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CS ๋…ผ๋ฌธ์ฝ๊ธฐ๋ผ๊ณ  ์ƒ๊ฐํ•˜๊ณ  ์—ฐ cs-adventure ์นดํ…Œ๊ณ ๋ฆฌ์ธ๋ฐ ์˜์™ธ๋กœ ์ฒ˜์Œ์ด ์ˆ˜์น˜ํ•ด์„? ์„ ์“ฐ๋Š” ์ตœ์ ํ™” ๊ด€๋ จ์ด ๋˜์—ˆ๋„ค์š”.

Introduction

์ด๋ฒˆ์— ์ฝ์€ ๋…ผ๋ฌธ์€ Active Contours Without Edges ๋ผ๋Š”, 2001๋…„์˜ ๋…ผ๋ฌธ์ž…๋‹ˆ๋‹ค. 2001๋…„ IEEE Transactions on Image Processing, Vol.10, No.2 ์— ๋ฐœํ‘œ๋œ ๋…ผ๋ฌธ์œผ๋กœ, ์ด ๋ถ„์•ผ - image processing - ์—์„œ๋Š” ์—„์ฒญ๋‚˜๊ฒŒ ์ค‘์š”ํ•œ ๋…ผ๋ฌธ์œผ๋กœ, ํ˜„์žฌ๊นŒ์ง€ 1๋งŒ 3์ฒœ ํšŒ ๊ฐ€๋Ÿ‰ ์ธ์šฉ๋˜์—ˆ์Šต๋‹ˆ๋‹ค.

๋ชฉํ‘œ๋Š” ์–ด๋–ค ์ด๋ฏธ์ง€๊ฐ€ ์ฃผ์–ด์กŒ์„ ๋•Œ, ์ด ์ด๋ฏธ์ง€์˜ ์™ธ๊ณฝ์„  โ€œContourโ€ ๋ฅผ ๋”ฐ๋Š” ๊ฒƒ์ž…๋‹ˆ๋‹ค. ํŠนํžˆ, ์—ฌ๊ธฐ์„œ๋Š” segmentation์ด๋ผ๊ณ  ํ•ด์„œ ๊ทธ๋ฆผ์˜ ํ”ฝ์…€์„ ๋ช‡๊ฐœ์˜ ํด๋ž˜์Šค๋กœ ๊ตฌ๋ถ„ํ•˜๋Š” ๋ฌธ์ œ๋ฅผ ํ•ด๊ฒฐํ•˜๋Š” ๊ฒƒ์œผ๋กœ ๋ณด๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ์˜ˆ๋ฅผ ๋“ค์–ด, ๋ฐฐ๊ฒฝ ์•ž์— ์‚ฌ๋žŒ์ด ์„œ ์žˆ๋‹ค๋ฉด, ์‚ฌ๋žŒ๊ณผ ๋ฐฐ๊ฒฝ์„ ๊ตฌ๋ถ„ํ•˜๋Š” ๋ฌธ์ œ๋ฅผ classification์ด๋ผ๊ณ  ํ•  ์ˆ˜ ์žˆ๊ฒ ์Šต๋‹ˆ๋‹ค. ๊ฝค ์˜ค๋ž˜ ์ „ (์ €๋„ ๋ฐœํ‘œ์ผ ๊ธฐ์ค€ 2001) ๋…ผ๋ฌธ์ด๋ฏ€๋กœ, ๋ณธ๊ฒฉ์ ์ธ Deep Learning์˜ ์‹œ๋Œ€๊ฐ€ ์˜ค๊ธฐ ์ „์˜ ๋ฐฉ๋ฒ•๋ก ์„ ๋ณผ ์ˆ˜ ์žˆ์—ˆ์Šต๋‹ˆ๋‹ค.

๋จผ์ €, ์šฉ์–ด๋ฅผ ๊ฐ„๋‹จํžˆ ์ •์˜ํ•ฉ๋‹ˆ๋‹ค. Segmentation๊ณผ Contour detection์€ ์›๋ž˜ ์•ฝ๊ฐ„ ๋‹ค๋ฅธ ๋ฌธ์ œ์ง€๋งŒ, ์—ฌ๊ธฐ์„œ๋Š” Segmentation์˜ ๋ฐฉ๋ฒ•์œผ๋กœ Level set (๋“ฑ๊ณ ์„ ) ์˜ Contour๋ฅผ ๋”ฐ๋Š” ๋ฐฉ๋ฒ•์„ ์ƒ๊ฐํ•˜๊ธฐ๋กœ ํ•ฉ๋‹ˆ๋‹ค. ์ด๋ฅผ ์œ„ํ•ด, ์˜ˆ๋ฅผ ๋“ค์–ด ์–ด๋–ค grayscale ์ด๋ฏธ์ง€๊ฐ€ ์ฃผ์–ด์ง€๋ฉด, ์ด ์„ ์„ ๋Œ€์ถฉ ์ƒ‰์ด ์ง„ํ•œ ์ชฝ๊ณผ ํ๋ฆฐ ์ชฝ์œผ๋กœ ๋‚˜๋ˆ„๊ธฐ ์œ„ํ•ด ์ง„ํ•œ ์ ๋“ค์ด ์ด๋ฃจ๋Š” Contour๋ฅผ ์ฐพ๊ณ ์ž ํ•œ๋‹ค๊ณ  ์ดํ•ดํ•˜๋ฉด ๋˜๊ฒ ์Šต๋‹ˆ๋‹ค.


Key Ideas

Energy functional

์ด ๋ฌธ์ œ์—์„œ๋Š”, Segmentation ๋ฌธ์ œ๋ฅผ Functional Optimization์˜ ๋ฌธ์ œ๋กœ ํ™˜์›ํ•ฉ๋‹ˆ๋‹ค. ์—ฌ๊ธฐ์„œ Functional์ด๋ž€, ์ •์˜์—ญ์ด ํ•จ์ˆ˜์˜ ์ง‘ํ•ฉ์ธ ํ•จ์ˆ˜๋ฅผ ๋งํ•ฉ๋‹ˆ๋‹ค.

