๋…ธ์ด์ฆˆ ์˜์ƒ

  • Nikon D5200์œผ๋กœ ์ €์กฐ๋„ ํ™˜๊ฒฝ์—์„œ ์ดฌ์˜ํ•œ 100๊ฐœ์˜ ์ •์  ์žฅ๋ฉด ์˜์ƒ ์‚ฌ์šฉ
  • ํ‰๊ท  ์˜์ƒ์„ ๊ธฐ์ค€์œผ๋กœ ์ง€์ƒ ์ง„์‹ค ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ๊ณ„์‚ฐ
  • ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์€ ์‹ค์ œ ๋…ธ์ด์ฆˆ ํ™˜๊ฒฝ์—์„œ๋„ ๊ฐ€์žฅ ์ •ํ™•ํ•œ ์ถ”์ • ์„ฑ๋Šฅ์„ ๋ณด์ž„

4.3 ์˜์ƒ ์žก์Œ ์ œ๊ฑฐ ์‘์šฉ

  • ์ œ์•ˆ๋œ ๋…ธ์ด์ฆˆ ์ถ”์ •์น˜๋ฅผ BM3D ์•Œ๊ณ ๋ฆฌ์ฆ˜์˜ ์ž…๋ ฅ ํŒŒ๋ผ๋ฏธํ„ฐ๋กœ ์‚ฌ์šฉ
  • TID2008 ๋ฐ BSDS500 ์˜์ƒ์—์„œ BM3D ์„ฑ๋Šฅ์„ ํ‰๊ฐ€
  • ๊ฒฐ๊ณผ์ ์œผ๋กœ, ์ถ”์ •๋œ ๋…ธ์ด์ฆˆ ์ˆ˜์ค€์„ ์‚ฌ์šฉํ•œ BM3D์˜ ์„ฑ๋Šฅ์€ ์‹ค์ œ ๋…ธ์ด์ฆˆ ์ˆ˜์ค€์„ ์‚ฌ์šฉํ•œ ๊ฒฝ์šฐ์™€ ๊ฑฐ์˜ ๋™์ผ
  • ์ด๋Š” ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์ด ์‹ค์ œ ๋””๋…ธ์ด์ง• ์‘์šฉ์—๋„ ๋งค์šฐ ์ ํ•ฉํ•จ์„ ๋ณด์—ฌ์คŒ

  1. ํ† ๋ก 

์ด ์—ฐ๊ตฌ๋Š” ๋‹จ์ผ ๋…ธ์ด์ฆˆ ์˜์ƒ์—์„œ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ํšจ์œจ์ ์œผ๋กœ ์ถ”์ •ํ•˜๋Š” ํ†ต๊ณ„์  ๋ฐฉ๋ฒ•์„ ์ œ์•ˆํ–ˆ์Šต๋‹ˆ๋‹ค. ํ•ต์‹ฌ์ ์ธ ๊ธฐ์—ฌ๋Š” ์˜์ƒ ํŒจ์น˜์˜ ๊ณต๋ถ„์‚ฐ ํ–‰๋ ฌ์˜ ๊ณ ์œ ๊ฐ’๊ณผ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ ์‚ฌ์ด์˜ ํ†ต๊ณ„์  ๊ด€๊ณ„๋ฅผ ์‹ฌ์ธต์ ์œผ๋กœ ๋ถ„์„ํ•œ ๊ฒƒ์ž…๋‹ˆ๋‹ค. ํŠนํžˆ, ์ด์ „์˜ ์ตœ์‹  ๋ฐฉ๋ฒ•๋“ค์ด ๊ฐ€์žฅ ์ž‘์€ ๊ณ ์œ ๊ฐ’์„ ๋…ธ์ด์ฆˆ ์ถ”์ •์น˜๋กœ ์‚ฌ์šฉํ•˜์—ฌ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ๊ณผ์†Œํ‰๊ฐ€ํ•˜๋Š” ๋ฌธ์ œ๋ฅผ ์ด๋ก ์ ์œผ๋กœ ๊ทœ๋ช…ํ•˜๊ณ , ์ด๋ฅผ Lemma 1๊ณผ Theorem 1์„ ํ†ตํ•ด ์ฆ๋ช…ํ–ˆ์Šต๋‹ˆ๋‹ค.

