image transforms instructed by : j. shanbezadeh email : [email protected] 1jamshid shanbehzadeh

Image Transforms

Instructed by :

J. ShanbezadehEmail :

[email protected]

1Jamshid Shanbehzadeh

mailto:[email protected]

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


Types of Image Transforms

• Unitary Transforms• Fourier Transforms• Cosine, Sine, Hartley Transforms• Hadamard, Haar• Wavelet Transforms• Ridglet, Curvelet, Contourlet


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)


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


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


Basis Inverse Transform Image (m1,m2)


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:


Inner Product


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


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


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


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.


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



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.


Reverse Transform

The inverse transform is f = Bf

B represents the inverse transformation matrix B = A-1


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


Orthogonal Matrix (Transform)

A real unitary matrix is called an orthogonal matrix.

For such a matrix,A-1 = AT



If the transform kernels are separable such that

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

RC AAA


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

Forward Separable Transforms

TRCFAAF


Inverse Separable Transforms

The inverse transformation is given by

F = BC F BRT

Where BC = AC-1 and BR = AR

-1


Forward Fourier Transform

پذیر جدایی دوبعدی کرنل


دوبعدی و بعدی یک تبدیل مقایسه


Inverse Fourier Transform

Fourier Transform :

Inverse Fourier Transform :


Fourier Transform (Separable)

و سینوسی صورت به را دوبعدی تبدیلنویسیم می :کسینوسی


Fourier Transform (Separable)


Fourier transform basis functions , N=16


پایه تصاویر مقادیر حقیقی پایه DFTقسمت تصاویر موهومی DFTقسمت


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


اصلی آن تصویر فوریه تبدیل


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

چرخش به فوریه تبدیل حساسیت


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.


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


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



Substitution u = - u and v = -v


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


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


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


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


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


Another form of amplitude compression is to take the logarithm of each component as given by

Where a and b are scaling constants


DIRECT REMAP CONTRAST ENHANCED LOG REMAP

Cam.pgm

An Ellipse

Displaying DFT Spectrum with Various Remap Methods


DIRECT REMAP CONTRAST ENHANCED LOG REMAP

House.pgm

A Rectangle

Displaying DFT Spectrum with Various Remap Methods


Cosine Transform

Forward Cosine Transform :

Inverse Cosine Transform :


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


Cosine transform basis functions, N=16.


Cosine transform basis images, N=4.


Cosine Transform


64: گسسته کسینوسی تبدیل محاسبه جهت پایه تصویر

مولفه ( به مربوط پایه )3و3تصویر



مولفه به مربوط پایه تصویر)3و7(

کسینوسی تبدیل پایه تصاویرگسسته


اصلی کسینوسی تصویر تبدیل


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


Sine Transform


Sine transform basis functions, N=15.


Two-dimensional Sine transform


Hartley Transform


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.


Hadamard Transform

A normalized N X N Hadamard matrix satisfies the relation :

HHT = 1


Hadamard Transform


Hadamard Transform Basis Function , N=16.

باالست بسیار تغییرات .حجم85Jamshid Shanbehzadeh

Hadamard Transform Basis Images, N=16.


Hadamard Transform


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


می متقارن و بوده کواریانس ماتریس حاصله ماتریسباشد.

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

Main Images


Basis Images


Images Generated From 10 Basis Images


Images Generated From 15 Basis Images


image transforms instructed by : j. shanbezadeh email : [email protected] 1jamshid shanbehzadeh

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