lectures/Komp_obr_SFedU/06_Pract.tex

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\title{ðÒÁËÔÉËÕÍ \No6: ÏÂÒÁÂÏÔËÁ ÉÚÏÂÒÁÖÅÎÉÊ}
\author{}\date{}\nocolon

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\section{òÁÂÏÔÁ Ó ÉÚÏÂÒÁÖÅÎÉÑÍÉ × Octave}
\subsection{FITS-ÆÁÊÌÙ}
îÁÉÂÏÌÅÅ ÐÏÐÕÌÑÒÎÏÊ ÕÔÉÌÉÔÏÊ ÐÒÏÓÍÏÔÒÁ FITS-ÆÁÊÌÏ× Ñ×ÌÑÅÔÓÑ \t{ds9}.

äÌÑ ÒÁÂÏÔÙ Ó ÏÂÙÞÎÙÍÉ ÉÚÏÂÒÁÖÅÎÉÑÍÉ × Octave ÉÓÐÏÌØÚÕÀÔÓÑ ÆÕÎËÃÉÉ \t{imread}, \t{iminfo},
\t{imshow}. ïÓÎÏ×ÎÙÅ ÆÕÎËÃÉÉ ÏÂÒÁÂÏÔËÉ ÉÚÏÂÒÁÖÅÎÉÊ ÎÁÈÏÄÑÔÓÑ × ÐÁËÅÔÅ \t{image}.
FITS-ÆÁÊÌÙ~--- ÐÁËÅÔ \t{fits}. õÚÎÁÔØ ×ÈÏÄÑÝÉÅ × ÓÏÓÔÁ× ÐÁËÅÔÁ ÆÕÎËÃÉÉ ÍÏÖÎÏ ÔÁË:
\begin{verbatim}
pkg describe -verbose fits
---
Package name:
fits
Version:
1.0.7
Short description:
The Octave-FITS package provides functions for  reading, and writing FITS
(Flexible Image Transport System) files.  This package uses the libcfitsio library.
Status:
Loaded
---
Provides:
Reading and writing FITS files
read_fits_image
save_fits_image
save_fits_image_multi_ext
\end{verbatim}

äÌÑ ÞÔÅÎÉÑ ÉÓÐÏÌØÚÕÅÍ \t{read\_fits\_image}:
\begin{verbatim}
[I H] = read_fits_image('stars.fit');
imshow(I,[1000 4000])
\end{verbatim}
îÅÏÂÈÏÄÉÍÏ ÕËÁÚÙ×ÁÔØ ÐÏÒÏÇÉ ÉÎÔÅÎÓÉ×ÎÏÓÔÅÊ ÄÌÑ \t{imshow}. âÏÌÅÅ ÔÏÞÎÏ ÍÏÖÎÏ ÏÐÒÅÄÅÌÉÔØ ÄÉÁÐÁÚÏÎÙ
ÉÚ ÓÔÁÔÉÓÔÉËÉ ÐÏ ÉÚÏÂÒÁÖÅÎÉÀ:
\begin{verbatim}
Imin = min(I(:))
 Imin =  1321
Imax(I(:))
 Imax =  52826
Imean = mean2(I)
 Imean =  2054.6
Istd = std2(I)
 Istd =  1275.5
Imed = median(I(:))
 Imed =  2009
Low = Imin; High = Imed + Istd;
imshow(I, [Low High]);
% or: imshow(I/High)
\end{verbatim}
ëÁË ×ÉÄÉÔÅ, ÉÚÏÂÒÁÖÅÎÉÅ ÐÏ×ÅÒÎÕÔÏ ×ÐÒÁ×Ï ÎÁ $90\degr$.
åÓÌÉ ÍÙ ÚÁÈÏÔÉÍ ÏÔÏÂÒÁÖÁÔØ ÉÚÏÂÒÁÖÅÎÉÑ ËÁË × ÏÒÉÇÉÎÁÌÅ, ÎÕÖÎÏ ÚÁÒÁÎÅÅ ÐÏ×ÅÒÎÕÔØ ÉÈ:
\begin{verbatim}
I = rot90(I);
imshow(I, [Low High])
\end{verbatim}
óÌÅÄÕÅÔ ÐÏÍÎÉÔØ, ÞÔÏ × Octave ËÏÏÒÄÉÎÁÔÙ ÉÚÏÂÒÁÖÅÎÉÑ ÎÁÞÉÎÁÀÔÓÑ Ó ×ÅÒÈÎÅÇÏ ÌÅ×ÏÇÏ ÕÇÌÁ (É ÏÓØ~Y
ÎÁÐÒÁ×ÌÅÎÁ ×ÎÉÚ), Á × ÓÔÁÎÄÁÒÔÅ FITS~--- Ó ÌÅ×ÏÇÏ ÎÉÖÎÅÇÏ (É ÏÓØ~Y ÎÁÐÒÁ×ÌÅÎÁ ××ÅÒÈ).

õÂÅÄÉÍÓÑ × ÜÔÏÍ:
\begin{verbatim}
I2=I;
I2([1:100],[1:100])=65000;
imshow(I2, [Low High]);
\end{verbatim}
óÒÁ×ÎÉÍ, ËÁË ÏÔÏÂÒÁÖÁÀÔÓÑ ËÏÏÒÄÉÎÁÔÙ ÐÉËÓÅÌÑ × ds9.

\subsection{òÁÂÏÔÁ Ó ÇÉÓÔÏÇÒÁÍÍÏÊ}
äÌÑ ÕÌÕÞÛÅÎÉÑ ÏÔÏÂÒÁÖÅÎÉÑ ÐÏÐÒÏÂÕÅÍ ÜË×ÁÌÉÚÏ×ÁÔØ ÇÉÓÔÏÇÒÁÍÍÕ:
\begin{verbatim}
HE = histeq(I/High, 65536);
imshow(HE);
% compare to HE = histeq(I/High, 256);
\end{verbatim}

ðÏÓÍÏÔÒÉÍ, ËÁË ÍÅÎÑÅÔÓÑ ÇÉÓÔÏÇÒÁÍÍÁ ÉÚÏÂÒÁÖÅÎÉÑ × ÚÁ×ÉÓÉÍÏÓÔÉ ÏÔ ÒÁÓÐÒÅÄÅÌÅÎÉÑ ÛÕÍÁ.
óÇÅÎÅÒÉÒÕÅÍ ÓÎÁÞÁÌÁ ÐÒÏÂÎÏÅ ÉÚÏÂÒÁÖÅÎÉÅ:
\begin{verbatim}
I = uint8(zeros(300,300));
I([1:50 250:end], :) = 200;
I([51:249], [1:30 270:end]) = 200;
I([101:199], [100:200]) = 100;
imshow(I)
imhist(I)
Iuniform = I + uint8(50*rand(size(I)));
imshow(Iuniform)
imhist(Iuniform)
Inorm = I + uint8(25*randn(size(I)));
imshow(Inorm)
Inorm = uint8(I + 25*randn(size(I)));
imshow(Inorm)
imhist(Inorm)
Ipoisson = imnoise(I, "poisson");
imshow(Ipoisson)
imhist(Ipoisson)
Isaltpepper = imnoise(I, "salt & pepper", 0.2);
imshow(Isaltpepper)
imhist(Isaltpepper)
Ispeckle = imnoise(I, "speckle", 0.03);
imshow(Ispeckle)
imhist(Ispeckle)
\end{verbatim}

óÒÁ×ÎÉÍ Ó ÜË×ÁÌÉÚÁÃÉÅÊ ÇÉÓÔÏÇÒÁÍÍÙ: \t{imshow(histeq(Inorm))}.

ðÏÐÒÏÂÕÅÍ ÐÒÉÍÅÎÉÔØ Ë ÉÚÏÂÒÁÖÅÎÉÀ ÇÁÍÍÁ-ËÏÒÒÅËÃÉÀ:
\begin{verbatim}
imshow(imadjust(Iuniform, [], [], 0.5))
imshow(imadjust(Iuniform, [], [], 5))
\end{verbatim}

ôÅÐÅÒØ ÐÏÐÒÏÂÕÅÍ ÐÒÅÏÂÒÁÚÏ×ÁÔØ ÑÒËÏÓÔØ ËÕÓÏÞÎÏ-ÌÉÎÅÊÎÏÊ ÆÕÎËÃÉÅÊ
$$\begin{cases}
I_{out} = 0.2 I_{in}, & I_{in} <  70,\\
I_{out} = 14 + 1.5 (I_{in}-70), & 70 \le I_{in} < 130, \\
I_{out} = 209 + 0.368*(I_{in}-130),& 130\le I_{in} < 256.
\end{cases}
$$
\begin{verbatim}
Ilinhyst = zeros(size(Inorm));
idx = find(Inorm < 70);
Ilinhyst(idx) = 0.2*Inorm(idx);
idx = find(Inorm >= 70 & Inorm < 130);
Ilinhyst(idx) = 14 + 1.5*(Inorm(idx)-70);
idx = find(Inorm >= 130);
Ilinhyst(idx) = 209 + 0.368*(Inorm(idx)-130);
Ilinhyst = uint8(Ilinhyst);
imshow(Ilinhyst);
imhist(Ilinhyst)
\end{verbatim}

ðÏÐÒÏÂÕÅÍ ÓÄÅÌÁÔØ ÓÔÕÐÅÎÞÁÔÏÅ ÐÒÅÏÂÒÁÚÏ×ÁÎÉÅ ÑÒËÏÓÔÉ: ÎÁ ÉÚÏÂÒÁÖÅÎÉÉ \t{Inorm} ÚÁÍÅÎÉÍ ×ÓÅ
ÉÎÔÅÎÓÉ×ÎÏÓÔÉ × ÄÉÁÐÁÚÏÎÅ $[50, 150]$ ÎÁ 100:
\begin{verbatim}
Ilinhyst = Inorm;
Ilinhyst(find(Inorm >= 50 & Inorm <= 150)) = 100;
imshow(Ilinhyst)
imhist(Ilinhyst)
\end{verbatim}

\subsection{æÉÌØÔÒÁÃÉÑ ÉÚÏÂÒÁÖÅÎÉÊ}
ðÏÐÒÏÂÕÅÍ ÐÒÏÓÔÕÀ ÍÅÄÉÁÎÎÕÀ ÆÉÌØÔÒÁÃÉÀ ÎÁ ÐÒÉÍÅÒÅ ÐÁÒÙ ÉÚÏÂÒÁÖÅÎÉÊ.
\begin{verbatim}
subplot(2,2,1)
subimage(Isaltpepper)
subplot(2,2,2)
subimage(medfilt2(Isaltpepper, [5 5])))
subplot(2,2,3)
subimage(Inorm)
subplot(2,2,4)
subimage(medfilt2(Inorm, [5 5]))
\end{verbatim}

÷ÉÄÉÍ, ÞÔÏ ÍÅÄÉÁÎÎÙÊ ÆÉÌØÔÒ ×ÐÏÌÎÅ ÎÅÐÌÏÈÏ ÓÐÒÁ×ÉÌÓÑ Ó ÛÕÍÏÍ <<ÓÏÌØ-ÐÅÒÅÃ>> É ÚÎÁÞÉÔÅÌØÎÏ ÈÕÖÅ~---
Ó ÇÁÕÓÓÏ×ÙÍ ÛÕÍÏÍ.

