image transforms instructed by : j. shanbezadeh email : [email protected] 1jamshid shanbehzadeh
TRANSCRIPT
Image Transforms
Instructed by :
J. ShanbezadehEmail :
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