2012 09 21_video_imageanalysis_lecture02
TRANSCRIPT
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Анализ изображений и видео
Наталья Васильева [email protected] HP Labs Russia
21 сентября 2012, Computer Science Center
Лекция 2: Основы пространственной и частотной обработки изображений
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2 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Обработка изображений
Обработка изображений Image Processing
Изображение Изображение
• На входе и выходе – изображения • Результат обработки «лучше» оригинала с точки зрения конкретного применения
• Лучше с эстетической точки зрения • Лучше для последующего анализа
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3 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Примеры
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4 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Примеры
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5 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
План лекции
• Пространственная область • Частотная область, преобразование Фурье • Обработка в пространственной области • Обработка в частотной области
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6 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Представление цифровых изображений (recap)
Цветное растровое изображение:
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7 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Пространственная область
= + +
= + +
f(x,y)
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8 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Представим «одномерную картинку»
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9 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
1-D изображение
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Частотное представление – основная идея
= ∑
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Преобразование Фурье для изображений – основная идея
• Любое изображение может быть представлено, как сумма синусов и косинусов различной амплитуды и частоты
• Частоты слагаемых характеризуют изображение:
• Яркость «сильно скачет» на небольших участках изображения – будут преобладать слагаемые с высокими частотами
• Яркость плавно изменяется – будут преобладать низкие частоты
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12 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Преобразование Фурье
= + +
f(x) F1*g1(x) F2*g2(x) F3*g3(x)
• Преобразование исходного представления изображения, как функции f(x), в частотное представление – набор Fi
• Преобразование обратимо
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13 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Преобразование Фурье
Прямое преобразование Фурье непрерывной фукнции одной переменной f(x):
Обратное преобразование Фурье:
g(x,u)
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Двумерный случай
Базисные функции: g(x, y, u, v)
Обратное преобразование:
Прямое преобразование
-1
-0.5
0
0.5
1
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Визуализация Фурье-спектра
• Фурье-спектр: набор всех |F(u,v)|
• Визуализация спектра – чем выше значение F(u,v), тем «светлее» точка с координатами (u,v)
• Светлый центр спектра – исходное изображение содержит в основном однородные области, без перепадов яркости
• Светлая периферия спектра – изображение содержит много локальных перепадов яркости
http://www1.idc.ac.il/toky/imageproc-10/lectures/fft_2d.pptx
u=0, v=0 u=1, v=0 u=2, v=0 u=-2, v=0 u=-1, v=0
u=0, v=1 u=1, v=1 u=2, v=1 u=-2, v=1 u=-1, v=1
u=0, v=2 u=1, v=2 u=2, v=2 u=-2, v=2 u=-1, v=2
u=0, v=-1 u=1, v=-1 u=2, v=-1 u=-2, v=-1 u=-1, v=-1
u=0, v=-2 u=1, v=-2 u=2, v=-2 u=-2, v=-2 u=-1, v=-2
U
V
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Визуализация Фурье-спектра
f(x,y) F(u,v)
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Примеры
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18 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Еще примеры
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Обработка в пространственной области
• Обработка в пространственной области – манипулирование пикселями изображения
• Например, инвертирование
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20 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Гистограммы
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21 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Гистограммы
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22 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Гистограммы - коррекция
• Линейное преобразование – линейное «растяжение» гистограммы, устойчивое растяжение
• Нелинейное преобразование • Эквализация (линеаризция) гистограммы
∑∑==
===
−==
k
i
ik
iixkk
kkx
nnxpxfy
Lknnxp
00)()(
1,..,1,0,)(
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23 © Copyright 2012 Hewlett-Packard Development Company, L.P. The information contained herein is subject to change without notice.
Результат эквализации гистограммы
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Результат эквализации гистограммы
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Пороговая бинаризация
Светлый объект на темном фоне
Два светлых объекта на темном фоне
Глобальная – порог единый для всех точек изображения Локальная или Динамическая – когда порог зависит от координат точки (x,y) Адаптивная – когда порог зависит от значения яркости в точке I(x,y)
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Глобальная бинаризация
• Выбор порога вручную
• Выбор порога автоматически 1. Случайно выбрать начальное значение порога T0 2. Сегментировать изображение по порогу T0: регионы
G1 и G2 из пикселей со значениями >T0 и ≤ T0 3. Вычислить средние значения µ1 and µ2 для регионов
G1 and G2 4. T1 = 0.5 (µ1 + µ2) 5. Повторять пока | Ti - Ti+1|< Tth
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Примеры бинаризации
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Выделение компонент связности
0 0 0
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Компоненты связности
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Фильтрация (свертка изображения с фильтром)
Операция свертки:
f – изображение w – ядро, фильтр g – результат свертки f*w
Свойства: • коммутативность: f*w = w*f • ассоциативность: f*(w1*w2)=(f*w1)*w2 • дистрибутивность по сложению: f*(w1+w2=f*w1 + f*w2 • kf*w = f*kw = k(f*w)
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Теорема о свертке
g = f * h g = f h
implies implies
G = F H G = F * H
Slide: http://www1.idc.ac.il/toky/imageproc-10/lectures/fft_2d.pptx
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Теорема о свертке
Problem in Frequency
Space
Original Problem
Solution in Frequency
Space
Solution of Original Problem
Relatively easy solution
Difficult solution
Fourier Transform
Inverse Fourier Transform
Slide: http://www1.idc.ac.il/toky/imageproc-10/lectures/fft_2d.pptx
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Сглаживание
• Линейные усредняющие фильтры – удаление «случайного шума»
• Фильтры, основанные на порядковых статистиках • Медианный фильтр (подавление шума «соль и перец»)
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Сглаживание фильтром Гаусса
Свертка с ядром Гаусса
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Сглаживание фильтром Гаусса: пример
Sigma =1.4 Size = 5
Sigma =2.8 Size = 10
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Выделение деталей
A point has been detected if |g| ≥ T, • T is a nonnegative threshold
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Обнаружение линий
• If |gi| > |gj| for all j≠i – the point is within line i. • Use one mask to detect lines of a given direction
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Выделение границ: примеры
Sobel Canny Исходное
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Обнаружение границ
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Градиент изображения Градиент направлен в сторону наибольшего изменения интенсивности
Направление градиента:
Величина градиента:
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Вычисление градиента изображения
Roberts: Prewitt: Sobel:
Дискретный случай:
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Пример
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Обнаружение контуров: вычисление производных
Вычисление второй производной: Лапласиан
•Маски Лапласиана:
2
2
2
22 ),(),(
yyxf
xyxff
∂∂
+∂
∂=∇
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Слайд: А. Конушин
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Слайд: А. Конушин
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Mexican hat
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Заключение
• Пространственная область • Частотная область, преобразование Фурье, теорема о свертке
• Обработка в пространственной и частотной областях • Гистограммы, бинаризация, выделение связных компонент, сглаживание, повышение резкости, выделение контуров