์ œ์•ˆ๋œ ๋น„๋ชจ์ˆ˜ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ ์ด๋Ÿฌํ•œ ๊ณผ์†Œํ‰๊ฐ€ ๋ฌธ์ œ๋ฅผ ํ•ด๊ฒฐํ•˜๊ธฐ ์œ„ํ•ด, ์ž‰์—ฌ ์ฐจ์› ๊ณ ์œ ๊ฐ’์˜ ํ†ต๊ณ„์  ํŠน์„ฑ(๊ฐ€์šฐ์‹œ์•ˆ ๋ถ„ํฌ)์„ ํ™œ์šฉํ•˜์—ฌ ์ด์ƒ์น˜๋ฅผ ์ œ๊ฑฐํ•˜๊ณ  ์‹ค์ œ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ์ถ”์ •ํ•ฉ๋‹ˆ๋‹ค. Algorithm 1์˜ ์ฐจ์› ์„ ํƒ ์ ˆ์ฐจ๋Š” ๊ณ ์œ ๊ฐ’ ์ง‘ํ•ฉ์˜ ํ‰๊ท ๊ณผ ์ค‘์•™๊ฐ’์„ ๋น„๊ตํ•˜์—ฌ ์ด์ƒ์น˜ ์กด์žฌ ์—ฌ๋ถ€๋ฅผ ํŒ๋‹จํ•˜๋ฉฐ, ์ด๋Š” Theorem 2์— ์˜ํ•ด ์ด๋ก ์ ์œผ๋กœ ์ •๋‹นํ™”๋ฉ๋‹ˆ๋‹ค.

์‹คํ—˜ ๊ฒฐ๊ณผ๋Š” ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์ด ๊ธฐ์กด์˜ ์ตœ์‹  ์•Œ๊ณ ๋ฆฌ์ฆ˜([19, 23])๋ณด๋‹ค ํ›จ์”ฌ ์šฐ์ˆ˜ํ•œ ์„ฑ๋Šฅ์„ ๋ณด์ž„์„ ๋ช…ํ™•ํžˆ ๋ณด์—ฌ์ค๋‹ˆ๋‹ค. TID2008๊ณผ BSDS500 ๋ฐ์ดํ„ฐ์…‹ ๋ชจ๋‘์—์„œ ๋ณธ ๋ฐฉ๋ฒ•์€ ์ •ํ™•๋„, ๊ฒฌ๊ณ ์„ฑ, ๊ทธ๋ฆฌ๊ณ  ์†๋„ ๋ฉด์—์„œ ๋›ฐ์–ด๋‚ฌ์Šต๋‹ˆ๋‹ค. ํŠนํžˆ, ํ…์Šค์ฒ˜๊ฐ€ ํ’๋ถ€ํ•˜์—ฌ ๋…ธ์ด์ฆˆ ์ถ”์ •์ด ์–ด๋ ค์šด BSDS500 ๋ฐ์ดํ„ฐ์…‹๊ณผ ์‹ค์ œ ํ™˜๊ฒฝ์—์„œ ์ดฌ์˜๋œ ๋…ธ์ด์ฆˆ ์˜์ƒ์—์„œ ๊ทธ ์šฐ์ˆ˜์„ฑ์ด ๋”์šฑ ๋‘๋“œ๋Ÿฌ์กŒ์Šต๋‹ˆ๋‹ค. ์‹คํ–‰ ์‹œ๊ฐ„ ์ธก๋ฉด์—์„œ, ๋ณธ ๋ฐฉ๋ฒ•์€ ๊ธฐ์กด ๋ฐฉ๋ฒ•๋“ค๋ณด๋‹ค ์•ฝ 8๋ฐฐ ๋นจ๋ผ ์‹ค์šฉ์ ์ธ ์‘์šฉ์— ๋งค์šฐ ์ ํ•ฉํ•ฉ๋‹ˆ๋‹ค.