ํŠนํžˆ, ์šฐ๋ฆฌ๋Š” ๊ฒฐ๊ณผ๋ฌผ์˜ ์™ธ๊ณฝ์„ ์ด Smoothํ•˜๊ธฐ๋ฅผ ์›ํ•˜๋ฏ€๋กœ, $X$์—์„œ Lipschitz Continuous ํ•œ ํ•จ์ˆ˜์˜ ์ง‘ํ•ฉ $\mathcal{L}$ ์—์„œ $\R$๋กœ ๊ฐ€๋Š” ํ•จ์ˆ˜์—ด์„ ์ƒ๊ฐํ•  ๊ฒƒ์ž…๋‹ˆ๋‹ค. ์—ฌ๊ธฐ์„œ Lipschitz ์—ฐ์†์ด๋ž€ ์—ฐ์†์„ฑ๋ณด๋‹ค ๋” ๊ฐ•ํ•œ ๊ฐœ๋…์œผ๋กœ, ์  $x, y$ ์™€ ์–ด๋–ค ์ƒ์ˆ˜ $K$์— ๋Œ€ํ•ด $\norm{f(x) - f(y)} \leq K \norm{x - y}$ ๋ฅผ ๋งŒ์กฑํ•˜๋Š” ํ•จ์ˆ˜๋“ค์„ ์˜๋ฏธํ•ฉ๋‹ˆ๋‹ค.

ํ•จ์ˆ˜์˜ Level set์— ๋Œ€ํ•ด ๋…ผ์˜ํ•˜๊ธฐ ์œ„ํ•ด, ์šฐ๋ฆฌ๋Š” ํ•จ์ˆ˜ $\phi \in \mathcal{L}$ ์— ๋Œ€ํ•ด, $\phi = 0$ ์ธ ์ ๋“ค์„ ์ด์€ ๊ณก์„ ์„ $C$๋ผ๊ณ  ์ •์˜ํ•ฉ๋‹ˆ๋‹ค. ๋˜ํ•œ, $\phi(x) > 0$ ์ธ ๊ณต๊ฐ„์„ $A$, $\phi(x) < 0$ ์ธ ๊ณต๊ฐ„์„ $B$๋ผ๊ณ  ์“ฐ๊ฒ ์Šต๋‹ˆ๋‹ค. ๋งˆ์ง€๋ง‰์œผ๋กœ, ์›๋ž˜์˜ ์ด๋ฏธ์ง€ ํ”ฝ์…€๊ฐ’์„ $u_0(x, y)$ ํ•จ์ˆ˜๋กœ ๋‚˜ํƒ€๋ƒ…๋‹ˆ๋‹ค.

์ด์ œ, ๋‹ค์Œ๊ณผ ๊ฐ™์€ Functional๋“ค์„ ์ •์˜ํ•ฉ๋‹ˆ๋‹ค.

  • $Len(C)$ : ๊ณก์„ ์˜ ๊ธธ์ด. ๊ณก์„ ์˜ ๊ธธ์ด๊ฐ€ ๊ธธ๋ฉด $\phi$ ๊ฐ€ ๋œ smoothํ•˜๊ธฐ ๋•Œ๋ฌธ์— (ํ•ด์„ํ•™์ ์ธ term์ด๋ผ๊ธฐ๋ณด๋‹ค๋Š”, ๊ธฐํ•˜์ ์ธ smoothํ•จ), ๋งค๋„๋Ÿฌ์šด ๊ณก์„ ์„ ๊ทธ๋ฆฌ๋„๋ก ํŽ˜๋„ํ‹ฐ๋ฅผ ํ†ตํ•ด incentivise ํ•ฉ๋‹ˆ๋‹ค.
  • $Area(A)$ : $\phi(x) > 0$ ์ธ ๋ถ€๋ถ„์˜ ๋„“์ด. ๊ธธ์ด์™€ ๊ธฐ๋ณธ์ ์ธ ์˜๋ฏธ๋Š” ๊ฐ™์Šต๋‹ˆ๋‹ค.
  • $\int_{A} \abs{u_0(x, y) - c_1}^2 \dd{x}\dd{y}$ : ์–ด๋–ค ์‹ค์ˆ˜๊ฐ’ $c_1$ ์„ ์žก์•„์„œ, $\phi$ ์•ˆ์ชฝ์—์„œ $u_0$ ์˜ ํ‰๊ท ์„ ๋‚˜ํƒ€๋‚ด๊ณ  ์‹ถ์Šต๋‹ˆ๋‹ค. ์ด๋•Œ ์ด ํ‰๊ท ๊ฐ’์ด ๊ฐ€๊ธ‰์  ์ •ํ™•ํ•˜๊ธฐ๋ฅผ ๋ฐ”๋ž€๋‹ค๋Š” ์˜๋ฏธ์ž…๋‹ˆ๋‹ค. ์ผ์ข…์˜, ์˜์—ญ ์•ˆ์—์„œ์˜ intensity์˜ ๋ถ„์‚ฐ์œผ๋กœ ์ƒ๊ฐํ•˜๋ฉด ๋ฉ๋‹ˆ๋‹ค. ๊ฐ€๊ธ‰์  ์˜์—ญ์„ ์ž˜ ์žก์•„์„œ ๋ฐ๊ธฐ์˜ ๋ถ„์‚ฐ์ด ์ž‘๊ฒŒ ์ž๋ฅธ๋‹ค๋Š” ์˜๋ฏธ๊ฐ€ ๋˜๊ฒ ์Šต๋‹ˆ๋‹ค.
  • $\int_{B} \abs{u_0(x, y) - c_2}^2 \dd{x}\dd{y}$ : $\phi$ ๋ฐ”๊นฅ์ชฝ์—์„œ๋„ ๋˜‘๊ฐ™์€ ์ž‘์—…์„ ํ•ฉ๋‹ˆ๋‹ค.