òÁÓÓÍÏÔÒÉÍ ÐÒÉÍÅÎÅÎÉÅ ÒÁÚÌÉÞÎÙÈ ÆÉÌØÔÒÏ×. äÌÑ ÎÁÞÁÌÁ ÓÏÚÄÁÄÉÍ ÍÁÔÒÉÃÙ ÆÉÌØÔÒÏ×:
\begin{verbatim}
fgauss = fspecial("gaussian", 5, 1);
flog = fspecial("log", 5, 1);
fsobelv = fspecial("sobel");
fsobelh = rot90(fsobelv);
\end{verbatim}

é ÐÒÉÍÅÎÉÍ ÉÈ Ë ÏÒÉÇÉÎÁÌØÎÏÍÕ ÉÚÏÂÒÁÖÅÎÉÀ, \t{Isaltpepper} É \t{Inorm}:
\begin{verbatim}
imshow(imfilter(I, fgauss, "symmetric"));
imshow(imfilter(Isaltpepper, fgauss, "symmetric"));
imshow(imfilter(Inorm, fgauss, "symmetric"));
imshow(histeq(imfilter(I, flog, "symmetric")));
imshow(histeq(imfilter(Isaltpepper, flog, "symmetric")));
imshow(histeq(imfilter(Inorm, flog, "symmetric")));
\end{verbatim}

ëÁË ×ÉÄÉÔÅ, Õ ÄÉÆÆÅÒÅÎÃÉÁÌØÎÏÇÏ ÆÉÌØÔÒÁ ÏÔÓÅËÁÅÔÓÑ ÏÔÒÉÃÁÔÅÌØÎÁÑ ËÏÍÐÏÎÅÎÔÁ~--- Ô.Ë. ÉÚÏÂÒÁÖÅÎÉÑ
ÉÍÅÀÔ ÔÉÐ \t{uint8\_t}, ÏÄÎÁËÏ, ÍÏÖÎÏ ÓÄÅÌÁÔØ ÔÁË:
\begin{verbatim}
imshow(histeq(imfilter(double(Isaltpepper), flog, "symmetric")));
imshow(histeq(imfilter(double(Inorm), flog, "symmetric")));
imshow(histeq(imfilter(double(I), flog, "symmetric")));
\end{verbatim}

Octave ÎÅÐÒÁ×ÉÌØÎÏ ÒÁÓÓÞÉÔÙ×ÁÅÔ ÚÎÁÞÅÎÉÅ ÆÉÌØÔÒÏ×, ÎÁÐÒÉÍÅÒ:
\begin{verbatim}
sum(flog(:))
ans = -0.024092
sum(fgauss(:))
ans =  1.00000
sum(fsobelv(:))
ans = 0
\end{verbatim}
õ ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÈ ÆÉÌØÔÒÏ× ÓÕÍÍÁ ÜÌÅÍÅÎÔÏ× ÄÏÌÖÎÁ ÂÙÔØ ÒÁ×ÎÁ ÎÕÌÀ (ÞÔÏÂÙ ËÏÎÓÔÁÎÔÎÙÅ ÕÒÏ×ÎÉ
ÐÒÅ×ÒÁÝÁÌÉÓØ × ÎÕÌØ), Á Õ ÕÓÒÅÄÎÑÀÝÉÈ~--- ÅÄÉÎÉÃÅ (ÞÔÏÂÙ ÓÏÈÒÁÎÑÔØ ÓÒÅÄÎÀÀ ÉÎÔÅÎÓÉ×ÎÏÓÔØ ÐÏ
ÉÚÏÂÒÁÖÅÎÉÀ). òÁÓÛÉÒÉÍ  ÒÁÚÍÅÒ ÍÁÔÒÉÃÙ ÆÉÌØÔÒÁ ÌÁÐÌÁÓÉÁÎÁ ÇÁÕÓÓÉÁÎÙ, ÞÔÏÂÙ ÕÌÕÞÛÉÔØ ÐÒÉÂÌÉÖÅÎÉÅ Ë
ÎÕÌÀ:
\begin{verbatim}
flog = fspecial("log", 17, 1);
imshow(histeq(imfilter(double(I), flog, "symmetric")));
imshow(histeq(imfilter(double(Isaltpepper), flog, "symmetric")));
imshow(histeq(imfilter(double(Inorm), flog, "symmetric")));
\end{verbatim}
ôÅÐÅÒØ ÎÁ ÐÒÉÍÅÒÅ ÐÅÒ×ÏÇÏ ÉÚÏÂÒÁÖÅÎÉÑ ÍÙ ×ÉÄÉÍ, ÞÔÏ ÆÉÌØÔÒ ÒÁÂÏÔÁÅÔ ÐÒÁ×ÉÌØÎÏ.

çÏÒÉÚÏÎÔÁÌØÎÁÑ É ×ÅÒÔÉËÁÌØÎÁÑ ËÏÍÐÏÎÅÎÔÙ ÇÒÁÄÉÅÎÔÁ:
\begin{verbatim}
ih = imfilter(double(Inorm), fsobelh, "symmetric");
iv = imfilter(double(Inorm), fsobelv, "symmetric");
imshow(histeq(iv)));
imshow(histeq(ih));
grad = sqrt(ih.*ih + iv.*iv);
imshow(histeq(grad));
\end{verbatim}

åÓÌÉ ÍÙ ÐÏÐÙÔÁÅÍÓÑ ÐÒÏÄÅÌÁÔØ ÔÏ ÖÅ ÓÁÍÏÅ ÄÌÑ \t{I}, Õ×ÉÄÉÍ, ÞÔÏ \t{histeq} × ÄÁÎÎÏÍ ÓÌÕÞÁÅ ÎÅ
ÓÒÁÂÏÔÁÅÔ, É ÏÔÏÂÒÁÖÁÔØ ÐÒÉÄÅÔÓÑ ËÏÍÁÎÄÏÊ
\begin{verbatim}
imshow(grad,[min(grad(:)) max(grad(:))]);
\end{verbatim}
úÄÅÓØ \t{histeq} ÎÅ ÓÒÁÂÏÔÁÌ ÉÚ-ÚÁ ÓÌÉÛËÏÍ <<ËÒÉ×ÏÊ>> ÇÉÓÔÏÇÒÁÍÍÙ: ÐÏÄÁ×ÌÑÀÝÅÅ ÂÏÌØÛÉÎÓÔ×Ï ÐÉËÓÅÌÅÊ
ÐÒÉÈÏÄÉÔÓÑ ÎÁ ÎÕÌÅ×ÏÊ ÕÒÏ×ÅÎØ.

äÌÑ ÓÍÑÇÞÅÎÉÑ ÏÂÝÅÇÏ ËÏÎÔÒÁÓÔÁ ÍÏÖÎÏ  ÉÓÐÏÌØÚÏ×ÁÔØ ÐÒÉÅÍ, ÎÁÚÙ×ÁÅÍÙÊ <<ÎÅÒÅÚËÉÍ ÍÁÓËÉÒÏ×ÁÎÉÅÍ>>
(ÐÒÉÍÅÎÑÅÍÙÊ ÅÝÅ × ÆÏÔÏÇÒÁÆÉÉ ÎÁ ÆÏÔÏÜÍÕÌØÓÉÀ). äÌÑ ÜÔÏÇÏ ×ÏÓÐÏÌØÚÕÅÍÓÑ ÆÏÒÍÕÌÏÊ
$$I_{out}=N\cdot I_{in} - F(I_{in}),$$
ÇÄÅ $F$~-- ÎÅËÏÔÏÒÙÊ ÕÓÒÅÄÎÑÀÝÉÊ ÆÉÌØÔÒ (× ÏÒÉÇÉÎÁÌÅ~--- ÒÁÓÆÏËÕÓÉÒÏ×ÁÎÎÏÅ ÐÏÚÉÔÉ×ÎÏÅ ÉÚÏÂÒÁÖÅÎÉÅ).
\begin{verbatim}
imshow(2*Inorm-imfilter(Inorm, fgauss, "symmetric"));
\end{verbatim}

ðÏÓÍÏÔÒÉÍ, ËÁË ÉÚÍÅÎÉÔÓÑ ÒÁÚÍÙÔÏÅ ÉÚÏÂÒÁÖÅÎÉÅ ÐÒÉ ÎÅÒÅÚËÏÍ ÍÁÓËÉÒÏ×ÁÎÉÉ:
\begin{verbatim}
ig = imfilter(I, fgauss, "symmetric");
subplot(1,2,1)
imshow(ig)
subplot(1,2,2)
imshow(histeq(2*ig-imfilter(ig, fgauss, "symmetric")))
\end{verbatim}

ïÞÉÓÔÉÍ ×ÓÅ É ×ÅÒÎÅÍÓÑ Ë FITS-ÆÁÊÌÁÍ. úÁÇÒÕÚÉÍ ÉÚÏÂÒÁÖÅÎÉÅ É ÐÏÓÔÒÏÉÍ ÄÉÆÆÅÒÅÎÃÉÁÌØÎÙÅ ÆÉÌØÔÒÙ Ó
ÂÏÌÅÅ ËÒÕÐÎÙÍ ÑÄÒÏÍ:
\begin{verbatim}
clear
[I H] = read_fits_image('stars.fit');
imshow(histeq(I,65536));
fgauss = fspecial("gaussian", 30, 5);
flog = fspecial("log", 37, 5);
\end{verbatim}

ðÏÐÒÏÂÕÅÍ ÔÅÐÅÒØ ÐÒÉÍÅÎÉÔØ Ë ÉÚÏÂÒÁÖÅÎÉÀ ÎÅÒÅÚËÏÅ ÍÁÓËÉÒÏ×ÁÎÉÅ É ÒÁÚÌÉÞÎÙÅ ÆÉÌØÔÒÙ:
\begin{verbatim}
imshow(histeq(I-imfilter(I, fgauss, "symmetric"), 65536));
imshow(histeq(imfilter(I, fgauss, "symmetric"), 65536));
imshow(histeq(imfilter(I, flog, "symmetric"), 65536));
imshow(histeq(2*I-imfilter(I, fgauss, "symmetric"), 65536));
imshow(histeq(medfilt2(I), 65536));
imshow(histeq(I-medfilt2(I, [15 15]), 65536))
\end{verbatim}
ðÏÓÌÅÄÎÉÊ ×ÁÒÉÁÎÔ Ñ×ÌÑÅÔÓÑ ÐÅÒ×ÙÍ ÐÒÉÂÌÉÖÅÎÉÅÍ Ë ×ÙÞÉÔÁÎÉÀ ÆÏÎÁ.

