Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain Joseph Fourier was born in 1768 (France) “the analytic theory of heat”, 1822(his book) Fourier’s contribution <<any function that periodically repeats itself can be expressed as the sum of sines/or cosines of different frequencies (Fourier series)>> <<even functions that are not periodic (but whose area under the curve is finite) can be expressed as integral of sines and or cosines (Fourier transform)

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.2. Introduction to the Fourier Transform and the Frequency domain 4-2-1. the One-Dimensional Fourier Transform and its inverse

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain روابط يک بعدي به سادگي قابل تعميم به دو بعد هستند

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain تبديل فوريه يک تابع گسسته يک بعدي از رابطه زير حاصل مي شود. يکي از خصوصيات مهم تبديل فوريه گسسته اين است که همواره موجود است و بر خلاف حالت پيوسته وجود آن به شرط خاصي نياز ندارد.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain تبديل فوريه يک تابع گسسته (رابطه اويلر) نکته مهم: براي محاسبه هر ترم از تبديل فوريه تمام مقادير f دخالت دارند. تبديل فوريه به يک منشور رياضي شباهت دارد که يک سيگنال را به مولفه هايش تفکيک مي کند.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain مولفه هاي تبديل فوريه مختلط هستند F(u) in polar coordinate Magnitude or spectrum of FT Phase angel or phase spectrum of FT Power spectrum or spectral density

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain Example 4.1: M=1024 K=16 A=1

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain Two-Dimensional DFT and its inverse

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain Two-Dimensional DFT and its inverse Magnitude or spectrum of FT Phase angel of FT Power spectrum

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain Two-Dimensional DFT and its inverse Due to the properties of exponentials, it is not difficult to show اين رابطه به اين معني است که مي توان تبديل فوريه يک تصوير را به وسط منتقل کرد DC component

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain براي نمايش بهتر فوريه تصوير از تبديل لگاريتمي استفاده شده است.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.2.3. Filtering in the Frequency domain Some basic properties of the frequency domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain Basics of filtering in the frequency domain Multiply the input image by (-1)x+y to center the transform Compute F(u,v), the DFT of the image Multiply F(u,v) by a filter function H(u,v) Compute the inverse DFT Obtain the real part of the result Multiply the result by (-1)x+y

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain Some basic filter and their properties Notch filter (a constant function with a hole at the origin) Low-pass filter High-pass filter

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.2.4. Correspondence between filtering in the spatial and frequency domains The most fundamental relationship between the spatial and frequency domains is established by a well-known result called the “convolution theorem”

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain براي تکميل بحث به اطلاعات ديگري نيز نياز داريم که در ادامه به آنها مي پردازيم

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain براي تکميل بحث به اطلاعات ديگري نيز نياز داريم که در ادامه به آنها مي پردازيم

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain LOW PASS FILTER Let H(u) denote a frequency domain, Gaussian filter function Corresponding filter in the spatial domain is 1)هر دو تابع گوسي هستند با مقادير حقيقي 2) اگر انحراف معيار تابع گوسي در حوزه فرکانس زياد باشد، انحراف معيار حوزه مکان کم خواهد بود و بر عکس

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain HIGH PASS FILTER Let H(u) denote a frequency domain, Gaussian filter function Corresponding filter in the spatial domain is

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.3 Smoothing Frequency-Domain Filters 4.3.1. Ideal Low-pass filter(ILPF)

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain D0 is Cut-off frequency فرکانس قطع فيلتر ايده آل توسط ادوات الکترونيک قابل پياده سازي نيست، اما بصورت نرم افزاري مي توان آن را پياده کرد

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain D0 is Cut-off frequency پيدا کردن نقطه قطع با استفاده از طيف قدرت امکان پذير است. Power Spectrum به ازاي شعاعهاي متفاوت، مقادير مختلفي براي آلفا بدست مي آيد.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain استفاده از فيلتر ايده آل پايين گذر بخاطر وجود خواص بلورينگ و Ringing عملي و ميسر نيست. اين خصوصيت در شکل صفحه قبل مشهود است. در شکل صفحه بعد دليل خصوصيت Ringing به تصوير کشيده شده است.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.3.2. Butterworth low-pass filters(BLPF) If H(u,v)=0.5 then D(u,v)=D0 n is order of filter

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain فيلتر باترورث با مرتبه 20، خصوصياتي بسيار نزديک به فيلتر ايده آل دارد.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.3.2. Gaussian Low-pass filter(GLPF) Where D0 is the cutoff frequency. When D(u,v)=D0, the filter is down to 0.607 of its maximum value.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.3.4.Additional Examples of Lowpass filter

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain نرم کردن تصوير ماهواره براي پردازش هاي بعدي

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4. Sharpening Frequency Domain Filters

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4.1. Ideal High-pass filter (IHPF)

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4.2. Butterworth High-pass filter (BHPF)

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4.2. Gaussian High-pass filter (GHPF)

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4.4. The Laplacian in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4.4. The Laplacian in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain We form an enhanced image g(x,y) by subtracting the laplacian from the original image:

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4.5. Unsharp masking, High-Boost filtering, and High Frequency Emphasis Filtering Unsharp masking: توليد تصوير شارپ از طريق کم کردن تصوير اصلي از تصوير بلور شده، لذا داريم: لذا داريم:

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.4.5. Unsharp masking, High-Boost filtering, and High Frequency Emphasis Filtering در حوزه فوريه: لذا داريم:

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain فيلتر high-boost در حوزه فوريه

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain High-frequency Emphasis filtering: گاهي اوقات لازم است که مقادير فرکانس بالا با اهميت بيشتري ارتقا يابد. در اين مواقع از اين تکنيک استفاده مي شود.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.5. Homomorphic filtering استفاده از مدل illumination-reflectance مي تواند در حذف نويز از تصاوير با رنج ديناميکي کم موثر باشد. اين فيلتر بر مبناي جداسازي مولفه هاي لومينانس و رفلکتانس و پردازش آنها عمل مي کند.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.5. Homomorphic filtering

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.5. Homomorphic filtering

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain 4.5. Homomorphic filtering

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain مولفه illumination داراي تغييرات مکاني کم است در حاليکه مولفه reflectance داراي تغييات شديد خصوصا در نواحي نزديک به لبه ها است. با استفاده از اين روش مولفه illumination حذف و مولفه reflectance تقويت مي شود.

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Chapter 4 Image Enhancement in the Frequency Domain