์ง๊ด€์ ์œผ๋กœ, ์ € ๋„ค ๊ฐ’ ๋ชจ๋‘ ์ž‘์•˜์œผ๋ฉด ์ข‹๊ฒ ๋‹ค๋Š”๊ฒƒ์„ ์•Œ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. ์•ž ๋‘๊ฐœ๊ฐ€ ์ž‘์œผ๋ ค๋ฉด ํ•จ์ˆ˜๊ฐ€ ๋Œ€์ถฉ ๊ณก์„ ์œผ๋กœ ์ญ‰ ๋งค๋„๋Ÿฝ๊ฒŒ ์ด์–ด์ ธ์•ผ ํ•˜๊ณ , ๋’ค ๋‘๊ฐœ๊ฐ€ ์ž‘์œผ๋ ค๋ฉด ๊ทธ ์•ˆ์ชฝ๊ณผ ๋ฐ”๊นฅ์ชฝ์— ์–ด๋–ค intensity ๊ฐ’์„ ์žก์•„์„œ ๊ทธ ๊ฐ’์— ๊ฐ€๊น๊ฒŒ ์ž˜๋ ค์•ผ ํ•ฉ๋‹ˆ๋‹ค.
์šฐ๋ฆฌ๋Š” ์ € ๋„ค Functional์˜ ์„ ํ˜•๊ฒฐํ•ฉ์„ โ€œEnergy Functionalโ€ ์ด๋ผ๊ณ  ๋ถ€๋ฅด๊ธฐ๋กœ ํ•˜๊ณ , ์ € ๊ฐ’์„ minimizeํ•˜๋Š” $c_1, c_2, \phi$ ๋ฅผ ์ฐพ๋Š” ๊ฒƒ์„ ๋ชฉํ‘œ๋กœ ํ•ฉ๋‹ˆ๋‹ค.

Integral formulation

๊ทธ๋Ÿฌ๋‚˜, ์ € ์‹์€ ์ €๋Œ€๋กœ๋Š” ์ƒ๋‹นํžˆ ๊ณ„์‚ฐํ•˜๊ธฐ๊ฐ€ ์–ด๋ ต์Šต๋‹ˆ๋‹ค. ์ข€๋” ๊ณ„์‚ฐ์„ ์ž˜ ํ•˜๊ธฐ ์œ„ํ•ด, ์‹์„ ์‚ด์ง ์กฐ์ ˆํ•ด ๋ด…์‹œ๋‹ค. ์ด๋ฅผ ์œ„ํ•ด, ํ—ค๋น„์‚ฌ์ด๋“œ ํ•จ์ˆ˜ $H$๋ฅผ ๋„์ž…ํ•ฉ๋‹ˆ๋‹ค. $H$๋Š” $x \geq 0$ ์ผ ๋•Œ 1, $x < 0$ ์ผ ๋•Œ 0์ธ ํ•จ์ˆ˜์ž…๋‹ˆ๋‹ค. ์ด๋ฅผ ๋„์ž…ํ•˜๋ฉด $H(\phi(x, y)) = 1$ iff $\phi(x, y) \geq 0$ ๊ฐ€ ์„ฑ๋ฆฝํ•ฉ๋‹ˆ๋‹ค.

  1. Length : ๊ธธ์ด๋Š” ์Šคํ† ํฌ์Šค ์ •๋ฆฌ์™€ ํ—ค๋น„์‚ฌ์ด๋“œ ํ•จ์ˆ˜์˜ ์ •์˜๋ฅผ ์ด์šฉํ•˜๋ฉด, ์•„๋ž˜์™€ ๊ฐ™์ด ์“ธ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. \(L(\phi) = Len(C) = \int_{\R^2} \abs{\nabla H(\phi(x, y))} \dd{x}\dd{y}\) ๋‹น์—ฐํžˆ ์ผ๋ฐ˜์ ์œผ๋กœ $H(\phi(x, y))$ ๋Š” ๋ฏธ๋ถ„์ด ๋ถˆ๊ฐ€๋Šฅํ•˜์ง€๋งŒ, ์šฐ๋ฆฌ๋Š” Heaviside์˜ ๋„ํ•จ์ˆ˜๋ฅผ Dirac-delta๋กœ ์“ฐ๊ณ  ์žˆ์œผ๋ฏ€๋กœ (in distribution function sense) ์ ๋ถ„์€ ์ž˜ ๋ฉ๋‹ˆ๋‹ค. 1
    ๋‚˜๋จธ์ง€ ์‹๋“ค๊ณผ ์ ๋ถ„ํ•˜๋Š” ๋ณ€์ˆ˜ ๋“ฑ์„ ๋งž์ถฐ์ฃผ๊ธฐ ์œ„ํ•ด, ์ด ์‹์„ ์กฐ๊ธˆ ๋ฐ”๊พธ์–ด ์•„๋ž˜์™€ ๊ฐ™์ด ์”๋‹ˆ๋‹ค. \(L(\phi) = Len(C) = \int_{\R^2} \delta(\phi(x, y))\abs{\nabla \phi(x, y)} \dd{x}\dd{y}\)
  2. Area : ๊ฐ„๋‹จํ•œ ๋‹ค๋ณ€์ˆ˜ ์ ๋ถ„์ž…๋‹ˆ๋‹ค. \(S(\phi) = Area(A) = \int_{\R^2} H(\phi(x, y)) \dd{x}\dd{y}\)
  3. ์—ญ์‹œ ๊ฐ„๋‹จํ•œ ๋‘ ๊ฐœ์˜ ๋‹ค๋ณ€์ˆ˜ ์ ๋ถ„์‹์„ ์“ธ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. \(F_i(\phi) = \int_{\R^2} \abs{u_0(x, y) - c_1}^2 H(\phi(x, y))\dd{x}\dd{y}\) \(F_o(\phi) = \int_{\R^2} \abs{u_0(x, y) - c_2}^2 (1 - H(\phi(x, y)))\dd{x}\dd{y}\)

