image transforms instructed by : j. shanbezadeh email : shanbehzadeh@gmail.com 1jamshid shanbehzadeh

Image Transforms

Instructed by :

J. ShanbezadehEmail :

Shanbehzadeh@gmail.com

1Jamshid Shanbehzadeh

Contents• Introduction• Applications of Image Transforms• Types of Image Transforms• 2-D Transform

– Basis Image (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis2Jamshid 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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis4Jamshid 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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis6Jamshid 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 N1*N2 image array F(n1,n2) :

هر • ازای می m2و m1به ساخته پایه تصویر یکشود.

•n1 وn2. هستند جدید فضای در تصویر پیکسلهای7Jamshid Shanbehzadeh

Basis Image (m1,m2)

8Jamshid Shanbehzadeh

Basis Image (m1,m2)

9Jamshid Shanbehzadeh

10Jamshid Shanbehzadeh

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

می پیمایش را تصویر کل یعنی.نمایند

دوبعدی و بعدی یک تبدیل مقایسه

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(n1,n2; m1,m2) : the Inverse Transform Kernel

معکوس بایستی تصاویر تبدیل در استفاده مورد کرنلباشد .پذیر

13Jamshid Shanbehzadeh

Basis Inverse Transform Image (m1,m2)

14Jamshid Shanbehzadeh

هر ازای هم n2و n1به بر پایه تصاویر اگر میشود، ساخته پایه تصویر یک. باشند 15Jamshid Shanbehzadehعمود

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 512 -1)(Addition)

Number of operations for all = 512 X 512 (512 X 512(Mul) +(512 X 512 -1)(Addition) )

تصاویر داخلی ضرب

19Jamshid Shanbehzadeh

Number Operations = 67,108,864(Multiplications)+1,032,192(additions)

Image Size(IS) =512 X 512Block Size(BS) =8 X 8Number of Blocks(NB) =128 X 128Size 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)}

بالکهایی به را تصاویر محاسبات، حجم کاهش براینماییم می :تقسیم

تصاویر بندی بلوک

است زیاد کاهش، علیرغم محاسبات .حجم

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

باشد کاهشمی بسیار محاسبات .حجم

21Jamshid Shanbehzadeh

Contents• Introduction• Applications of Image Transforms• Types of Image Transforms• 2-D Transform

– Basis Image (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis22Jamshid 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(m1,m2), giving

Separable Transforms

24Jamshid Shanbehzadeh

Separable Transforms

25Jamshid Shanbehzadeh

Contents• Introduction• Applications of Image Transforms• Types of Image Transforms• 2-D Transform

– Basis Image (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis26Jamshid Shanbehzadeh

Forward TransformF 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 byF=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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis28Jamshid 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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis30Jamshid 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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis32Jamshid Shanbehzadeh

Orthogonal Matrix (Transform)

A real unitary matrix is called an orthogonal matrix.

For such a matrix,A-1 = AT

33Jamshid Shanbehzadeh

Contents• Introduction• Applications of Image Transforms• Types of Image Transforms• 2-D Transform

– Basis Image (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis34Jamshid Shanbehzadeh

Separable Transforms

If the transform kernels are separable such that

Where AR and AC are row and column unitary transform matrices.

RC AAA

35Jamshid Shanbehzadeh

The transformed image matrix can be obtained from the image matrix by

Forward Separable Transforms

TRCFAAF

36Jamshid Shanbehzadeh

Inverse Separable Transforms

The inverse transformation is given by

F = BC F BRT

Where BC = AC-1 and BR = AR

-1

37Jamshid Shanbehzadeh

Contents• Introduction• Applications of Image Transforms• Types of Image Transforms• 2-D Transform

– Basis Image (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis38Jamshid Shanbehzadeh

Forward Fourier Transform

پذیر جدایی دوبعدی کرنل

39Jamshid Shanbehzadeh

دوبعدی و بعدی یک تبدیل مقایسه

40Jamshid Shanbehzadeh

Inverse Fourier Transform

Fourier Transform :

Inverse Fourier Transform :

41Jamshid Shanbehzadeh

Fourier Transform (Separable)

و سینوسی صورت به را دوبعدی تبدیلنویسیم می :کسینوسی

42Jamshid Shanbehzadeh

Fourier Transform (Separable)