BM3D์™€ ๊ฐ™์€ ๋ฐ๋…ธ์ด์ง• ์•Œ๊ณ ๋ฆฌ์ฆ˜์— ๋ณธ ๋ฐฉ๋ฒ•์œผ๋กœ ์ถ”์ •๋œ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์ž…๋ ฅ์œผ๋กœ ์‚ฌ์šฉํ–ˆ์„ ๋•Œ, ์ฐธ ๋…ธ์ด์ฆˆ ๋ ˆ๋ฒจ์„ ์‚ฌ์šฉํ–ˆ์„ ๋•Œ์™€ ๊ฑฐ์˜ ๋™์ผํ•œ ๋ฐ๋…ธ์ด์ง• ์„ฑ๋Šฅ์„ ๋‹ฌ์„ฑํ–ˆ์Šต๋‹ˆ๋‹ค. ์ด๋Š” ๋ณธ ๋…ธ์ด์ฆˆ ์ถ”์ • ๋ฐฉ๋ฒ•์˜ ์‹ค์šฉ์  ๊ฐ€์น˜๋ฅผ ๊ฐ•๋ ฅํ•˜๊ฒŒ ๋’ท๋ฐ›์นจํ•ฉ๋‹ˆ๋‹ค.

๋ณธ ์—ฐ๊ตฌ์˜ ํ•œ๊ณ„์ ์€ ๋…ธ์ด์ฆˆ๊ฐ€ ์˜ํ‰๊ท  ๊ฐ€๋ฒ•์  ๊ฐ€์šฐ์‹œ์•ˆ ๋ถ„ํฌ๋ฅผ ๋”ฐ๋ฅธ๋‹ค๊ณ  ๊ฐ€์ •ํ•œ ๊ฒƒ์ž…๋‹ˆ๋‹ค. ์‹ค์ œ ํ™˜๊ฒฝ์—์„œ์˜ ๋…ธ์ด์ฆˆ๋Š” ๋” ๋ณต์žกํ•œ ํŠน์„ฑ์„ ๊ฐ€์งˆ ์ˆ˜ ์žˆ์œผ๋ฉฐ, ์ด๋Š” ํ–ฅํ›„ ์—ฐ๊ตฌ์˜ ๋ฐฉํ–ฅ์ž…๋‹ˆ๋‹ค.


  1. ๊ฒฐ๋ก 

๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ๋‹จ์ผ ๋…ธ์ด์ฆˆ ์˜์ƒ์œผ๋กœ๋ถ€ํ„ฐ ๋…ธ์ด์ฆˆ ๋ถ„์‚ฐ์„ ์ž๋™์œผ๋กœ ์ถ”์ •ํ•˜๋Š” ํšจ์œจ์ ์ธ ๋ฐฉ๋ฒ•์„ ์ œ์•ˆํ–ˆ์Šต๋‹ˆ๋‹ค. ์ด ๋ฐฉ๋ฒ•์€ ๋‹ค์–‘ํ•œ ์ปดํ“จํ„ฐ ๋น„์ „ ์•Œ๊ณ ๋ฆฌ์ฆ˜์— ๋งค์šฐ ์ค‘์š”ํ•ฉ๋‹ˆ๋‹ค. ์ œ์•ˆ๋œ ๋ฐฉ๋ฒ•์˜ ์„ฑ๋Šฅ์€ ์ด๋ก ์ ์œผ๋กœ ๋ณด์žฅ๋˜์—ˆ์œผ๋ฉฐ, ๋‘ ๊ฐ€์ง€ ์ตœ์‹  ๋ฐฉ๋ฒ•๊ณผ ๋น„๊ตํ•˜์—ฌ ๋Œ€๋ถ€๋ถ„์˜ ๊ฒฝ์šฐ ๊ฐ€์žฅ ์ ์€ ์‹คํ–‰ ์‹œ๊ฐ„์œผ๋กœ ์ตœ๊ณ ์˜ ์ •ํ™•๋„๋ฅผ ๋‹ฌ์„ฑํ•จ์„ ๋ณด์—ฌ์ฃผ์—ˆ์Šต๋‹ˆ๋‹ค.

์ฝ์–ด์ฃผ์…”์„œ ๊ฐ์‚ฌํ•ฉ๋‹ˆ๋‹ค :)



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