ํ•˜๋‚˜ ๊ด€์ฐฐํ•  ์ˆ˜ ์žˆ๋Š” ๊ฒƒ์€, $c_1$๊ณผ $c_2$๋Š” $\phi$๋ฅผ ๊ณ ์ •ํ•˜๊ณ  ์ตœ์ ํ™”ํ•  ์ˆ˜ ์žˆ๋‹ค๋Š” ์ ์ž…๋‹ˆ๋‹ค. ๊ตฌ์ฒด์ ์œผ๋กœ, ์ฒซ๋ฒˆ์งธ ์‹์„ ๋‹ค์‹œ \(\int_{A} \abs{u_0(x, y) - c_1}^2 \dd{x}\dd{y}\) ์ด๋ ‡๊ฒŒ ๋Œ๋ ค๋†“๊ณ  ๋ณด๋ฉด, $c_1$์€ ์ž๋ช…ํ•˜๊ฒŒ $u_0$์˜ $A$์—์„œ์˜ โ€˜ํ‰๊ท โ€™ ์ด ๋˜์–ด์•ผ ํ•ฉ๋‹ˆ๋‹ค. (์ ๋ถ„์„ ํ†ตํ•ด ํ•จ์ˆ˜์˜ ํ‰๊ท ์„ ๊ตฌํ•˜๋Š” ๋ฐฉ๋ฒ•์€ standard ํ•˜๋ฏ€๋กœ ์ƒ๋žต) ๋”ฐ๋ผ์„œ, ์•ž์œผ๋กœ $c_1, c_2$๋Š” $\phi$๋กœ๋ถ€ํ„ฐ ๊ฐ„๋‹จํ•œ ์ ๋ถ„์„ ํ†ตํ•ด ๊ณ„์‚ฐ ๊ฐ€๋Šฅํ•˜๋ฏ€๋กœ, ์œ„ ์‹์„ $\phi$๋กœ๋งŒ ์ตœ์†Œํ™”ํ•œ๋‹ค๊ณ  ๋ฌธ์ œ๋ฅผ ๋‹จ์ˆœํ™”ํ•˜๊ฒ ์Šต๋‹ˆ๋‹ค.
๋˜ํ•œ, ์‹ค์ œ ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ 1, 2, 3์— ๊ฐ๊ฐ ์ ๋‹นํ•œ ์ƒ์ˆ˜๋ฅผ ๋ถ™์—ฌ์„œ ๊ณ„์‚ฐํ•ฉ๋‹ˆ๋‹ค. ํŠนํžˆ, (3) ์˜ $F_i$ ์™€ $F_o$์— ๋‹ค๋ฅธ ์ƒ์ˆ˜๋ฅผ ๋ถ™์—ฌ์„œ ๊ณ„์‚ฐํ•˜๋Š”๋ฐ, ์‹ค์ œ๋กœ๋Š” ์›๋ณธ ๋…ผ๋ฌธ์˜ ์ €์ž๋“ค๋„ ์ƒ์ˆ˜๋ฅผ ๋Œ€์ถฉ ์žก์•˜๊ณ , ์ด ์ƒ์ˆ˜๋ฅผ ์–ด๋–ป๊ฒŒ ์žก์•„์•ผ ํ•˜๋Š”์ง€์— ๋Œ€ํ•ด์„œ๋Š” ๋ณ„๋กœ ๋…ผ์ฆ์ด ์—†์—ˆ์œผ๋ฏ€๋กœ ์ €๋Š” ์—ฌ๊ธฐ์„œ (1) * $\mu$ + (2) * $\nu$ + (3) * $\lambda$ ๋กœ ๋†“๊ณ  ๊ณ„์‚ฐํ•˜๊ฒ ์Šต๋‹ˆ๋‹ค. ์™ธ๊ณฝ์„ ์ด ๋งค๋„๋Ÿฌ์šด ๊ฒƒ์ด ์ค‘์š”ํ•˜๋ฉด $\mu, \nu$ ๋ฅผ ๋†’๊ฒŒ ์žก๊ณ , ์ƒ‰์˜ ์ •ํ™•๋„๊ฐ€ ์ค‘์š”ํ•˜๋ฉด $\lambda$๋ฅผ ๋†’๊ฒŒ ์žก์œผ๋ฉด ๋ฉ๋‹ˆ๋‹ค.

Regularization & Euler-Lagrange

์ดํ›„์˜ ์—ฐ์‚ฐ์—์„œ ๊ฐ€์žฅ ํฐ ๋ฌธ์ œ ์ค‘ ํ•˜๋‚˜๋Š”, $H$ ์™€ $\delta$๋Š” ๋ฏธ๋ถ„์ด ๋ถˆ๊ฐ€๋Šฅํ•œ ํ•จ์ˆ˜๋ผ๋Š” ๋ฌธ์ œ๊ฐ€ ์žˆ์Šต๋‹ˆ๋‹ค. ์ด ๋ฌธ์ œ๋ฅผ ํ•ด๊ฒฐํ•˜๊ธฐ ์œ„ํ•ด, ์šฐ๋ฆฌ๋Š” Regularization์ด๋ผ๋Š” ๋ฐฉ๋ฒ•์„ ์ ์šฉํ•ฉ๋‹ˆ๋‹ค.