ðÏÓÍÏÔÒÉÍ, ËÁË ÍÏÖÎÏ ÎÁÊÔÉ ÐÒÉÂÌÉÖÅÎÎÕÀ ÍÁÓËÕ ÄÌÑ ÐÏÉÓËÁ Ú×ÅÚÄ:
\begin{verbatim}
Imed = medfilt2(I, [3 3]);
 % [3 3] is default, we can omit it here
ImedLG = imfilter(Imed, flog, "symmetric");
imshow(ImedLG, [-0.1, 0.1]);
mask = logical(zeros(size(I)));
\end{verbatim}

ôÅÐÅÒØ ÎÕÖÎÏ ÐÏÄÏÂÒÁÔØ ÐÏÄÈÏÄÑÝÉÊ ÐÏÒÏÇÏ×ÙÊ ÕÒÏ×ÅÎØ:
\begin{verbatim}
mask = logical(zeros(size(I))); mask(find(ImedLG < 0)) = 1;
imshow(mask)
mask = logical(zeros(size(I))); mask(find(ImedLG < -0.005)) = 1;
imshow(mask)
\end{verbatim}
õÖÅ ÔÁËÏÅ ÇÒÕÂÏÅ ÐÒÉÂÌÉÖÅÎÉÅ ÐÏÚ×ÏÌÉÌÏ ÏÂÎÁÒÕÖÉÔØ ÎÅËÏÔÏÒÏÅ ËÏÌÉÞÅÓÔ×Ï Ú×ÅÚÄ ÎÁ ÉÚÏÂÒÁÖÅÎÉÉ. åÓÌÉ
ÉÓÐÏÌØÚÏ×ÁÔØ ÂÏÌÅÅ ÔÏÞÎÙÊ (<<ÒÏÂÁÓÔÎÙÊ>>) ÍÅÔÏÄ ×ÙÄÅÌÅÎÉÑ ÆÏÎÁ (Á ÚÄÅÓØ ÆÏÎ ÄÏÓÔÁÔÏÞÎÏ ÐÅÒÅËÏÛÅÎ),
ÐÏÌÕÞÁÔÓÑ ÒÅÚÕÌØÔÁÔÙ ËÕÄÁ ÌÕÞÛÅ.

ïÔÏÂÒÁÚÉÍ ÏÒÉÇÉÎÁÌ, ÚÁÍÅÎÉ× ÎÕÌÑÍÉ ×ÓÅ ÔÏÞËÉ, ÇÄÅ ÍÁÓËÁ ÉÓÔÉÎÎÁ, Á ÚÁÔÅÍ~--- ÎÁÏÂÏÒÏÔ:
\begin{verbatim}
J=I; J(mask)=0; % hide stars
imshow(histeq(J, 65536));
J=I; J(~mask)=0; % hide background
imshow(J);
\end{verbatim}
îÁÛÁ ÇÒÕÂÁÑ ÍÁÓËÁ ÏÐÒÅÄÅÌÉÌÁ ×ÓÅ Ú×ÅÚÄÙ, Á ÔÁËÖÅ ÎÅËÏÔÏÒÏÅ ËÏÌÉÞÅÓÔ×Ï ÛÕÍÏ×, ÎÏ ÐÒÉ ÐÏÍÏÝÉ
ÍÏÒÆÏÌÏÇÉÞÅÓËÉÈ ÏÐÅÒÁÃÉÊ ÍÙ ÍÏÖÅÍ ÕÄÁÌÉÔØ ÍÅÌËÉÅ ÏÂßÅËÔÙ (ÐÒÁ×ÄÁ, ÎÅ ÆÁËÔ, ÞÔÏ ÎÅ ÐÏÔÅÒÑÅÍ ÔÁËÉÍ
ÏÂÒÁÚÏÍ ÓÌÁÂÙÅ Ú×ÅÚÄÙ; ËÒÏÍÅ ÔÏÇÏ, ÍÙ ×ÉÄÉÍ ÎÁ ÐÅÒ×ÏÍ ÉÚÏÂÒÁÖÅÎÉÉ, ÞÔÏ ÏÓÏÂÏ ÑÒËÉÅ Ú×ÅÚÄÙ
ÍÁÓËÉÒÏ×ÁÎÙ ÎÅ ÐÏÌÎÏÓÔØÀ, ÎÁÛ ÓÐÏÓÏ ÄÏÓÔÁÔÏÞÎÏ ÐÌÏÈ É ÇÏÄÉÔÓÑ ÌÉÛØ ÄÌÑ ÏÞÅÎØ  ÇÒÕÂÏÊ ÁÓÔÒÏÍÅÔÒÉÉ).

\subsection{íÏÒÆÏÌÏÇÉÞÅÓËÉÅ ÏÐÅÒÁÃÉÉ, ÐÏÉÓË Ó×ÑÚÎÙÈ ÏÂÌÁÓÔÅÊ}
ðÏÐÒÏÂÕÅÍ ÐÏÄÞÉÓÔÉÔØ ÎÅÍÎÏÇÏ ÎÁÛÕ ÍÁÓËÕ.
\begin{verbatim}
imshow(imopen(mask, strel('disk', 1, 0)));
imshow(imopen(mask, strel('disk', 2, 0)));
imshow(imopen(mask, strel('disk', 3, 0))); % use this
mask = imopen(mask, strel('disk', 3, 0));
\end{verbatim}

ôÅÐÅÒØ ÍÏÖÅÍ ÁÎÁÌÏÇÉÞÎÏ ÐÒÏÓÍÏÔÒÅÔØ ÍÁÓËÉÒÏ×ÁÎÎÙÅ ÉÚÏÂÒÁÖÅÎÉÑ:
\begin{verbatim}
J=I; J(mask)=0; % hide stars
imshow(histeq(J, 65536));
J=I; J(~mask)=0; % hide background
imshow(J);
\end{verbatim}

îÏ ÎÁÍ ÉÎÔÅÒÅÓÎÅÊ ×ÙÄÅÌÉÔØ ×ÓÅ Ú×ÅÚÄÙ ÉÚ ÉÚÏÂÒÁÖÅÎÉÑ. éÓÐÏÌØÚÕÅÍ ÐÏÉÓË Ó×ÑÚÎÙÈ ÏÂÌÁÓÔÅÊ:
\begin{verbatim}
[L NUM]= bwlabel(mask, 8);
NUM
NUM =  200
imshow(L, [1 NUM])
\end{verbatim}
ëÁË ×ÉÄÉÍ, ÎÕÍÅÒÁÃÉÑ ÏÂßÅËÔÏ× ÉÄÅÔ ÓÌÅ×Á-ÎÁÐÒÁ×Ï, Ó×ÅÒÈÕ-×ÎÉÚ. þÔÏÂÙ ×ÙÄÅÌÉÔØ ËÏÎËÒÅÔÎÙÊ ÏÂßÅËÔ, ÍÙ
ÍÏÖÅÍ ÐÒÉÍÅÎÉÔØ ÍÁÓËÕ ÔÁËÏÇÏ ÔÉÐÁ:
\begin{verbatim}
J = zeros(size(I)); idx = find(L == 100); J(idx) = I(idx);
imshow(J);
\end{verbatim}
íÙ ×ÙÄÅÌÉÌÉ ÏÄÎÕ Ú×ÅÚÄÕ. íÏÖÅÍ ×ÙÄÅÌÉÔØ ÔÏÎËÕÀ ÐÏÌÏÓËÕ ÆÏÎÁ ×ÏËÒÕÇ ÎÅÅ:
\begin{verbatim}
mask100 = logical(zeros(size(I)));
mask100(idx) = 1;
mask100 = xor(imdilate(mask100, strel('disk',2,0)), mask100);
imshow(mask100);
\end{verbatim}

é ÐÏÓÞÉÔÁÔØ ÐÏ ÎÅÊ ÓÔÁÔÉÓÔÉËÕ:
\begin{verbatim}
bg100 = I(mask100);
mean(bg100)
ans =  2063.1
std(bg100)
ans =  61.711
\end{verbatim}

îÏ ÄÌÑ ×ÙÞÉÓÌÅÎÉÑ ÐÏÌÏÖÅÎÉÊ ÃÅÎÔÒÏÉÄÏ× ÕÄÏÂÎÅÊ ÉÓÐÏÌØÚÏ×ÁÔØ ÄÒÕÇÕÀ ÆÕÎËÃÉÀ ÐÏÉÓËÁ Ó×ÑÚÎÙÈ ÏÂÌÁÓÔÅÊ:
\t{bwconncomp}.
\begin{verbatim}
K = bwconncomp(mask, 8);
p = regionprops(K);
\end{verbatim}
ðÏ ÕÍÏÌÞÁÎÉÀ × \t{p} (ÜÔÏ~--- ÍÁÓÓÉ× ÓÔÒÕËÔÕÒ) ÓÏÄÅÒÖÁÔÓÑ ÐÏÌÑ \t{Area} (ÐÌÏÝÁÄØ ÏÂßÅËÔÁ),
\t{BoundingBox} (ËÏÏÒÄÉÎÁÔÙ ÂÏËÓÁ, × ËÏÔÏÒÏÍ ÓÏÄÅÒÖÉÔÓÑ ÏÂßÅËÔ: ÌÅ×ÙÊ ×ÅÒÈÎÉÊ ÕÇÏÌ É ÒÁÚÍÅÒ) É
\t{Centroid} (ÇÅÏÍÅÔÒÉÞÅÓËÉÊ ÃÅÎÔÒ ÏÂßÅËÔÁ). ðÏ ÐÅÒ×ÙÍ Ä×ÕÍ ÐÁÒÁÍÅÔÒÁÍ ÍÙ ÍÏÖÅÍ ÏÔÆÉÌØÔÒÏ×ÁÔØ
ÓÌÉÛËÏÍ ÍÅÌËÉÅ É ÓÌÉÛËÏÍ ËÒÕÐÎÙÅ ÏÂßÅËÔÙ, Á ÔÁËÖÅ ÏÂßÅËÔÙ, Õ ËÏÔÏÒÙÈ ÏÔÎÏÛÅÎÉÅ ÛÉÒÉÎÙ Ë ×ÙÓÏÔÅ
ÓÉÌØÎÏ ÏÔÌÉÞÁÅÔÓÑ ÏÔ ÅÄÉÎÉÃÙ (Ô.Å.  ÔÁÍ ×ÎÕÔÒÉ~--- Ñ×ÎÏ ÎÅ ËÒÕÇ). ôÒÅÔÉÊ ÐÁÒÁÍÅÔÒ ÐÏÚ×ÏÌÑÅÔ ÓÕÄÉÔØ
Ï ÐÒÉÍÅÒÎÏÍ ÒÁÓÐÏÌÏÖÅÎÉÉ ÃÅÎÔÒÁ ÏÂßÅËÔÁ (ÂÏÌÅÅ ÔÏÞÎÏ, ËÏÎÅÞÎÏ, ÍÙ ÍÏÖÅÍ ×ÙÞÉÓÌÉÔØ ÅÇÏ ÌÉÛØ ×ÚÑ×
ÄÁÎÎÙÅ Ó ÏÒÉÇÉÎÁÌØÎÏÇÏ ÉÚÏÂÒÁÖÅÎÉÑ, ×ÙÞÔÑ ÆÏÎ É ÐÏÓÞÉÔÁ× ÔÏÞÎÙÊ ÃÅÎÔÒÏÉÄ).