43Jamshid Shanbehzadeh

Fourier transform basis functions , N=16

44Jamshid Shanbehzadeh

پایه تصاویر مقادیر حقیقی پایه DFTقسمت تصاویر موهومی DFTقسمت

45Jamshid Shanbehzadeh

اصلی .)تصویر است ( یافته انتقال وسط به مبدا اصلی تصویر فوریه تبدیل اندازه

ها اندازه لگاریتم از استفاده با اصلی تصویر فوریه تبدیل فوریه اندازه تبدیل فاز

46Jamshid Shanbehzadeh

اصلی آن تصویر فوریه تبدیل

47Jamshid Shanbehzadeh

اصلی تصویر نمونه تصاویر دو فوریه تصاویر تبدیل یافته فوریه چرخش تبدیلتصاویر

چرخش به فوریه تبدیل حساسیت

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

ضرب با فوریه x+y)1در (-f(x,y)یعنی تبدیل مربع f(x,y)مبدا مرکز به.N X Nفرکانسی شود می داده انتقال متناظرش

50Jamshid Shanbehzadeh

The Fourier transform in vector-space form : F = Aff = A*TF

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

52Jamshid Shanbehzadeh

Fourier Transform Properties

53Jamshid Shanbehzadeh

Fourier Transform Properties

Substitution u = - u and v = -v

54Jamshid Shanbehzadeh

55Jamshid Shanbehzadeh

a) Original image b) 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 image Magnitude 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

تصاویر، دادن شیفت باتبدیل از حاصل فاز درمی ایجاد تغییر فوریه

ثابت آن اندازه اما شودماند .خواهد

57Jamshid Shanbehzadeh

g) Original image Magnitude 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 0.1),(0.0 kjF

Where is the clipping Factor and is the maximum coefficient magnitude.

0.10.0 c maxf

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 REMAP CONTRAST ENHANCED LOG REMAP

Cam.pgm

An Ellipse

Displaying DFT Spectrum with Various Remap Methods

62Jamshid Shanbehzadeh

DIRECT REMAP CONTRAST 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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis64Jamshid 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: گسسته کسینوسی تبدیل محاسبه جهت پایه تصویر

مولفه ( به مربوط پایه )3و3تصویر

مولفه ( به مربوط پایه )7و7تصویر

مولفه ( به مربوط پایه )5و5تصویر

مولفه به مربوط پایه تصویر)3و7(

کسینوسی تبدیل پایه تصاویرگسسته

70Jamshid Shanbehzadeh

اصلی کسینوسی تصویر تبدیل

71Jamshid Shanbehzadeh

ضرایب از قسمتی توسط تصویر DCTبازسازیضلع) به مثلث یک در واقع ضرائبی توسط شده بازسازی ) 0/125پیکسل (128الف با ضرایب

1/1071=MSEتوسط) شده بازسازی و 64ب اول اول (64سطر ) 0/4375ستون با MSE=1/4708ضرایب

توسط) شده بازسازی و 32پ اول اول (32سطر ) 0/2343ستون با MSE=1/2087ضرایب

پ)ب) الف) 72Jamshid Shanbehzadeh

ضرایب از قسمتی توسط تصویر DCTبازسازی

• 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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis74Jamshid 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 (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis78Jamshid Shanbehzadeh

Hartley Transform

79Jamshid Shanbehzadeh

Contents• Introduction• Applications of Image Transforms• Types of Image Transforms• 2-D Transform

– Basis Image (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis80Jamshid 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 :

HHT = 1

82Jamshid Shanbehzadeh

Hadamard Transform

83Jamshid Shanbehzadeh

Hadamard Transform

84Jamshid Shanbehzadeh

Hadamard Transform Basis Function , N=16.

باالست بسیار تغییرات .حجم85Jamshid Shanbehzadeh

Hadamard Transform Basis Images, N=16.

86Jamshid Shanbehzadeh

87Jamshid Shanbehzadeh

88Jamshid Shanbehzadeh

Hadamard Transform

89Jamshid Shanbehzadeh

Contents• Introduction• Applications of Image Transforms• Types of Image Transforms• 2-D Transform

– Basis Image (m1,m2)

– Reverse 2-D Transform• Basis Inverse Transform Image (m1,m2)

– 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 Analysis90Jamshid Shanbehzadeh

Principal Component Analysis

• K= تصاوير تعداد• = تصاوير mXnابعاد

),,,,,( 21 ki xxxxX تصوير پيکسلهای ابعاد N = mXnتعدادتصوير

تصویر نوع یک به مخصوص است تبدیلیچهره تصویر خاصمثال

91Jamshid Shanbehzadeh

92Jamshid Shanbehzadeh

می متقارن و بوده کواریانس ماتریس حاصله ماتریسباشد.

آنگاه باشند، ناهمبسته اصلی ماتریس از سطر دو اگر. شد خواهد صفر نظیر عنصر 93Jamshid Shanbehzadeh

94Jamshid Shanbehzadeh

Main Images

95Jamshid Shanbehzadeh

Basis Images

96Jamshid Shanbehzadeh

Images Generated From 10 Basis Images

97Jamshid Shanbehzadeh

Images Generated From 15 Basis Images

98Jamshid Shanbehzadeh