Regularization์€ ์ „ํ˜€ ์–ด๋ ต์ง€ ์•Š์€๋ฐ, $H$ ๋Œ€์‹  $\epsilon$ ์ด๋ผ๋Š” factor์— dependentํ•œ, ๊ทธ๋ฆฌ๊ณ  $H_\epsilon \to H$ as $\epsilon \to 0$ ํ•จ์ˆ˜ $H_\epsilon$ ์œผ๋กœ ๋Œ€์ฒดํ•˜๊ณ , ๊ทธ ๋„ํ•จ์ˆ˜๋ฅผ $\delta_\epsilon$ ์œผ๋กœ ์“ฐ๋ฉด ๋ฉ๋‹ˆ๋‹ค. ๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ๋‹ค์Œ์˜ $H_\epsilon$ ์„ ์ œ์‹œํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. \(H_\epsilon(z) = \frac{1}{2} \left(1 + \frac{2}{\pi} \arctan\left(\frac{z}{\epsilon}\right)\right)\) ์ด ์‹๊ณผ ๊ทธ ๋„ํ•จ์ˆ˜ $\delta_\epsilon$ ์„ ์ด์šฉ, ๋ชจ๋“  $H$ ์™€ $\delta$๋ฅผ ๋Œ€์ฒดํ•˜๋ฉด ๋ฉ๋‹ˆ๋‹ค.
์ด์ œ, ์—ฌ๊ธฐ๊นŒ์ง€ ์˜ค๋ฉด์„œ ์šฐ๋ฆฌ๊ฐ€ ์ตœ์ข…์ ์œผ๋กœ ๋ฌด์—‡์„ ์ตœ์†Œํ™”ํ•˜๋Š”์ง€ ๋ณด๊ฒ ์Šต๋‹ˆ๋‹ค. ์—ฌ๊ธฐ์„œ๋ถ€ํ„ฐ๋Š” $F_i, F_o$ ๋“ฑ๋„ ๋ชจ๋‘ ์œ„ ์‹์— ๋”ฐ๋ผ relaxation ๋œ ๊ฒƒ์œผ๋กœ ์ฝ์–ด์•ผ ํ•ฉ๋‹ˆ๋‹ค. \(\int_{\R^2} \mu L(\phi) + \nu S(\phi) + \lambda(F_i(\phi) + F_o(\phi)) \dd{x}\dd{y}\) ์ด ์‹์„ ์ตœ์†Œํ™”ํ•˜๋Š” ํ•จ์ˆ˜ $\phi$๋ฅผ ์ฐพ๋Š” ๋Œ€ํ‘œ์ ์ธ ๋ฐฉ๋ฒ•์€ ๋ณ€๋ถ„๋ฒ• ์ž…๋‹ˆ๋‹ค. ๋ณ€๋ถ„๋ฒ•์œผ๋กœ ๋”์ฐํ•œ ๊ณ„์‚ฐ์„ ํ†ตํ•ด ์˜ค์ผ๋Ÿฌ-๋ผ๊ทธ๋ž‘์ฃผ ๋ฐฉ์ •์‹์„ ์œ ๋„ํ•˜๋ฉด 2 3, ์•„๋ž˜์™€ ๊ฐ™์€ ํŽธ๋ฏธ๋ถ„ ๋ฐฉ์ •์‹์„ ์–ป์Šต๋‹ˆ๋‹ค. \(\delta_\epsilon \left(\mu\nabla\cdot\left(\frac{\nabla \phi}{\abs{\nabla \phi}}\right) - \nu - \lambda(u_0 - c_1)^2 + \lambda(u_0 - c_2)^2\right) = 0 \tag{PDE}\)

Partial Differential Equation

์šฐ๋ฆฌ๋Š” ์ด๋Ÿฐ ํŽธ๋ฏธ๋ถ„๋ฐฉ์ •์‹์€ ํ’€ ๋ฐฉ๋ฒ•์ด ์—†๊ธฐ ๋•Œ๋ฌธ์—, ๋งˆ์ง€๋ง‰์œผ๋กœ ์ˆ˜์น˜ํ•ด์„์„ ์ ์šฉํ•ฉ๋‹ˆ๋‹ค. ๊ตฌ์ฒด์ ์œผ๋กœ, Finite Differnce method๋ฅผ ์ด์šฉํ•ด์•ผ ํ•ฉ๋‹ˆ๋‹ค.

ํŽธ๋ฏธ๋ถ„๋ฐฉ์ •์‹์„ iterative ํ•˜๊ฒŒ ํ’€๊ธฐ ์œ„ํ•ด, evolving ํ•˜๋Š” ํ•ด $\phi$ ๋ฅผ ์ƒ๊ฐํ•ฉ๋‹ˆ๋‹ค. ์ด๋Ÿฐ ๋ฐฉ๋ฒ•์„ ์“ฐ๋Š” ์ด์œ ๋Š” $c_1$ ๊ณผ $c_2$๋ฅผ $\phi$๋กœ๋ถ€ํ„ฐ ๊ณ„์‚ฐํ•ด์•ผ ํ•ด์„œ ์œ„ ์‹์„ ๊ณ„์‚ฐํ•  ์ˆ˜ ์—†์œผ๋ฏ€๋กœ, ๋‹ค์Œ๊ณผ ๊ฐ™์€ ์•Œ๊ณ ๋ฆฌ์ฆ˜์„ ๋Œ๋ฆฌ๊ฒ ๋‹ค๋Š” ์˜๋ฏธ์ž…๋‹ˆ๋‹ค.