íÏÖÎÏ ÎÁÒÉÓÏ×ÁÔØ ÎÅÓËÏÌØËÏ ÂÏËÓÏ×:
\begin{verbatim}
rectangle('Position', p(1).BoundingBox, 'EdgeColor','g','LineWidth',2)
rectangle('Position', p(3).BoundingBox, 'EdgeColor','g','LineWidth',2)
rectangle('Position', p(100).BoundingBox, 'EdgeColor','g','LineWidth',2)
\end{verbatim}

\begin{verbatim}
centroids = cat(1, p.Centroid);
plot(centroids(:,1), centroids(:,2), 'ko');
\end{verbatim}

ôÅÐÅÒØ ÓÒÁ×ÎÉÍ ËÏÏÒÄÉÎÁÔÙ ÃÅÎÔÒÏÉÄÁ ÉÚÏÂÒÁÖÅÎÉÑ Ú×ÅÚÄÙ Ó ËÏÏÒÄÉÎÁÔÁÍÉ ÃÅÎÔÒÏÉÄÁ ÍÁÓËÉ. ïÂÒÁÔÉÔÅ
×ÎÉÍÁÎÉÅ, ÞÔÏ \t{BoundingBox} ÉÍÅÅÔ ËÏÏÒÄÉÎÁÔÙ <<Ó ÐÏÌÏ×ÉÎËÁÍÉ>>:
\begin{verbatim}
p(100).BoundingBox
502.500   710.500    14.000    14.000
imshow(mask100);
rectangle('Position', p(100).BoundingBox, 'EdgeColor','g','LineWidth',2)
\end{verbatim}
üÔÏ Ó×ÑÚÁÎÏ Ó ÔÅÍ, ÞÔÏ ÐÒÉ ÏÔÏÂÒÁÖÅÎÉÉ ÉÚÏÂÒÁÖÅÎÉÊ ÃÅÌÙÍ ËÏÏÒÄÉÎÁÔÁÍ ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÃÅÎÔÒ ÐÉËÓÅÌÑ.
ëÏÎÔÕÒ, ÏÐÉÓÙ×ÁÀÝÉÊ ÎÁÛÕ ÏÂÌÁÓÔØ ÉÎÔÅÒÅÓÁ, ÎÅ ÄÏÌÖÅÎ ÐÅÒÅÓÅËÁÔØ ÐÉËÓÅÌÉ, ÉÚ-ÚÁ ÜÔÏÇÏ ËÏÏÒÄÉÎÁÔÙ ÅÇÏ
ÌÅ×ÏÇÏ ÕÇÌÁ ÎÁ 0.5\,ÐÉËÓÅÌÑ ÍÅÎØÛÅ, Á ÛÉÒÉÎÁ~--- ÎÁ 1\,ÐÉËÓÅÌØ ÂÏÌØÛÅ.
óÌÅÄÏ×ÁÔÅÌØÎÏ, ÄÌÑ ÔÏÇÏ, ÞÔÏÂÙ ÐÏÌÕÞÉÔØ ÞÁÓÔØ ÉÚÏÂÒÁÖÅÎÉÑ, ÓÏÄÅÒÖÁÝÕÀ ÔÏÌØËÏ ÐÉËÓÅÌÉ ÚÏÎÙ ÉÎÔÅÒÅÓÁ,
ÎÁÍ ÎÕÖÎÏ ÂÕÄÅÔ ÓÄÅÌÁÔØ ÐÒÅÏÂÒÁÚÏ×ÁÎÉÅ:
\begin{verbatim}
Bstart = p(100).BoundingBox(1:2) + 0.5;
Bsize = p(100).BoundingBox(3:4) - 1;
rectangle('Position', [Bstart Bsize], 'EdgeColor','r','LineWidth',2);
\end{verbatim}
ôÅÐÅÒØ ÎÁÛ ÂÏËÓ ×ËÌÀÞÁÅÔ ×ÓÅ ÐÉËÓÅÌÉ ÉÚÏÂÒÁÖÅÎÉÑ Ú×ÅÚÄÙ. ÷ÙÞÉÓÌÉÍ ÓÕÍÍÁÒÎÙÊ ÐÏÔÏË É ÃÅÎÔÒÏÉÄ:
\begin{verbatim}
bg = mean(bg100); % mean background
[y x] = ind2sub(size(I), idx);
% coordinates of pixels in mask of 100th star
M = sum(I(idx)) - numel(idx)*bg
% energy in circle
M =  55148.54545
xc=0; yc=0; for n=1:numel(x); i=I(idx(n))-bg; xc+=i*x(n); yc+=i*y(n);
printf("%d(%d,%d)\n",i,x(n),y(n)); endfor; xc/=M; yc/=M;
xc
 % X centroid position
xc =  509.36
yc  % Y centroid position
yc =  718.41
p(100).Centroid
 % centroid by mask
ans =
509.63   717.57
imshow(histeq(I, 65536)); % take a look at real image
% real centroid:
rectangle('Position', [xc-1 yc-1 2 2], 'EdgeColor','g','LineWidth',2,
    "Curvature", 1)
% centroid of mask:
rectangle('Position', [p(100).Centroid-1 2 2], 'EdgeColor','r','LineWidth',2,
    "Curvature", 1)
\end{verbatim}

íÙ ×ÉÄÉÍ, ÞÔÏ ÎÁÓÔÏÑÝÉÊ ÃÅÎÔÒ ÔÑÖÅÓÔÉ ÉÚÏÂÒÁÖÅÎÉÑ Ú×ÅÚÄÙ ÄÏÓÔÁÔÏÞÎÏ ÓÉÌØÎÏ ÏÔÌÉÞÁÅÔÓÑ ÏÔ
ÇÅÏÍÅÔÒÉÞÅÓËÏÇÏ ÃÅÎÔÒÁ ÍÁÓËÉ. ïÓÏÂÅÎÎÏ ÓÉÌØÎÏ ÜÔÏ ÐÒÏÑ×ÌÑÅÔÓÑ × ÓÌÕÞÁÅ ÄÅÆÏÒÍÁÃÉÊ ÉÚÏÂÒÁÖÅÎÉÑ
(×ÓÌÅÄÓÔ×ÉÅ ÓÄ×ÉÇÁ ÉÚÏÂÒÁÖÅÎÉÑ, ËÏÍÙ ÉÌÉ ÐÒÏÞÉÈ ÁÂÅÒÒÁÃÉÊ).

÷ÙÞÉÓÌÑÔØ ÂÏÌØÛÏÅ ËÏÌÉÞÅÓÔ×Ï ÈÁÒÁËÔÅÒÉÓÔÉË ÕÄÏÂÎÅÊ ÉÎÁÞÅ. îÁÐÒÉÍÅÒ, ÔÁË:
\begin{verbatim}
regioninfo = regionprops(L,I,'MajorAxisLength','MinorAxisLength','PixelIdxList',
    'PixelValues');
\end{verbatim}
ÍÙ ÐÏÌÕÞÉÍ ÚÎÁÞÅÎÉÑ ÂÏÌØÛÏÊ É ÍÁÌÏÊ ÐÏÌÕÏÓÅÊ ÜÌÌÉÐÓÁ Ó ÉÄÅÎÔÉÞÎÙÍÉ ÎÁÛÅÊ ÆÉÇÕÒÏÊ ÍÏÍÅÎÔÁÍÉ (ÞÔÏ
ÐÏÚ×ÏÌÉÔ ÓÕÄÉÔØ Ï <<ËÒÕÇÌÏÓÔÉ>> ËÁÖÄÏÇÏ ÏÂßÅËÔÁ), ÓÐÉÓÏË ÉÎÄÅËÓÏ× ÐÉËÓÅÌÅÊ É ÓÐÉÓÏË ÉÎÔÅÎÓÉ×ÎÏÓÔÅÊ
ÐÉËÓÅÌÅÊ ÎÁ ÉÚÏÂÒÁÖÅÎÉÉ. åÓÌÉ ÂÙ ÆÏÎ Õ ÎÁÛÅÇÏ ÏÂßÅËÔÁ ÂÙÌ ÉÄÅÁÌØÎÏ ÇÌÁÄËÉÍ, ÍÏÖÎÏ ÂÙÌÏ ÂÙ ÄÏÂÁ×ÉÔØ
ÅÝÅ Ó×ÏÊÓÔ×Ï \t{'WeightedCentroid'}, ÞÔÏÂÙ ÓÒÁÚÕ ÐÏÌÕÞÉÔØ ËÏÏÒÄÉÎÁÔÙ ÃÅÎÔÒÁ ËÁÖÄÏÊ Ú×ÅÚÄÙ, ÎÏ, Õ×Ù,
ÄÌÑ ËÁÖÄÏÊ Ú×ÅÚÄÙ ÐÒÉÄÅÔÓÑ ÓÞÉÔÁÔØ ÆÏÎ ÁÎÁÌÏÇÉÞÎÏ ×ÙÛÅÓËÁÚÁÎÎÏÍÕ~--- ÞÅÒÅÚ ËÏÌØÃÁ. äÌÑ ÔÅÓÎÙÈ ÐÏÌÅÊ
ÐÏÔÒÅÂÕÅÔÓÑ ÕÂÅÄÉÔØÓÑ, ÞÔÏ ÚÏÎÁ ×ÏËÒÕÇ Ú×ÅÚÄÙ Ó×ÏÂÏÄÎÁ ÏÔ ÄÒÕÇÏÊ Ú×ÅÚÄÙ (ÐÒÉ ÐÏÍÏÝÉ ×ÓÅ ÔÅÈ ÖÅ
ÍÏÒÆÏÌÏÇÉÞÅÓËÉÈ ÏÐÅÒÁÃÉÊ).

áÒÇÕÍÅÎÔÏÍ \t{regionprops} ÍÏÖÅÔ ×ÙÓÔÕÐÁÔØ É ÓÔÒÕËÔÕÒÁ, ÐÏÌÕÞÅÎÎÁÑ ÐÒÉ ÐÏÍÏÝÉ \t{bwconncomp}.

