Digital Image Processing Unit-2

7/29/2019 Digital Image Processing Unit-2

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Department: ELECTRONICS &COMMU

Unit: II

Topic name: Image Transforms - 1

Books referred: 01. Digital Image P02. www.wikipedi

03. www.google.c

Image Transforms:

The Fourier Transform is a

image into its sine and cosine com

in the Fourier or frequency domai

Fourier domain image, each point

image.

The Fourier Transform is u

filtering, image reconstruction and

The DFT (Discrete Fourier

not contain all frequencies forming

describe the spatial domain image.

in the spatial domain image, i.e. th

For a square image of size

Where f(a,b) is the imag

function corresponding to each poithe value of each point F(k,l) is o

base function and summing the res

The basic functions are

represents the DC-component of t

1,N-1) represents the highest frequ

In a similar way, the Fourie

Fourier transform is given by:

To obtain the result for th

image point. However, because th

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important image processing tool which is used to

ponents. The output of the transformation repres

n, while the input image is the spatial domain eq

represents a particular frequency contained in the

ed in a wide range of applications, such as image

image compression.

ransform) is the sampled Fourier Transform and

an image, but only a set of samples which is large

The number of frequencies corresponds to the n

image in the spatial and Fourier domain is of the s

N, the two-dimensional DFT is given by:

in the spatial domain and the exponential ter

nt F(k,l) in the Fourier space. The equation can betained by multiplying the spatial image with the

ult.

ine and cosine waves with increasing frequenc

he image which corresponds to the average brigh

ency.

r image can be re-transformed to the spatial doma

e above equations, a double sum has to be calcu

Fourier Transform is separable, it can be written a

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decompose an

nts the image

ivalent. In the

spatial domain

nalysis, image

herefore does

nough to fully

mber of pixels

ame size.

is the basis

interpreted as:corresponding

ies, i.e. F(0,0)

tness and F(N-

in. The inverse

lated for each

,

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LCE/7.5.1/RC 01

TEACHING NOTES

Department: ELECTRONICS &COMMUNICATION ENGINEERING

Unit: II Date:

Topic name: Image Transforms - 3 No. of marks allotted by JNTUK:

Books referred: 01. Digital Image Processing by R C Gonzalez and R E Woods02. www.wikipedia.org

03. www.google.com

The Hadamard transform Hm is a 2m 2m matrix, the Hadamard matrix (scaled by a

normalization factor), that transforms 2m

real numbers xn into 2m real numbers Xk. The Hadamard

transform can be defined in two ways: recursively, or by using the binary (base-2) representation of

the indices n and k.

Recursively, we define the 1 1 Hadamard transform H0 by the identity H0 = 1, and then

define Hm for m > 0 by:

Where the 1/2 is a normalization i.e., sometimes omitted. Thus, other than this

normalization factor, the Hadamard matrices are made up entirely of 1 and 1.

Equivalently, we can define the Hadamard matrix by its (k, n)-th entry by writing,

And

Where the kj and nj are the binary digits (0 or 1) of n and k, respectively. In this case, we

have:

This is exactly the multidimensional DFT, normalized to be unitary, if the inputs and outputs

are regarded as multidimensional arrays indexed by the nj and kj, respectively.

Some examples of the Hadamard matrices follow:

(This H1 is precisely the size-2 DFT. It can also be regarded as the Fourier transform on the

two-element additive group of Z/ (2))

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Topic name: Discrete Cosine Transform No. of marks allotted by JNTUK:


03. www.google.com

Where is bitwise dot product of the binary representations of the numbers i and j. For

example,

Agreeing with the above (ignoring the overall constant). Note that the first row, first column

of the matrix is denoted by H00

The rows of the Hadamard matrices are the Walsh functions.

Discrete Cosine Transform:

A discrete cosine transform (DCT) expresses a sequence of finitely many data points in terms

of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous

applications in science and engineering, from lossy compression of audio and images (where small

high-frequency components can be discarded), to spectral methods for the numerical solution of

partial differential equations. The use of cosine rather than sine functions is critical in these

applications: for compression, it turns out that cosine functions are much more efficient (as

explained below, fewer are needed to approximate a typical signal), whereas for differential

equations the cosines express a particular choice of boundary conditions.

In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform

(DFT), but using only real numbers. DCTs are equivalent to DFTs of roughly twice the length,

operating on real data with even symmetry (since the Fourier transform of a real and even function is

real and even), where in some variants the input and/or output data are shifted by half a sample.

There are eight standard DCT variants, of which four are common.The most common variant of discrete cosine transform is the type-II DCT, which is often

called simply "the DCT"; its inverse, the type-III DCT, is correspondingly often called simply "the

inverse DCT" or "the IDCT". Two related transforms are the discrete sine transforms (DST), which is

equivalent to a DFT of real and odd functions, and the modified discrete cosine transforms (MDCT),

which is based on a DCT of overlapping data.


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Department: ELECTRONICS &COMMU

Unit: II

Topic name: Haar Transform & Hot

Books referred: 01. Digital Image P02. www.wikipedi

03. www.google.c

Haar Transform:

The Haar transform is the s

a function against the Haar wave

cross-multiplies a function against

The Haar transform is der

shown below:

The Haar transform can be

matrix act as samples of finer and f

Compare with the Walsh tr

Hotelling Transform:

In statistics, principal com

formally it is a transform used f

characteristics of the dataset that

'most important', but this is not ne

PCA is also called the Kar

specialty of being the optimal line

However this comes at the price

discrete cosine transform. Unlike o

vectors. Its basis vectors depend o

The principal component

E(x)=0)

(See arg max for the nota

found by subtracting the first k-1 p

And by substituting this as the new

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implest of the wavelet transforms. This transform c

let with various shifts and stretches, like the Fou

sine wave with two phases and many stretches.

ived from the Haar matrix. An example of a 4x4

thought of as a sampling process in which rows o

iner resolution.

ansform, which is also 1/1, but is non-localized.

onents analysis (PCA) is a technique to simplify a

or reducing dimensionality in a dataset while r

contribute most to its variance. These characterist

essarily the case, depending on the application.

unen-Love transform or the Hotelling transfor

ar transform for keeping the subspace that has la

of greater computational requirement, e.g. if co

ther linear transforms, the PCA does not have a fi

the data set.

w1 of a dataset x can be defined as (assuming

tion) With the first k-1 component, the k-th com

incipal components from x:

dataset to find a principal component in:

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ross-multiplies

rier transform

Haar matrix is

the transform

dataset; more

etaining those

ics may be the

. PCA has the

rgest variance.

pared to the

ed set of basis

ero mean, i.e.

ponent can be

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Unit: II Date:

Topic name: Hotelling Transform - 2 No. of marks allotted by JNTUK:


03. www.google.com

A simpler way to calculate the components wi uses the covariance matrix of x, the

measurement vector. By finding the eigenvalues and eigenvectors of the covariance matrix, we find

that the eigenvectors with the largest eigenvalues correspond to the dimensions that have the

strongest correlation in the dataset. The original measurements are finally projected onto the

reduced vector space.

Related (or even more similar than related?) is the calculus of empirical orthogonal functions

(EOF).

Another method of dimension reduction is a self-organizing map.

If PCA is used in pattern recognition an often useful alternative is the linear discriminant

analysis that takes into account the class separability, which is not the case for PCA.


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Digital Image Processing Unit-2

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