  1. $\phi_0$ ์„ ์ •ํ•ฉ๋‹ˆ๋‹ค.
  2. $c_1, c_2$ ๋ฅผ $\phi_n$ ์œผ๋กœ๋ถ€ํ„ฐ ๊ณ„์‚ฐํ•ฉ๋‹ˆ๋‹ค.
  3. โ€œPDEโ€ ๋ฅผ ํ•œ๋ฒˆ ํ’€์–ด์„œ $\phi_{n+1}$ ์„ ๊ณ„์‚ฐํ•ฉ๋‹ˆ๋‹ค.
  4. $\phi_{n+1}$ ์ด $\phi_n$ ๊ณผ ๋งŽ์ด ๋‹ค๋ฅด๋‹ค๋ฉด, (2) ๋กœ ๋Œ์•„๊ฐ€์„œ ๋ฐ˜๋ณตํ•ฉ๋‹ˆ๋‹ค.

Finite Difference method๋Š” ๋ณ„๋กœ ์–ด๋ ต์ง€ ์•Š์€๋ฐ, $\Delta_{-}^{x}$ ๊ฐ™์€ ์‹์œผ๋กœ $x, y$ ๋ฐฉํ–ฅ $+, -$ ๋กœ ๋„ค๊ฐœ์˜ time differnece๋ฅผ ์ •์˜ํ•˜๊ณ  ์—ฌ๋Ÿฌ ๊ณต์‹๋“ค์„ ์ ์šฉํ•˜๊ธฐ๋งŒ ํ•˜๋ฉด ๋ฉ๋‹ˆ๋‹ค. ํŽธ๋ฏธ๋ฐฉ์ด ๋ณต์žกํ•˜๊ฒŒ ์ƒ๊ฒผ์ง€๋งŒ ๊ฐ term์€ ๊ทธ๋ ‡๊ฒŒ ์–ด๋ ต์ง€ ์•Š์Šต๋‹ˆ๋‹ค.

๋จผ์ €, $\nu - \lambda(u_0 - c_1)^2 + \lambda(u_0 - c_2)^2$ ๋ถ€๋ถ„์€ ์ž๋ช…ํ•ฉ๋‹ˆ๋‹ค. ($c_1, c_2$ ๋„ $\phi_n$ ์œผ๋กœ๋ถ€ํ„ฐ ๊ตฌํ–ˆ์œผ๋ฏ€๋กœ) Divergence ๋ถ€๋ถ„์ด ๋ฌธ์ œ์ธ๋ฐ, ๊ทธ ๋ถ€๋ถ„์€ ์ด๋ฏธ ๊ณผ๊ฑฐ์˜ ์„ ํ–‰ ์—ฐ๊ตฌ ๋…ผ๋ฌธ (๋งํฌ) ์— ์˜ํ•ด ์–ด๋–ป๊ฒŒ ํ•ด์•ผ ํ•˜๋Š”์ง€ ์ž˜ ์•Œ๋ ค์ ธ ์žˆ๋‹ค๊ณ  ํ•ฉ๋‹ˆ๋‹ค. ๊ตฌ์ฒด์ ์œผ๋กœ, ์ถฉ๋ถ„ํžˆ ์ž‘์€ $\Delta t$๋ฅผ ์žก์•„์„œ ์ด๋ ‡๊ฒŒ ์“ฐ๋ฉด ๋ฉ๋‹ˆ๋‹ค.

formula

์ด์ œ ์ˆ˜์น˜ํ•ด์„ ๋ฌธ์ œ๋„ ํ•ด๊ฒฐ๋˜์—ˆ๊ณ , ์œ„ ์‹์„ ๊ทธ๋ƒฅ ์—ด์‹ฌํžˆ ๊ณ„์‚ฐํ•˜๋ฉด ๋ฉ๋‹ˆ๋‹ค.

Reinitialization

Level Set์„ ๊ตฌํ•˜๊ณ ์ž Dirac delta ๊ฐ™์€ ํ•จ์ˆ˜๋“ค์„ ์‚ฌ์šฉํ•  ๋•Œ, re-initialization์ด๋ผ๋Š” ๊ณผ์ •์„ ๊ฑฐ์น˜์ง€ ์•Š์œผ๋ฉด level set์ด ์ง€๋‚˜์น˜๊ฒŒ flatํ•ด์ง€๋Š” ๊ฒฝํ–ฅ์„ฑ์ด ์žˆ๋‹ค๊ณ  ํ•ฉ๋‹ˆ๋‹ค. ๊ทธ ์ด์œ ๋Š” ์šฐ๋ฆฌ๊ฐ€ ์ •์ƒ์ ์ธ $\delta$ ๊ฐ€ ์•„๋‹Œ $\delta_\epsilon$ ๊ฐ™์€ ๋น„์Šทํ•œ ํ•จ์ˆ˜๋“ค๋กœ ๋„˜์–ด๊ฐ€์„œ ์ƒ๊ธด ๋ฌธ์ œ์ธ๋ฐ, ์ด๋ฅผ ํ•ด๊ฒฐํ•˜๊ธฐ ์œ„ํ•ด $\phi$๋ฅผ ๋งค๋ฒˆ ์ˆ˜์ •ํ•ด์ค„ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. ๋…ผ๋ฌธ์—์„œ ์ €์ž๋“ค์€ โ€˜๋ฐ˜๋“œ์‹œ ํ•„์š”ํ•˜์ง€ ์•Š๋‹คโ€™ ๊ณ  ์“ฐ๊ณ  ์žˆ๊ณ , ์‹ค์ œ๋กœ๋„ ์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์—์„œ๋Š” Reinitialize๋ฅผ ํ•˜์ง€ ์•Š์•„๋„ ๊ฒฐ๊ณผ๊ฐ€ ์–ด๋–จ๋•Œ๋Š” ์ž˜ ๋‚˜์˜ค๋Š” ๊ฒƒ ๊ฐ™์Šต๋‹ˆ๋‹ค๋งŒ, ์ถ”๊ฐ€ํ•˜๊ณ  ์‹ถ๋‹ค๋ฉด ๋‹ค์Œ์˜ Evolution equation์„ ํ’€๋ฉด ๋ฉ๋‹ˆ๋‹ค. \(\pdv{\psi}{t} = sign(\phi(x, y, t))(1 - \abs{\nabla \psi})\) ์—ฌ๊ธฐ์„œ $\phi(x, y, t)$ ๋Š”, ์•ž์„œ $\phi_n$ ์„ ์‹ค์ œ๋กœ๋Š” $t = n\Delta t$์—์„œ์˜ $\phi(x, y, t)$ ๊ฐ’์œผ๋กœ ๋ณด๊ณ  ์žˆ๊ธฐ ๋•Œ๋ฌธ์— ํ•„์š”ํ•œ ์ง€์ ์—์„œ๋Š” ๊ณ„์‚ฐํ•  ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค. ์šฐ๋ฆฌ๋Š” ์–ด์ฐจํ”ผ ์ด ์‹๋„ ์ˆ˜์น˜ํ•ด์„์œผ๋กœ ํ’€ ๊ฒƒ์ด๋ฏ€๋กœ $\phi(x, y, t)$ ์˜ ๊ฐ’์€ ํ•ด์„์ ์œผ๋กœ ๊ตฌํ•  ํ•„์š” ์—†์Šต๋‹ˆ๋‹ค.