ôÅÐÅÒØ ÎÁÊÄÅÍ ×ÓÅ ÏÂßÅËÔÙ ÂÏÌÅÅ-ÍÅÎÅÅ ËÒÕÇÌÏÊ ÆÏÒÍÙ:
\begin{verbatim}
roundness = [regioninfo.MajorAxisLength] ./ [regioninfo.MinorAxisLength];
staridx = find(roundness < 1.1 & roundness > 0.9);
 % indexes of "stars"
imshow(ismember(L, staridx)); % show filtered image
\end{verbatim}

ôÅÐÅÒØ, ÅÓÌÉ ÐÒÏÊÔÉÓØ ÉÔÅÒÁÔÉ×ÎÏ ÐÏ ËÁÖÄÏÍÕ ÏÂßÅËÔÕ Ó ÎÏÍÅÒÏÍ \t{staridx}, ÍÏÖÎÏ ÐÏÓÞÉÔÁÔØ ÄÌÑ ÎÅÇÏ
ÉÎÄÉ×ÉÄÕÁÌØÎÕÀ ÍÁÓËÕ ÆÏÎÁ (ËÁË ÂÙÌÏ ÐÏËÁÚÁÎÏ ×ÙÛÅ~--- ÞÅÒÅÚ ÄÉÌÁÔÁÃÉÀ), ÏÐÒÅÄÅÌÉÔØ ÔÁËÉÍ ÏÂÒÁÚÏÍ
ÕÒÏ×ÅÎØ ÆÏÎÁ É ÉÓÐÏÌØÚÏ×ÁÔØ ÅÇÏ ÄÌÑ ×ÙÞÉÓÌÅÎÉÑ ÐÒÏÞÉÈ ÐÁÒÁÍÅÔÒÏ×: ÐÏÌÎÏÊ ÉÎÔÅÎÓÉ×ÎÏÓÔÉ ×ÎÕÔÒÉ ÍÁÓËÉ
É ËÏÏÒÄÉÎÁÔ ÃÅÎÔÒÏÉÄÁ.

åÓÌÉ ÍÙ ÄÏÂÁ×ÉÍ × \t{regionprops} Ó×ÏÊÓÔ×Ï \t{'Area'}, Õ ÎÁÓ ÐÏÑ×ÉÔÓÑ ×ÏÚÍÏÖÎÏÓÔØ ÄÏ ÐÏÌÕÞÅÎÉÑ
ÎÏÍÅÒÏ× ÏÂßÅËÔÏ×, ×ÏÚÍÏÖÎÏ Ñ×ÌÑÀÝÉÈÓÑ Ú×ÅÚÄÁÍÉ, ÏÔÓÏÒÔÉÒÏ×ÁÔØ ÉÈ ÐÏ ÕÂÙ×ÁÎÉÀ ÐÌÏÝÁÄÉ (ÓËÏÒÅÊ
×ÓÅÇÏ~--- É ÐÏ ÕÂÙ×ÁÎÉÀ ÑÒËÏÓÔÉ):
\begin{verbatim}
[~, i] = sort([regioninfo(staridx).Area], 'descend'); % get sorted indexes
staridxsorted = staridx(i);
\end{verbatim}
é ÍÏÖÅÍ ÏÔÏÂÒÁÚÉÔØ ÏËÒÅÓÔÎÏÓÔÉ ÐÅÒ×ÙÈ 10 ÐÏ ÐÌÏÝÁÄÉ ÍÁÓËÉ:
\begin{verbatim}
imshow(histeq(I, 65536));
for i=1:10;
c = hsv2rgb([i/10. 1 1]);
rectangle('Position', p(staridxsorted(i)).BoundingBox, 'EdgeColor', c,
    'LineWidth',2);
endfor
for i = 1:10; ltxt{i} = sprintf("star #%d", i); endfor
legend(ltxt)
\end{verbatim}
÷ÉÄÉÍ, ÞÔÏ Õ ÑÒËÉÈ Ú×ÅÚÄ ÍÁÓËÁ ÐÏÔÅÒÑÌÁ ÐÒÉÌÉÞÎÕÀ ÞÁÓÔØ ËÒÙÌØÅ×, Á ÔÁËÖÅ ÏÄÎÁ ÄÏÓÔÁÔÏÞÎÏ ÑÒËÁÑ
Ú×ÅÚÄÁ ÂÙÌÁ ÐÒÏÐÕÝÅÎÁ.
âÏÌÅÅ ÎÁÄÅÖÎÙÍ ÂÕÄÅÔ ÏÔÓÏÒÔÉÒÏ×ÁÔØ ÐÏ ÚÎÁÞÅÎÉÀ ÉÎÔÅÎÓÉ×ÎÏÓÔÉ.

\section{ðÅÒ×ÉÞÎÁÑ ÏÂÒÁÂÏÔËÁ ÓÎÉÍËÏ×}
îÁÉÂÏÌÅÅ ÒÁÓÐÒÏÓÔÒÁÎÅÎÎÙÍÉ Ó×ÅÔÏÐÒÉÅÍÎÉËÁÍÉ × ÁÓÔÒÏÆÉÚÉËÅ ÏÐÔÉÞÅÓËÏÇÏ ÄÉÁÐÁÚÏÎÁ Ñ×ÌÑÀÔÓÑ
ðúó-ÍÁÔÒÉÃÙ. óÒÅÄÉ ÐÒÏÞÉÈ Ó×ÅÔÏÐÒÉÅÍÎÉËÏ× ÜÔÏÇÏ ÄÉÁÐÁÚÏÎÁ ÏÎÉ ÉÍÅÀÔ ÎÁÉÌÕÞÛÉÅ ÈÁÒÁËÔÅÒÉÓÔÉËÉ.
ïÄÎÁËÏ, ÅÓÔØ Õ ÎÉÈ É ÎÅÄÏÓÔÁÔËÉ: × ÔÏÎËÉÈ ÐÏÄÌÏÖËÁÈ ×ÏÚÍÏÖÎÁ ÉÎÔÅÒÆÅÒÅÎÃÉÑ (ÏÓÏÂÅÎÎÏ ÂÌÉÖÅ Ë
éë-ÄÉÁÐÁÚÏÎÕ), ÐÒÉ×ÏÄÑÝÁÑ Ë ×ÏÚÎÉËÎÏ×ÅÎÉÑ <<ÆÒÉÎÇÏ×>>; Á ÎÁÌÉÞÉÅ ÈÏÔÑ ÂÙ ÏÄÎÏÇÏ ÏÞÅÎØ ÑÒËÏÇÏ
ÉÓÔÏÞÎÉËÁ ÐÒÉ×ÏÄÉÔ Ë <<ÒÁÓÔÅËÁÎÉÀ ÚÁÒÑÄÁ>>, ÕÓÕÇÕÂÌÑÀÝÅÍÕÓÑ ÐÒÉ ÓÞÉÔÙ×ÁÎÉÉ ÉÚÏÂÒÁÖÅÎÉÑ.
ëíïð-Ó×ÅÔÏÐÒÉÅÍÎÉËÉ × ÚÎÁÞÉÔÅÌØÎÏÊ ÍÅÒÅ ÉÚÂÁ×ÌÅÎÙ ÏÔ ÐÒÏÂÌÅÍ ÒÁÓÔÅËÁÎÉÑ ÚÁÒÑÄÁ, ÎÏ × ÎÉÈ ÅÓÔØ
ÄÒÕÇÁÑ ÐÒÏÂÌÅÍÁ: ËÁÖÄÙÊ ÐÉËÓÅÌØ ÐÏ ÓÕÔÉ ÓÎÁÂÖÅÎ ÓÏÂÓÔ×ÅÎÎÙÍ ÕÓÉÌÉÔÅÌÅÍ, × ÒÅÚÕÌØÔÁÔÅ ÔÏÞÎÁÑ
ÆÏÔÏÍÅÔÒÉÑ Ó ëíïð ÐÒÅ×ÒÁÝÁÅÔÓÑ × ÓÐÌÏÛÎÏÊ ÁÄ!

ðÒÏÃÅÓÓ ÐÏÌÕÞÅÎÉÑ ÉÚÏÂÒÁÖÅÎÉÊ ÎÁ ðúó ÍÏÖÎÏ × ÕÐÒÏÝÅÎÎÏÍ ×ÉÄÅ Ó×ÅÓÔÉ Ë ÆÏÒÍÕÌÅ:
$$ ADU_{x,y}= \gamma F_{x,y}(e_{x,y} + D_{x,y})T +  B_{x,y} + R,$$
(ÚÄÅÓØ ÏÐÕÝÅÎ ÐÒÏÃÅÓÓ ÄÉÓËÒÅÔÉÚÁÃÉÉ, ÒÁÓÔÅËÁÎÉÑ ÚÁÒÑÄÁ É Ô.Ð.), ÇÄÅ $ADU$~-- ÏÔÓÞÅÔÙ, ÓÞÉÔÁÎÎÙÅ Ó
ÐÉËÓÅÌÑ Ó ËÏÏÒÄÉÎÁÔÁÍÉ $(x,y)$; $\gamma$~-- <<gain>>, ËÏÜÆÆÉÃÉÅÎÔ ÐÒÅÏÂÒÁÚÏ×ÁÎÉÑ ÆÏÔÏÜÌÅËÔÒÏÎÏ× ×
$ADU$ (ÚÁ×ÉÓÉÔ ÏÔ ÄÌÉÎÙ ×ÏÌÎÙ ÉÚÌÕÞÅÎÉÑ); $F$~-- ÐÏÇÌÏÝÅÎÉÅ ÉÚÌÕÞÅÎÉÑ ×ÓÌÅÄÓÔ×ÉÅ ÎÅÉÄÅÁÌØÎÏÓÔÉ
ÏÐÔÉÞÅÓËÏÊ ÓÉÓÔÅÍÙ (ËÏÍÐÅÎÓÉÒÕÅÔÓÑ ÄÅÌÅÎÉÅÍ ÎÁ <<ÐÌÏÓËÉÅ ÐÏÌÑ>>); $e$~-- ËÏÌÉÞÅÓÔ×Ï ÆÏÔÏÜÌÅËÔÒÏÎÏ×,
×ÏÚÎÉËÛÉÈ ÐÒÉ ÐÏÐÁÄÁÎÉÉ Ë×ÁÎÔÁ × ÄÁÎÎÕÀ ÑÞÅÊËÕ (ÚÁ×ÉÓÉÔ ÏÔ ÄÌÉÎÙ ×ÏÌÎÙ), ÏÎÏ ÚÁÛÕÍÌÑÅÔÓÑ ÍÎÏÖÅÓÔ×ÏÍ
ÆÁËÔÏÒÏ×: ÆÏÔÏÎÎÙÍ ÛÕÍÏÍ, ÔÕÒÂÕÌÅÎÔÎÏÓÔØÀ ÁÔÍÏÓÆÅÒÙ É Ô.Ä., É Ô.Ð.; $D$~--
<<ÔÅÍÎÏ×ÏÊ ÔÏË>>, ÏÐÒÅÄÅÌÑÀÝÉÊÓÑ ÜÌÅËÔÒÏÎÁÍÉ, ×ÏÚÎÉËÁÀÝÉÍÉ × ÒÅÚÕÌØÔÁÔÅ ÔÅÐÌÏ×ÙÈ ÐÒÏÃÅÓÓÏ× × ÔÅÌÅ
ÐÏÌÕÐÒÏ×ÏÄÎÉËÁ; $B$~-- <<bias>>, ÎÁÞÁÌØÎÏÅ ÓÍÅÝÅÎÉÅ (ÄÁÖÅ ÐÒÉ ÎÕÌÅ×ÏÊ ÜËÓÐÏÚÉÃÉÉ~--- ÓÏÂÓÔ×ÅÎÎÏ,
ÔÁË ËÁÄÒÙ <<bias>> É ÐÏÌÕÞÁÀÔ) ÄÌÑ ÕÓÔÒÁÎÅÎÉÑ ÎÅÌÉÎÅÊÎÏÓÔÉ; $T$~-- ×ÒÅÍÑ ÎÁËÏÐÌÅÎÉÑ ÓÉÇÎÁÌÁ
(ÜËÓÐÏÚÉÃÉÑ); $R$~-- ÛÕÍ ÓÞÉÔÙ×ÁÎÉÑ.

