Image Transforms Instructed by : J. Shanbezadeh 1Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –B–Basis Image (m 1,m 2 ) –R–Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2–2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –F–Forward Separable Transform –I–Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 2Jamshid Shanbehzadeh

Applications of Image Transforms Extracting Features from Images –In Fourier Transform, the average dc term is proportional to the average image amplitude Image Compression –Dimensionality Reduction 3Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 4Jamshid Shanbehzadeh

Types of Image Transforms Unitary Transforms Fourier Transforms Cosine, Sine, Hartley Transforms Hadamard, Haar Wavelet Transforms Ridglet, Curvelet, Contourlet 5Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 6Jamshid Shanbehzadeh

F(0,0)=f(0,0).A(0,0,0,0)+f(0,1)A(0,1,0,0)+f(0,2)A(0,2,0,0)+….+f(0,N2-1)A(0,N2-1,0,0) 2-D Transforms : the Forward Transform Kernel Forward transform of the N 1 *N 2 image array F(n 1,n 2 ) : به ازای هر m 1 و m 2 یک تصویر پایه ساخته می شود. n 1 و n 2 پیکسلهای تصویر در فضای جدید هستند. 7Jamshid Shanbehzadeh

Basis Image (m 1,m 2 ) 8Jamshid Shanbehzadeh

Basis Image (m 1,m 2 ) 9 Jamshid Shanbehzadeh

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پیکسلهای تصویر اصلی را در پیکسلهای تصویر پایه، نظیر به نظیر در یکدیگر ضرب داخلی می نماییم. 11Jamshid Shanbehzadeh

یعنی کل تصویر را پیمایش می نمایند. مقایسه تبدیل یک بعدی و دوبعدی 12Jamshid Shanbehzadeh

Reverse 2-D Transforms A reverse or inverse transformation provides a mapping from the transform domain to the image space as given by : B(n 1,n 2 ; m 1,m 2 ) : the Inverse Transform Kernel کرنل مورد استفاده در تبدیل تصاویر بایستی معکوس پذیر باشد. 13Jamshid Shanbehzadeh

Basis Inverse Transform Image (m 1,m 2 ) 14Jamshid Shanbehzadeh

به ازای هر n1 و n2 یک تصویر پایه ساخته میشود، اگر تصاویر پایه بر هم عمود باشند. 15Jamshid Shanbehzadeh

2-D Unitary Transforms The transformation is unitary if the following orthonormality conditions are met: 16Jamshid Shanbehzadeh

Inner Product 17Jamshid Shanbehzadeh

Inner Product 18Jamshid Shanbehzadeh

Image Size(IS) =512 X 512 Number of Operations = IS X IS(Mul)+(IS X IS-1) (Addition) for one element =512 X 512(Mul) +(512 X )(Addition) Number of operations for all = 512 X 512 ( 512 X 512(Mul) +(512 X )(Addition) ) ضرب داخلی تصاویر 19Jamshid Shanbehzadeh

Number Operations = 67,108,864(Multiplications)+1,032,192(additions) Image Size(IS) =512 X 512 Block Size(BS) =8 X 8 Number of Blocks(NB) =128 X 128 Size of Basis Image(SBI) =8 X 8 Number of Operations = NB X {BS X SBI(Mul)+(BS-1) (Addition)} =128 X 128{(64 X 64)Mult+63(Addition)} برای کاهش حجم محاسبات، تصاویر را به بلاکهایی تقسیم می نماییم: بلوک بندی تصاویر حجم محاسبات علیرغم کاهش، زیاد است. 20Jamshid Shanbehzadeh

16,384X(512(Mult)+448(additions))=8,388,608(Multi)+7,340,032(Additions) If we perform matrix multiplication, then we have for two N X N matrixes: Number of operations (NO)= N X N {N(Mul) + (N-1)(addition)} Number of Image Blocks (NIB) = Image Size/(NXN) Total Number of Operations(TNO)=NIB X NO Matrix multiplication حجم محاسبات بسیار کاهش می باشد. 21Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 22Jamshid Shanbehzadeh

Separable Transforms The transformation is said to be separable if its kernels can be written in the form Where the kernel subscripts indicate row and column one-dimensional transform operations. 23Jamshid Shanbehzadeh

A separable two-dimensional unitary transform can be computed in two steps: First, a one-dimensional transform is taken along each column of the image, yielding Next, a second one-dimensional unitary transform is taken along each row of P(m 1,m 2 ), giving Separable Transforms 24Jamshid Shanbehzadeh

Separable Transforms 25Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 26Jamshid Shanbehzadeh

Forward Transform F and f denote the matrix and vector representations of a signal array. F and f be the matrix and vector forms of the transformed signal. The two-dimensional unitary transform is given by F=Af Where A is the forward transformation matrix. 27Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 28Jamshid Shanbehzadeh