์ด ํ…Œํฌ๋‹‰์— ๊ด€ํ•ด์„œ๋Š” ๋‹ค๋ฅธ ๋…ผ๋ฌธ์„ ํ†ตํ•ด ๋” ์•Œ๊ฒŒ ๋˜๋ฉด ๋ณด์ถฉํ•ด ๋ณผ ์ƒ๊ฐ์ž…๋‹ˆ๋‹ค. ์•„์ง์€ ์‹ค์ œ ์˜ˆ์‹œ๊ฐ€ ์—†์–ด์„œ์ธ์ง€ ์™œ ์ด๊ฒŒ ํ•„์š”ํ•œ์ง€, ์–ด๋–ค ์˜๋ฏธ์ธ์ง€ ์ž˜ ์™€๋‹ฟ์ง€ ์•Š๋„ค์š”.


Conclusion

์ด ์•Œ๊ณ ๋ฆฌ์ฆ˜์€ Noisy image์—์„œ๋„ ์ƒ๊ฐ๋ณด๋‹ค ํ›Œ๋ฅญํ•œ ์„ฑ๋Šฅ์„ ๋ณด์—ฌ์ฃผ๊ณ , vector-valued (= ์ปฌ๋Ÿฌ ์ด๋ฏธ์ง€) ๊ฐ™์€ ํ™•์žฅ๋„ ๊ทธ๋ ‡๊ฒŒ ์–ด๋ ต์ง€ ์•Š์Šต๋‹ˆ๋‹ค. ํŠนํžˆ, ์ด๋Ÿฐ ์‹์˜ Energy Functional์„ ์ž˜ ์ •์˜ํ•˜๊ธฐ์— ๋”ฐ๋ผ์„œ ๋ฒ”์šฉ์„ฑ์ด ๊ต‰์žฅํžˆ ๋†’๊ณ  ์›ํ•˜๋Š” Feature๊ฐ€ ์žˆ๋‹ค๋ฉด ์ถ”๊ฐ€๋กœ embed ํ•  ์ˆ˜๋„ ์žˆ์–ด์„œ ํ™•์žฅ์„ฑ๋„ ๋†’์Šต๋‹ˆ๋‹ค. ํŠนํžˆ, ์›๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” curvature ๊ฐ™์€ ์ •๋ณด๋“ค์„ ์ถ”๊ฐ€๋กœ ์ด์šฉํ•˜๋Š” ์ผ€์ด์Šค๋“ค๋„ ์ œ์‹œ๋˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค.


Thoughts

  1. ์œ„ ์•Œ๊ณ ๋ฆฌ์ฆ˜์—์„œ๋Š” Parameter ๊ฐ€ ์ƒ๋‹นํžˆ ์ค‘์š”ํ•ด ๋ณด์ž…๋‹ˆ๋‹ค. $\mu, \nu, \lambda$ ์˜ ์–ด๋–ค ์กฐํ•ฉ์ด ์ข‹์€ ๊ฒฐ๊ณผ๋ฅผ ๋‚ด๋Š”์ง€์— ๋Œ€ํ•ด์„œ๋Š” ์›๋ณธ ๋…ผ๋ฌธ์—์„œ๋Š” ๋ณ„๋กœ Discussํ•˜์ง€ ์•Š์•˜๋Š”๋ฐ, ์‹คํ—˜์ ์œผ๋กœ ํ™•์ธํ•ด์•ผ ํ•˜๋Š” ๊ฑธ๊นŒ์š”? $\mu, \nu$ ๋Š” ์–ด๋–ป๊ฒŒ ์‹คํ—˜์ ์œผ๋กœ ๊ฒ€์ฆํ•  ์ˆ˜ ์žˆ์–ด ๋ณด์ด๋Š”๋ฐ, $\lambda$๋Š” ์ข€ ์˜ค๋ฐ”์ธ๊ฒƒ ๊ฐ™์Šต๋‹ˆ๋‹ค. ๊ฒ€์ƒ‰์„ ์ข€ ํ•ด๋ณด๋‹ˆ, ๋‹ค๋ฅธ ๋…ผ๋ฌธ ๋ช‡ํŽธ์—์„œ ์ด๋ฏธ์ง€์˜ ์–ด๋–ค computableํ•œ ์„ฑ์งˆ๋“ค๋กœ๋ถ€ํ„ฐ parameter๋ฅผ ์ž๋™์œผ๋กœ ํŠœ๋‹ํ•˜๋Š” ๋…ผ๋ฌธ๋“ค์ด ์žˆ์—ˆ์Šต๋‹ˆ๋‹ค.
  2. Functional Optimization์€ ์ผ๋ฐ˜์ ์ธ optimization์˜ ๋ฐฉ๋ฒ•๋ก ๋“ค๊ณผ๋Š” ์ข€ ๋‹ค๋ฅด๋‹ค๋ณด๋‹ˆ ์–ด๋ ต์Šต๋‹ˆ๋‹ค. ๋“ฃ๊ธฐ๋กœ๋Š” Banach space์œ„์—์„œ์˜ Lagrange Multiplier๊ฐ™์€ ํ•ด๊ดดํ•œ๊ฒŒ ์žˆ๋‹ค๊ณ  ํ•ฉ๋‹ˆ๋‹ค. Functional๋„ ๊ฒฐ๊ตญ Banach space๋‚˜ Hilbert Space ๊ฐ™์€ ์ข‹์€ ๊ณต๊ฐ„ ์œ„์—์„œ ์–ด๋–ค ํ•จ์ˆ˜๋ฅผ ์ตœ์ ํ™”ํ•˜๋Š” ๊ฑฐ๋‹ˆ๊นŒ, ์ผ๋ฐ˜์ ์ธ ์ตœ์ ํ™”์™€ ๋น„์Šทํ•œ ์•„์ด๋””์–ด๋“ค์ด ์žˆ๋Š” ๊ฑธ๊นŒ์š”?