éÚÂÁ×ÉÔØÓÑ ÏÔ ÛÕÍÁ ÓÞÉÔÙ×ÁÎÉÑ ÎÅ×ÏÚÍÏÖÎÏ, ÏÄÎÁËÏ, ËÁË ÍÙ ÐÏÍÎÉÍ, ÅÓÌÉ ÕÓÒÅÄÎÉÔØ $N$~ËÁÄÒÏ×, ÔÏ
$R$~ÕÍÅÎØÛÉÔÓÑ × $\sqrt{N}$~ÒÁÚ. ëÒÏÍÅ ÔÏÇÏ, ÞÅÍ ÂÏÌØÛÅ ÜËÓÐÏÚÉÃÉÑ, ÔÅÍ ÂÏÌØÛÉÍ ÂÕÄÅÔ ×ËÌÁÄ ÐÒÏÞÉÈ
ÛÕÍÏ×. ïÄÎÁËÏ, ÐÒÉ ÍÁÌÙÈ ÜËÓÐÏÚÉÃÉÑÈ ÍÏÖÅÔ ÐÒÏÑ×ÌÑÔØÓÑ <<ÜÆÆÅËÔ ÚÁÔ×ÏÒÁ>>, ×ÙÚ×ÁÎÎÙÊ ÔÅÍ, ÞÔÏ ×ÒÅÍÑ
ÏÔËÒÙ×ÁÎÉÑ\slash ÚÁËÒÙ×ÁÎÉÑ ÚÁÔ×ÏÒÁ ËÏÎÅÞÎÏ.

þÔÏÂÙ ÓËÏÒÒÅËÔÉÒÏ×ÁÔØ ÉÎÔÅÎÓÉ×ÎÏÓÔØ ÎÁ ÕÒÏ×ÅÎØ ÓÍÅÝÅÎÉÑ, ÍÙ ÍÏÖÅÍ ÓÄÅÌÁÔØ ËÁË ÍÏÖÎÏ ÂÏÌØÛÅ ËÁÄÒÏ× Ó
ÎÕÌÅ×ÏÊ ÜËÓÐÏÚÉÃÉÅÊ (<<bias>>), ÐÏÌÕÞÉÔØ ÍÅÄÉÁÎÕ ÓÔÏÐËÉ ÉÚÏÂÒÁÖÅÎÉÊ É ×ÙÞÅÓÔØ ÉÚ ÉÚÏÂÒÁÖÅÎÉÑ. ðÒÉ
ÜÔÏÍ ÛÕÍ bias'Á ÔÁËÖÅ ÂÕÄÅÔ ÕÍÅÎØÛÅÎ × $\sqrt{N}$~ÒÁÚ.

äÌÑ ËÏÒÒÅËÃÉÉ ÎÁ ÔÅÐÌÏ×ÏÊ ÛÕÍ ÎÁÍ ÎÅÏÂÈÏÄÉÍÏ ÎÁËÏÐÉÔØ ËÁË ÍÏÖÎÏ ÂÏÌØÛÅ <<ÄÁÒËÏ×>> (× ÉÄÅÁÌÅ ÉÈ
ÜËÓÐÏÚÉÃÉÑ ÄÏÌÖÎÁ ÂÙÔØ ÔÁËÏÊ ÖÅ, ËÁË ÜËÓÐÏÚÉÃÉÑ ËÁÄÒÁ ÄÁÎÎÙÈ, Á ÕÓÌÏ×ÉÑ ÄÏÌÖÎÙ ÂÙÔØ ÁÂÓÏÌÀÔÎÏ
ÔÁËÉÍÉ ÖÅ: ÔÅÍÐÅÒÁÔÕÒÁ ÞÉÐÁ, ÏËÒÕÖÁÀÝÅÊ ÓÒÅÄÙ É Ô.Ä., É Ô.Ð.). ðÒÉ ÚÁËÒÙÔÏÍ ÚÁÔ×ÏÒÅ ÍÙ ÐÏÌÕÞÉÍ:
$$ ADU_{x,y}= \gamma D_{x,y}T +  B_{x,y} + R,$$
Ô.Å. <<Á×ÔÏÍÁÔÉÞÅÓËÉ>> ÓÀÄÁ ÖÅ ×ËÌÀÞÅÎ ÕÒÏ×ÅÎØ ÓÍÅÝÅÎÉÑ. óÌÅÄÏ×ÁÔÅÌØÎÏ, ÅÓÌÉ ÎÁÛÉ <<ÄÁÒËÉ>> ÉÍÅÀÔ
ÔÕ ÖÅ ÓÁÍÕÀ ÜËÓÐÏÚÉÃÉÀ, ÞÔÏ É ÏÓÎÏ×ÎÏÊ ÎÁÕÞÎÙÊ ËÁÄÒ, ÔÏ ÄÌÑ ÕÄÁÌÅÎÉÑ ÔÅÐÌÏ×ÙÈ ÛÕÍÏ× ÄÏÓÔÁÔÏÞÎÏ
×ÙÞÅÓÔØ ÍÅÄÉÁÎÎÏÅ ÕÓÒÅÄÎÅÎÉÅ ÓÔÏÐËÉ <<ÄÁÒËÏ×>> ÉÚ ÎÁÛÅÇÏ ËÁÄÒÁ. ïÄÎÁËÏ, ÎÅ ×ÓÅÇÄÁ ÜÔÏ ×ÏÚÍÏÖÎÏ:
ÓËÁÖÅÍ, ÐÒÉ ÐÏÌÕÞÅÎÉÉ ÓÐÅËÔÒÁ ×ÙÓÏËÏÇÏ ÒÁÚÒÅÛÅÎÉÑ ÜËÓÐÏÚÉÃÉÑ ÍÏÖÅÔ ÄÌÉÔØÓÑ ÎÅÓËÏÌØËÏ ÞÁÓÏ×!
åÓÔÅÓÔ×ÅÎÎÏ, ÍÙ ÎÅ ÍÏÖÅÍ ÐÏÌÕÞÉÔØ ÄÁÖÅ ÄÅÓÑÔËÁ ÔÅÍÎÏ×ÙÈ ËÁÄÒÏ× Ó ÔÁËÏÊ ÜËÓÐÏÚÉÃÉÅÊ (Ô.Ë. É ÕÓÌÏ×ÉÑ
ÐÏÍÅÎÑÔØÓÑ ÕÓÐÅÀÔ, É ÎÁÞÎÅÔÓÑ ÎÏ×ÁÑ ÎÏÞØ). ÷ ÐÏÄÏÂÎÙÈ ÓÌÕÞÁÑÈ ÐÒÉÂÅÇÁÀÔ Ë ÈÉÔÒÏÓÔÉ: ÔÅÍÎÏ×ÏÊ ÔÏË
ÐÒÏÐÏÒÃÉÏÎÁÌÅÎ ×ÒÅÍÅÎÉ ÜËÓÐÏÚÉÃÉÉ. üÔÏ ÏÚÎÁÞÁÅÔ, ÞÔÏ ÍÏÖÎÏ ÓÄÅÌÁÔØ ÎÅÓËÏÌØËÏ ÓÅÒÉÊ ËÏÒÏÔËÉÈ
ÜËÓÐÏÚÉÃÉÊ (ÓËÁÖÅÍ, ÏÄÎÁ, ÐÑÔØ, ÄÅÓÑÔØ É Ä×ÁÄÃÁÔØ ÍÉÎÕÔ) × ÔÅÞÅÎÉÅ ÄÎÑ, Á ÚÁÔÅÍ ÜËÓÔÒÁÐÏÌÉÒÏ×ÁÔØ ÉÈ.
÷ÏÔ × ÜÔÏÍ ÓÌÕÞÁÅ ÄÌÑ ÐÏÌÕÞÅÎÉÑ ÔÁËÏÇÏ <<ÉÓËÕÓÓÔ×ÅÎÎÏÇÏ ÄÁÒËÁ>> ÎÅÏÂÈÏÄÉÍÏ ÂÕÄÅÔ, ÐÏÍÉÍÏ ×ÓÅÈ
ÏÓÔÁÌØÎÙÈ ËÁÄÒÏ×, ÓÎÉÍÁÔØ ÅÝÅ É bias'Ù. äÌÑ ËÁÖÄÏÊ ÜËÓÐÏÚÉÃÉÉ~$T$ ÍÙ ×ÙÞÉÓÌÑÅÍ ÔÅÍÎÏ×ÏÊ ÔÏË ËÁË
$D_T=M(D_{Ti})-M(B_i)$, ÇÄÅ $M$~-- ÍÅÄÉÁÎÁ. ðÒÉ ÐÏÍÏÝÉ ÌÉÎÅÊÎÏÊ ÜËÓÔÒÁÐÏÌÑÃÉÉ ÍÙ ÐÏÌÕÞÁÅÍ
ÚÎÁÞÅÎÉÅ~$D$ ÄÌÑ ÚÁÄÁÎÎÏÊ ÜËÓÐÏÚÉÃÉÉ. îÏ Ô.Ë. ÜÔÏ ÚÎÁÞÅÎÉÅ ÕÖÅ ÉÚÂÁ×ÌÅÎÏ ÏÔ ÓÍÅÝÅÎÉÑ, ÄÌÑ ËÏÒÒÅËÃÉÉ
ÎÁÍ ÎÅÏÂÈÏÄÉÍÏ ÂÕÄÅÔ ÉÚ ÉÚÏÂÒÁÖÅÎÉÑ ×ÙÞÅÓÔØ ÎÅ ÔÏÌØËÏ~$D$, ÎÏ É~$B$!
üÔÏ~--- ÅÄÉÎÓÔ×ÅÎÎÙÊ ÓÌÕÞÁÊ, ËÏÇÄÁ ÎÁÍ ÎÅÏÂÈÏÄÉÍÏ ÂÕÄÅÔ ÐÏÌÕÞÉÔØ É ÓÔÏÐËÕ ËÁÄÒÏ× ÓÍÅÝÅÎÉÑ (ËÁË
ÇÏ×ÏÒÉÌÏÓØ ×ÙÛÅ, ÅÓÌÉ <<ÔÅÍÎÏ×ÙÅ>> ÓÎÑÔÙ Ó ÔÏÊ ÖÅ ÜËÓÐÏÚÉÃÉÅÊ, ÞÔÏ É ÏÓÎÏ×ÎÏÊ ËÁÄÒ, ÔÏ ÓÍÅÝÅÎÉÅ
Á×ÔÏÍÁÔÉÞÅÓËÉ ËÏÒÒÅËÔÉÒÕÅÔÓÑ ÐÒÉ ×ÙÞÉÔÁÎÉÉ <<ÔÅÍÎÏ×ÙÈ>> ÉÚ <<ÏÂßÅËÔÁ>>).