Reverse Transform The inverse transform is f = Bf B represents the inverse transformation matrix B = A -1 29Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 30Jamshid Shanbehzadeh

Unitary Matrix (Transform) For a unitary transformation, the matrix inverse is given by A -1 = A *T A is said to be a unitary matrix 31Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 32Jamshid Shanbehzadeh

Orthogonal Matrix (Transform) A real unitary matrix is called an orthogonal matrix. For such a matrix, A -1 = A T 33Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 34Jamshid Shanbehzadeh

Separable Transforms If the transform kernels are separable such that Where A R and A C are row and column unitary transform matrices. 35Jamshid Shanbehzadeh

The transformed image matrix can be obtained from the image matrix by Forward Separable Transforms F 36Jamshid Shanbehzadeh

Inverse Separable Transforms The inverse transformation is given by F = B C F B R T Where B C = A C -1 and B R = A R -1 37Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 38Jamshid Shanbehzadeh

Forward Fourier Transform کرنل دوبعدی جدایی پذیر 39Jamshid Shanbehzadeh

مقایسه تبدیل یک بعدی و دوبعدی 40Jamshid Shanbehzadeh

Inverse Fourier Transform Fourier Transform : Inverse Fourier Transform : 41Jamshid Shanbehzadeh

Fourier Transform (Separable) تبدیل دوبعدی را به صورت سینوسی و کسینوسی می نویسیم: 42Jamshid Shanbehzadeh

Fourier Transform (Separable) 43Jamshid Shanbehzadeh

Fourier transform basis functions, N=16 44Jamshid Shanbehzadeh

قسمت حقیقی مقادیر تصاویر پایه DFT قسمت موهومی تصاویر پایه DFT 45Jamshid Shanbehzadeh

تصویر اصلی اندازه تبدیل فوریه تصویر اصلی (مبدا به وسط انتقال یافته است.) اندازه تبدیل فوریه تصویر اصلی با استفاده از لگاریتم اندازه ها فاز تبدیل فوریه 46Jamshid Shanbehzadeh

تصویر اصلی تبدیل فوریه آن 47Jamshid Shanbehzadeh

دو نمونه تصویر اصلی تبدیل فوریه تصاویر چرخش یافته تصاویرتبدیل فوریه تصاویر حساسیت تبدیل فوریه به چرخش 48Jamshid Shanbehzadeh

Fourier Transform Properties The spectral component at the origin of the Fourier domain is equal to N times the spatial average of the image plane. 49Jamshid Shanbehzadeh

Zero-frequency term at the center Multiplying the image function by factor (-1) j+k یعنی با ضرب f(x,y) در (-1) x+y مبدا تبدیل فوریه f(x,y) به مرکز مربع فرکانسی N X N متناظرش انتقال داده می شود. 50Jamshid Shanbehzadeh

The Fourier transform in vector-space form : F = Af f = A* T F f and F are vectors obtained by column scanning the matrices f and F. F and f denote the matrix and vector representations of an image array. F and f be the matrix and vector forms of the transformed image. Fourier transform in vector-space 51Jamshid Shanbehzadeh

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Fourier Transform Properties 53Jamshid Shanbehzadeh

Fourier Transform Properties Substitution u = - u and v = -v 54Jamshid Shanbehzadeh

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a) Original imageb) Phase only image c) Contrast enhanced version of image (b) to show detail Phase data contains information about where objects are in the image Fourier Transform Phase Information 56Jamshid Shanbehzadeh

a) Original imageMagnitude of the Fourier spectrum of (a) Phase of the Fourier spectrum of (a) d) Original image shifted by 128 rows and 128 columns Magnitude of the Fourier spectrum of (d) Phase of the Fourier spectrum of (d) Translation Property با شیفت دادن تصاویر، در فاز حاصل از تبدیل فوریه تغییر ایجاد می شود اما اندازه آن ثابت خواهد ماند. 57Jamshid Shanbehzadeh

g) Original imageMagnitude of the Fourier spectrum of (g) Phase of the Fourier spectrum of (g) These images illustrate that when an image is translated, the phase changes, even though magnitude remains the same. Translation Property 58Jamshid Shanbehzadeh

a) Original image b) Fourier spectrum image of original image c) Original image rotated by 90 degrees d) Fourier spectrum image of rotated image Rotation results in Corresponding Rotations with Image and Spectrum Rotation Property 59Jamshid Shanbehzadeh

The test image has been scaled over unit range Where is the clipping Factor and is the maximum coefficient magnitude. 60Jamshid Shanbehzadeh

Another form of amplitude compression is to take the logarithm of each component as given by Where a and b are scaling constants 61Jamshid Shanbehzadeh

DIRECT REMAPCONTRAST ENHANCED LOG REMAP Cam.pgm An Ellipse Displaying DFT Spectrum with Various Remap Methods 62Jamshid Shanbehzadeh