  1. ํ•ด์„๊ฐœ๋ก ์„ ๋ฐฐ์šฐ๊ณ  ๋‚˜์„œ๋ถ€ํ„ฐ ๋””๋ž™-๋ธํƒ€๋ฅผ ์ ๋ถ„์— ํ™œ์šฉํ•˜๋Š”๊ฒŒ ์˜คํžˆ๋ ค ์ •๋ง ์ดํ•ด๊ฐ€ ์•ˆ ๊ฐ”์—ˆ๋Š”๋ฐ, ์ด ๊ฐœ๋…์€ Measure, Distribution function ๋“ฑ ์‹คํ•ด์„ํ•™ ๋ฐ ๊ทธ ์ด์ƒ์˜ ํ•ด์„ํ•™์„ ๋ฐฐ์šฐ๋ฉด ๋‹ค์‹œ make sense ํ•ฉ๋‹ˆ๋‹ค. ์ž ์‹œ ๊ณตํ•™์ˆ˜ํ•™์˜ ๊ด€์ ์œผ๋กœ ๋Œ์•„๊ฐ€์„œ ์ด ์‹์„ ๋ฐ›์•„๋“ค์ด๊ธฐ๋กœ ํ•ฉ๋‹ˆ๋‹ค. ์–ด์ฐจํ”ผ, ์ด ์ ๋ถ„์„ ์‹ค์ œ๋กœ ๊ณ„์‚ฐํ•  ๊ฒƒ์€ ์•„๋‹ˆ๋‹ˆ๊นŒ์š”.ย โ†ฉ

  2. ๋ณ€๋ถ„๋ฒ•๊ณผ ์˜ค์ผ๋Ÿฌ-๋ผ๊ทธ๋ž‘์ฃผ์— ๊ด€ํ•œ ๋‚ด์šฉ์€ ๋ฏธ์ ๋ถ„ํ•™ II์—์„œ ์ •๋ง ์ดˆ๋ณด์ ์œผ๋กœ ๋ฐฐ์šด ์ดํ›„๋กœ ์ฒ˜์Œ์ด๋ผ์„œ, ๊ฑฐ์˜ ๋‹ค ๊นŒ๋จน์—ˆ๋˜ ๋‚ด์šฉ์„ ๋‹ค์‹œ ๋ณด๊ณ  ์ถ”๊ฐ€๋กœ ์ด๊ฒƒ์ €๊ฒƒ ๋” ๊ณต๋ถ€ํ•ด์•ผ ํ–ˆ์Šต๋‹ˆ๋‹ค. ์ €๋ณด๋‹ค ์ด๋Ÿฐ๊ฑธ ํ›จ์”ฌ ๋งŽ์ด ์จ๋จน๋Š” UNIST ๊ธฐ๊ณ„๊ณตํ•™๊ณผ(+๋ฌผ๋ฆฌํ•™ ๋ถ€์ „)์˜ ์ง€์ธ์—๊ฒŒ ๊ธ‰ํžˆ ์˜ค์ผ๋Ÿฌ-๋ผ๊ทธ๋ž‘์ฃผ ๋ฐฉ์ •์‹์— ๋Œ€ํ•ด ๋ฐฐ์› ์Šต๋‹ˆ๋‹ค.ย โ†ฉ

  3. ๋ฐ”๋กœ ์œ„์—์„œ ์–ธ๊ธ‰๋œ ์ง€์ธ์ด ๋˜์ ธ์ค€ ๊ณ ์ „์—ญํ•™ ๊ฐ•์˜๋…ธํŠธ(์ œ๊ฐ€ ์ด๋Ÿฐ๊ฑธ ๋ณด๊ฒŒ ๋ ์ค„์€ ๋ชฐ๋ž๋„ค์š”โ€ฆ) ๋ฅผ ๋‹ค์‹œ ๋ณด๋ฉด์„œ, ์‹์„ ์ฒ˜์Œ๋ถ€ํ„ฐ ๋‹ค ์œ ๋„ํ•ด ๋ดค๋Š”๋ฐ ์ง„์งœ ์ •์‹ ์ด ์•„๋“ํ•ฉ๋‹ˆ๋‹ค.ย โ†ฉ