úÎÁÞÉÔÅÌØÎÏ ÂÏÌÅÅ ÓÌÏÖÎÏÊ Ñ×ÌÑÅÔÓÑ ËÏÒÒÅËÃÉÑ ÎÁ ÎÅÏÄÎÏÒÏÄÎÏÓÔÉ ÏÐÔÉÞÅÓËÏÊ ÓÉÓÔÅÍÙ (×ÐÌÏÔØ ÄÏ
ÐÙÌÉÎÏË ÎÁ ÆÉÌØÔÒÁÈ É ÐÒÏÞÉÈ ÏÐÔÉÞÅÓËÉÈ ÐÏ×ÅÒÈÎÏÓÔÑÈ), $F$. ôÅÒÍÉÎ <<ÐÌÏÓËÏÓÔØ>> × ÄÁÎÎÏÍ ÓÌÕÞÁÅ
ÎÏÓÉÔ Ä×ÏÑËÉÊ ÓÍÙÓÌ: ÄÌÑ ÐÒÑÍÙÈ ÓÎÉÍËÏ× ÜÔÏ~--- ÎÅÏÄÎÏÒÏÄÎÏÓÔØ ÏÓ×ÅÝÅÎÎÏÓÔÉ ÐÏ ËÁÄÒÕ. äÌÑ
ÓÐÅËÔÒÁÌØÎÙÈ~--- ÅÝÅ É ÎÅÏÄÎÏÒÏÄÎÏÓÔØ ËÒÉ×ÏÊ ÓÐÅËÔÒÁÌØÎÏÊ ÞÕ×ÓÔ×ÉÔÅÌØÎÏÓÔÉ. ïÔ×ÌÅÞÅÍÓÑ ÐÏËÁ ÏÔ
ÓÐÅËÔÒÏÓËÏÐÉÉ. ïÓÎÏ×ÎÙÍÉ ÆÁËÔÏÒÁÍÉ, ÉÓËÁÖÁÀÝÉÍÉ ÉÎÔÅÎÓÉ×ÎÏÓÔØ ÏÂßÅËÔÏ× × ËÁÄÒÅ, Ñ×ÌÑÀÔÓÑ
×ÉÎØÅÔÉÒÏ×ÁÎÉÅ (ÜËÒÁÎÉÒÏ×ÁÎÉÅ ÂÏËÏ×ÙÈ ÐÕÞËÏ× ÁÐÅÒÔÕÒÏÊ ÏÐÔÉÞÅÓËÏÇÏ ÔÒÁËÔÁ) É ÐÒÏÞÉÅ ÎÅÏÄÎÏÒÏÄÎÙÅ
Ó×ÏÊÓÔ×Á ÏÐÔÉÞÅÓËÉÈ ÓÉÓÔÅÍ. ÷ ÉÄÅÁÌØÎÏÊ ÓÉÔÕÁÃÉÉ ÄÌÑ ËÏÍÐÅÎÓÉÒÏ×ÁÎÉÑ~$F$ ÎÁÍ ÎÅÏÂÈÏÄÉÍÏ ÐÏÌÕÞÉÔØ
ÓÎÉÍÏË ÁÂÓÏÌÀÔÎÏ ÏÄÎÏÒÏÄÎÏ ÚÁÓ×ÅÞÅÎÎÏÇÏ ÐÒÏÔÑÖÅÎÎÏÇÏ ÏÂßÅËÔÁ Ó ÕÇÌÏ×ÙÍÉ ÒÁÚÍÅÒÁÍÉ, ÚÎÁÞÉÔÅÌØÎÏ
ÐÒÅ×ÏÓÈÏÄÑÝÉÍÉ ÐÏÌÅ ÚÒÅÎÉÑ ÎÁÛÅÊ ÏÐÔÉÞÅÓËÏÊ ÓÉÓÔÅÍÙ. ÷ ÓÌÕÞÁÅ ÔÅÌÅÓËÏÐÏ× Ó ÍÁÌÙÍ ÐÏÌÅÍ ÚÒÅÎÉÑ ÄÌÑ
ÄÁÎÎÙÈ ÃÅÌÅÊ È×ÁÔÁÅÔ ×ÅÞÅÒÎÉÈ ÉÌÉ ÕÔÒÅÎÎÉÈ ÓÎÉÍËÏ× ÎÅÂÁ ÐÒÉ ÍÁËÓÉÍÁÌØÎÏ ÒÁÓÓÅÑÎÎÏÍ ÓÏÌÎÅÞÎÏÍ Ó×ÅÔÅ
×Ï ×ÒÅÍÑ ÇÒÁÖÄÁÎÓËÉÈ ÓÕÍÅÒÅË. ðÒÉ ÂÏÌØÛÏÍ ÐÏÌÅ ÚÒÅÎÉÑ ÎÅÏÂÈÏÄÉÍÏ ÉÓÐÏÌØÚÏ×ÁÔØ ÓÐÅÃÉÁÌØÎÙÅ ÉÓÔÏÞÎÉËÉ
Ó×ÅÔÁ Ó ËÁË ÍÏÖÎÏ ÂÏÌÅÅ ÒÁ×ÎÏÍÅÒÎÙÍ ÏÓ×ÅÝÅÎÉÅÍ. ðÏÓÌÅ ÐÏÌÕÞÅÎÉÑ <<ÐÌÏÓËÉÈ>> ËÁÄÒÏ× ÓÌÅÄÕÅÔ
×ÙÞÉÓÌÉÔØ ÍÅÄÉÁÎÕ ÉÚ ÉÈ ÓÔÏÐËÉ, ×ÙÞÅÓÔØ <<ÔÅÍÎÏ×ÙÅ>>, ÚÁÔÅÍ ÎÏÒÍÉÒÏ×ÁÔØ ÎÁ ÍÁËÓÉÍÕÍ
ÉÎÔÅÎÓÉ×ÎÏÓÔÉ, Á ÄÁÌÅÅ~--- ÒÁÚÄÅÌÉÔØ ÐÏÄÇÏÔÏ×ÌÅÎÎÙÊ (Ó ×ÙÞÅÔÏÍ <<ÔÅÍÎÏ×ÙÈ>>) ÎÁÕÞÎÙÊ ËÁÄÒ ÎÁ
ÐÏÌÕÞÅÎÎÏÅ ÉÚÏÂÒÁÖÅÎÉÅ. ðÒÉ ÜÔÏÍ ÍÙ
ÐÏÌÕÞÁÅÍ:
$$I_{x,y} = \frac{\gamma F_{x,y}e_{x,y}T + R}{\gamma F_{x,y}T + R}\approx  e_{x,y}T.$$
ëÏÎÅÞÎÏ, <<ÐÌÏÓËÉÅ>> ËÁÄÒÙ ÍÙ ÍÏÖÅÍ ÓÎÉÍÁÔØ Ó ÌÀÂÏÊ ÐÏÄÈÏÄÑÝÅÊ ÜËÓÐÏÚÉÃÉÅÊ. îÏ ÄÌÑ ËÏÒÒÅËÃÉÉ ÎÁ
ÔÅÍÎÏ×ÙÅ ÔÏËÉ, ÅÓÌÉ ÜËÓÐÏÚÉÃÉÑ <<ÐÌÏÓËÉÈ>> ÏÔÌÉÞÁÅÔÓÑ ÏÔ ÜËÓÐÏÚÉÃÉÉ <<ÎÁÕÞÎÙÈ>> ËÁÄÒÏ×, ÎÁÍ
ÎÅÏÂÈÏÄÉÍÏ ÂÕÄÅÔ ÓÄÅÌÁÔØ ÅÝÅ ÏÄÉÎ ÎÁÂÏÒ <<ÔÅÍÎÏ×ÙÈ>>.

ëÁË ÐÒÉÍÅÒ ÍÅÄÉÁÎÎÏÇÏ ÕÓÒÅÄÎÅÎÉÑ ÓÔÏÐËÉ ÉÚÏÂÒÁÖÅÎÉÊ, ÓÇÅÎÅÒÉÒÕÅÍ ÚÁÛÕÍÌÅÎÎÙÅ ÉÚÏÂÒÁÖÅÎÉÑ (ÍÏÖÎÏ
ÉÓÐÏÌØÚÏ×ÁÔØ É \t{awgn}):
\begin{verbatim}
for N=1:10; II(:,:,N) = I+100*randn(size(I)); endfor
imshow(histeq(II(:,:,2),65536))
\end{verbatim}

ðÏÄÏÂÎÙÍ ÏÂÒÁÚÏÍ ÆÏÒÍÉÒÕÀÔÓÑ ÎÁ ðúó ÒÅÁÌØÎÙÅ ÉÚÏÂÒÁÖÅÎÉÑ (ÒÁÚ×Å ÞÔÏ ÛÕÍÙ ÍÙ ÎÅÓËÏÌØËÏ
ÐÒÅÕ×ÅÌÉÞÉÌÉ). éÍÅÎÎÏ ÉÚ-ÚÁ ÛÕÍÏ× ÄÌÑ ÐÏÌÕÞÅÎÉÑ ËÁÞÅÓÔ×ÅÎÎÙÈ ËÁÄÒÏ× dark É flat ÎÅÏÂÈÏÄÉÍÏ ÓÄÅÌÁÔØ
ÉÈ ËÁË ÍÏÖÎÏ ÂÏÌØÛÅ! ðÏÓÌÅ ÞÅÇÏ ÓÔÏÐËÁ ÉÚÏÂÒÁÖÅÎÉÊ ÕÓÒÅÄÎÑÅÔÓÑ:
\begin{verbatim}
Imed = median(II, 3);
subplot(2,2,1)
imshow(histeq(II(:,:,2),65536))
subplot(2,2,2)
imshow(histeq(II(:,:,5),65536))
subplot(2,2,3)
imshow(histeq(Imed,65536))
subplot(2,2,4)
imshow(histeq(I,65536))
\end{verbatim}

äÁÖÅ ÄÌÑ ÛÕÍÁ ÔÁËÏÇÏ ÕÒÏ×ÎÑ ÍÙ ÐÒÉÂÌÉÚÉÌÉÓØ Ë ÏÒÉÇÉÎÁÌÕ ÄÏÓÔÁÔÏÞÎÏ ÎÅÐÌÏÈÏ ×ÓÅÇÏ-ÔÏ ÎÁ ÄÅÓÑÔÉ
ÉÚÏÂÒÁÖÅÎÉÑÈ. åÓÌÉ ÖÅ ÉÈ ÓÄÅÌÁÔØ ÈÏÔÑ ÂÙ ÓÏÔÎÀ, ×ÏÓÓÔÁÎÏ×ÌÅÎÉÅ ÂÕÄÅÔ ÅÝÅ ÌÕÞÛÅ. á ÕÞÉÔÙ×ÁÑ ÔÏ, ÞÔÏ
ÍÅÄÉÁÎÎÙÊ ÆÉÌØÔÒ ÈÏÒÏÛÏ ÕÄÁÌÑÅÔ ×ÙÂÒÏÓÙ, ÔÁËÏÊ ÓÐÏÓÏÂ ÏÔÌÉÞÎÏ ÏÞÉÝÁÅÔ ÏÐÏÒÎÙÅ ËÁÄÒÙ ÏÔ ËÏÓÍÉÞÅÓËÉÈ
ÞÁÓÔÉÃ.