DIRECT REMAPCONTRAST ENHANCED LOG REMAP House.pgm A Rectangle Displaying DFT Spectrum with Various Remap Methods 63Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 64Jamshid Shanbehzadeh

Cosine Transform Forward Cosine Transform : Inverse Cosine Transform : 65Jamshid Shanbehzadeh

The DCT has been used historically in image compression, such as JPEG In computer imaging we often represent the basis matrices as images, called basis images, where we use various gray values to represent the different values in the basis matrix The basis images are separable Cosine Transform 66Jamshid Shanbehzadeh

Cosine transform basis functions, N=16. 67Jamshid Shanbehzadeh

Cosine transform basis images, N=4. 68Jamshid Shanbehzadeh

Cosine Transform 69Jamshid Shanbehzadeh

64 تصویر پایه جهت محاسبه تبدیل کسینوسی گسسته: تصویر پایه مربوط به مولفه (3و3) تصویر پایه مربوط به مولفه (7و7) تصویر پایه مربوط به مولفه (5و5) تصویر پایه مربوط به مولفه (7و3) تصاویر پایه تبدیل کسینوسی گسسته 70 Jamshid Shanbehzadeh

تصویر اصلی تبدیل کسینوسی 71Jamshid Shanbehzadeh

بازسازی تصویر توسط قسمتی از ضرایب DCT الف) بازسازی شده توسط ضرائبی واقع در یک مثلث به ضلع 128 پیکسل (0/125 ضرایب) با 1/1071=MSE ب) بازسازی شده توسط 64 سطر اول و 64 ستون اول (0/4375 ضرایب) با 1/4708=MSE پ) بازسازی شده توسط 32 سطر اول و 32 ستون اول (0/2343 ضرایب) با 1/2087=MSE الف)ب) پ) 72Jamshid Shanbehzadeh

بازسازی تصویر توسط قسمتی از ضرایب DCT Leads to Ringing effect Good reconstruction results in applying Cosine Transform in compression 73Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 74Jamshid Shanbehzadeh

Sine Transform 75Jamshid Shanbehzadeh

Sine transform basis functions, N=15. 76Jamshid Shanbehzadeh

Two-dimensional Sine transform 77Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 78Jamshid Shanbehzadeh

Hartley Transform 79Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 80Jamshid Shanbehzadeh

Hadamard Transform The Hadamard Transform is based on the Hadamard matrix, which is a square array of plus and minus 1s whose rows and columns are orthogonal. 81Jamshid Shanbehzadeh

Hadamard Transform A normalized N X N Hadamard matrix satisfies the relation : HH T = 1 82Jamshid Shanbehzadeh

Hadamard Transform 83Jamshid Shanbehzadeh

Hadamard Transform 84Jamshid Shanbehzadeh

Hadamard Transform Basis Function, N=16. حجم تغییرات بسیار بالاست. 85Jamshid Shanbehzadeh

Hadamard Transform Basis Images, N=16. 86Jamshid Shanbehzadeh

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Hadamard Transform 89Jamshid Shanbehzadeh

Contents Introduction Applications of Image Transforms Types of Image Transforms 2-D Transform –Basis Image (m 1,m 2 ) –Reverse 2-D Transform Basis Inverse Transform Image (m 1,m 2 ) –2-D Unitary Transform Separable Transform Forward Transform Reverse Transform Unitary Matrix(Transform) Orthogonal Matrix(Transform) Separable Transform –Forward Separable Transform –Inverse Separable Transform Fourier Transform –Forward Fourier Transform –Inverse Fourier Transform –Fourier Transform(Separable) –Fourier Transform Basis Functions –Fourier Transform Properties –Fourier Transform Phase Information –Translation Property –Rotation Property Cosine transform –Basis functions –Basis Images –Cosine symmetry Sine Transform –Basis functions –2-D sine transform Hartley Transform Hadamard Transform –Basis Functions –Basis Images Principle Components Analysis 90Jamshid Shanbehzadeh

Principal Component Analysis K= تعداد تصاوير =ابعاد تصاويرmXn تعداد پيکسلهای تصوير N = mXn ابعاد تصوير تبدیلی است مخصوص به یک نوع تصویر خاص مثلا تصویر چهره 91Jamshid Shanbehzadeh

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ماتریس حاصله ماتریس کواریانس بوده و متقارن می باشد. اگر دو سطر از ماتریس اصلی ناهمبسته باشند، آنگاه عنصر نظیر صفر خواهد شد. 93Jamshid Shanbehzadeh

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Main Images 95Jamshid Shanbehzadeh

Basis Images 96Jamshid Shanbehzadeh

Images Generated From 10 Basis Images 97Jamshid Shanbehzadeh

Images Generated From 15 Basis Images 98Jamshid Shanbehzadeh