é ÄÌÑ ÉÚÏÂÒÁÖÅÎÉÊ ÏÂßÅËÔÁ ÍÙ ÍÏÖÅÍ ×ÙÐÏÌÎÑÔØ ÐÏÄÏÂÎÙÅ ÍÁÎÉÐÕÌÑÃÉÉ ÄÌÑ ÓÎÉÖÅÎÉÑ ÛÕÍÁ ËÁÖÄÏÇÏ
$e_{x,y}$. îÏ, × ÜÔÏÍ ÓÌÕÞÁÅ ÐÒÏÃÅÓÓ
×ÏÓÓÔÁÎÏ×ÌÅÎÉÑ ÂÕÄÅÔ ÎÁÍÎÏÇÏ ÓÌÏÖÎÅÊ, Ô.Ë., ×Ï-ÐÅÒ×ÙÈ, ÔÅÌÅÓËÏÐ ÉÍÅÅÔ ÎÅÉÄÅÁÌØÎÏÅ ÓÏÐÒÏ×ÏÖÄÅÎÉÅ (É
ÏÂßÅËÔÙ ÎÅ ÔÏÌØËÏ ÎÅÍÎÏÇÏ <<ÒÁÚÍÁÚÙ×ÁÀÔÓÑ>>, ÎÏ ÅÝÅ É ÓÍÅÝÁÀÔÓÑ ÏÔ ÜËÓÐÏÚÉÃÉÉ Ë ÜËÓÐÏÚÉÃÉÉ);
×Ï-×ÔÏÒÙÈ, × ÐÒÏÃÅÓÓÅ ÐÏÌÕÞÅÎÉÑ ÉÚÏÂÒÁÖÅÎÉÊ ÍÏÇÕÔ ÉÚÍÅÎÑÔØÓÑ ÕÓÌÏ×ÉÑ (seeing, ÔÅÍÐÅÒÁÔÕÒÁ É ÐÒ.), ×
ÒÅÚÕÌØÔÁÔÅ ÞÅÇÏ PSF ÉÚÏÂÒÁÖÅÎÉÊ ÔÁËÖÅ ÂÕÄÅÔ ÍÅÎÑÔØÓÑ (ÓÌÅÄÏ×ÁÔÅÌØÎÏ, ÐÒÉÄÅÔÓÑ ÓÄÅÌÁÔØ ÎÅËÏÔÏÒÕÀ
ÄÅËÏÎ×ÏÌÀÃÉÀ É ÐÒÉ×ÅÄÅÎÉÅ ÉÚÏÂÒÁÖÅÎÉÑ Ë ÅÄÉÎÏÊ ËÏÏÒÄÉÎÁÔÎÏÊ ÓÉÓÔÅÍÅ, Á ÜÔÏ ÞÒÅ×ÁÔÏ ÏÛÉÂËÁÍÉ
ÉÎÔÅÒÐÏÌÑÃÉÉ).

\section{úÁÄÁÎÉÅ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ×ÙÐÏÌÎÅÎÉÑ}
÷ ÄÁÎÎÙÈ Ë ÚÁÄÁÎÉÀ ÓÏÄÅÒÖÁÔÓÑ ËÁÄÒÙ object, dark É flat. ÷ÙÐÏÌÎÉÔÅ ÓÌÅÄÕÀÝÅÅ.
\begin{enumerate}
\item éÚ <<ÎÁÕÞÎÙÈ>> ËÁÄÒÏ× ×ÙÂÅÒÉÔÅ ÏÄÉÎ Ó ÎÁÉÌÕÞÛÉÍ ËÁÞÅÓÔ×ÏÍ ÉÚÏÂÒÁÖÅÎÉÊ.
\item ÷ÙÞÉÓÌÉÔÅ ÍÅÄÉÁÎÎÙÅ ÚÎÁÞÅÎÉÑ ÔÅÍÎÏ×ÙÈ É ÐÌÏÓËÉÈ, ÐÏÌÕÞÉÔÅ ÎÏÒÍÉÒÏ×ÁÎÎÏÅ ÉÚÏÂÒÁÖÅÎÉÅ.
\item ðÏÄÂÅÒÉÔÅ ÐÏÄÈÏÄÑÝÉÊ ÆÉÌØÔÒ, ËÏÔÏÒÙÊ ÐÏÚ×ÏÌÉÔ ÐÏÌÕÞÉÔØ ÍÁÓËÕ, ×ÙÄÅÌÑÀÝÕÀ ÉÚÏÂÒÁÖÅÎÉÑ Ú×ÅÚÄ.
\item ðÏÄÇÏÔÏ×ØÔÅ m-ÆÁÊÌÙ ÄÌÑ ×ÙÐÏÌÎÅÎÉÑ ÒÕÔÉÎÎÙÈ ÏÐÅÒÁÃÉÊ: ×ÙÞÉÓÌÅÎÉÑ ÍÁÓËÉ ËÏÌØÃÁ ÆÏÎÏ×ÙÈ
ÐÉËÓÅÌÅÊ, ÏÐÒÅÄÅÌÅÎÉÑ ÕÒÏ×ÎÑ ÆÏÎÁ, ÏÓ×ÅÝÅÎÎÏÓÔÉ ÏÔ Ú×ÅÚÄÙ É ËÏÏÒÄÉÎÁÔ ÅÅ ÃÅÎÔÒÏÉÄÁ.
\item ïÔÓÏÒÔÉÒÕÊÔÅ ÍÁÓËÉ ÐÏ ÕÍÅÎØÛÅÎÉÀ ÐÌÏÝÁÄÉ, ÄÌÑ ÄÁÌØÎÅÊÛÉÈ ×ÙÞÉÓÌÅÎÉÊ ÉÓÐÏÌØÚÕÊÔÅ ÐÅÒ×ÙÅ
ÐÑÔØÓÏÔ.
\item ÷ÙÞÉÓÌÉÔÅ ÏÓ×ÅÝÅÎÎÏÓÔÉ ÏÔ Ú×ÅÚÄ × ÐÒÅÄÅÌÁÈ ÐÏÌÕÞÅÎÎÙÈ ÍÁÓÏË. ïÔÓÏÒÔÉÒÕÊÔÅ ÐÏ ÕÍÅÎØÛÅÎÉÀ
ÏÓ×ÅÝÅÎÎÏÓÔÉ. ÷ÙÂÅÒÉÔÅ ÐÅÒ×ÙÅ 20.
\item ÷ÙÐÏÌÎÉÔÅ ÁÓÔÒÏÍÅÔÒÉÀ ÄÌÑ ÜÔÉÈ ÓÁÍÙÈ ÑÒËÉÈ Ú×ÅÚÄ ËÁÄÒÁ, ÐÏÌÕÞÉ× ËÏÏÒÄÉÎÁÔÙ~$(x,y)$ ÃÅÎÔÒÏÉÄÏ×.
\item ðÒÏ×ÅÒØÔÅ ÐÏÌÕÞÅÎÎÙÅ ×ÅÌÉÞÉÎÙ (ÐÒÅÏÂÒÁÚÕÊÔÅ ADU × Ú×ÅÚÄÎÙÅ ×ÅÌÉÞÉÎÙ É ÐÒÏ×ÅÄÉÔÅ ÎÏÒÍÉÒÏ×ËÕ ÎÁ
ÓÔÁÎÄÁÒÔÎÙÅ ÏÂßÅËÔÙ ÉÚ ËÁÔÁÌÏÇÏ×; ÄÌÑ ËÏÏÒÄÉÎÁÔ ÃÅÎÔÒÏÉÄÏ× ÕÞÔÉÔÅ, ÞÔÏ ÉÚÏÂÒÁÖÅÎÉÑ ÐÏ×ÅÒÎÕÔÙ ÎÁ
$90\degr$ É ÏÓØ~$Y$ ÎÁÐÒÁ×ÌÅÎÁ ×ÎÉÚ) ÐÏ ËÁÔÁÌÏÇÁÍ. ïÐÒÅÄÅÌÉÔÅ ÐÏÇÒÅÛÎÏÓÔØ Ó×ÏÉÈ ÒÁÓÞÅÔÏ× (Ú×ÅÚÄÎÙÈ
×ÅÌÉÞÉÎ É ËÏÏÒÄÉÎÁÔ).
\end{enumerate}
äÌÑ ÏÐÒÅÄÅÌÅÎÉÑ ËÁÔÁÌÏÖÎÙÈ ×ÅÌÉÞÉÎ ×ÏÓÐÏÌØÚÕÊÔÅÓØ aladin, xephem, stellarium (É Ô.Ð.) ÉÌÉ on-line
ÓÅÒ×ÉÓÁÍÉ ËÁÔÁÌÏÇÏ×. ðÒÅÏÂÒÁÚÏ×ÁÔØ ËÏÏÒÄÉÎÁÔÙ~$(x,y)$ ÎÁ ÉÚÏÂÒÁÖÅÎÉÉ × $(\alpha, \delta)$ ÍÏÖÎÏ ÐÒÉ
ÐÏÍÏÝÉ ÕÔÉÌÉÔÙ \t{xy2sky} ÉÚ ÐÁËÅÔÁ \t{wcstools} (ÌÉÂÏ ÐÒÉ ÐÏÍÏÝÉ \t{ds9}). îÅ ÚÁÂÙ×ÁÊÔÅ Ï
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