19.Define discrete cosine transform.

19.Define discrete cosine transform.

   December 28, 2013       legend

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 The 1-D discrete cosine transform is defined as

For u = 0, 1, 2, . . , N-1. Similarly the inverse DCT is defined as

 

For u = 0, 1, 2, . . , N-1

Where α is

 

 The corresponding 2-D DCT pair is

 For u, v = 0, 1, 2, . . , N-1, and

For x, y= 0, 1, 2, . . , N-1

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18.What are the properties of Slant transform?

18.What are the properties of Slant transform?

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 Properties of Slant transform

 (i) The slant transform is real and orthogonal.

 S = S*

 S^-1 = S^T

(ii) The slant transform is fast, it can be implemented in (N log₂N) operations on an N x 1 vector.

(iii) The energy deal for images in this transform is rated in very good to excellent range.

(iv) The mean vectors for slant transform matrix S are not sequentially ordered for n ≥ 3.

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17.Write about Slant transform.

17.Write about Slant transform.

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 The Slant transform matrix of order N x N is the recursive expression S_n is given by

 Where I_m is the identity matrix of order M x M, and

 The coefficients are

 and

 The slant transform for N = 4 will be

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16.Explain the basic principle of Hotelling transform.

16.Explain the basic principle of Hotelling transform.

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Hotelling transform:

The basic principle of hotelling transform is the statistical properties of vector
representation. Consider a population of random vectors of the form,

And the mean vector of the population is defined as the expected value of x i.e.,

m_x = E{x}

The suffix m represents that the mean is associated with the population of x vectors. The
expected value of a vector or matrix is obtained by taking the expected value of each elememt.
The covariance matrix C_x in terms of x and m_x is given as

C_x = E{(x-m_x) (x-m_x)^T} 

T denotes the transpose operation. Since, x is n dimensional, {(x-m_x) (x-m_x)^T} will be of
n x n dimension. The covariance matrix is real and symmetric. If elements xi and xj are
uncorrelated, their covariance is zero and, therefore, c_ij = c_ji = 0.

For M vector samples from a random population, the mean vector and covariance matrix
can be approximated from the samples by

 and

 

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15.State distributivity and scaling property of 2D DFT

15.State distributivity and scaling property of 2D DFT

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Distributivity:

From the definition of the continuous or discrete transform pair,

and, in general,

In other words, the Fourier transform and its inverse are distributive over addition but not over
multiplication.

Scaling:

For two scalars a and b,

af (x, y) <=> aF(u, v)

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14.State and prove the translation property of 2D-DFT

14.State and prove the translation property of 2D-DFT

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The translation properties of the Fourier transform pair are

and

Where the double arrow indicates the correspondence between a function and its Fourier
Transform,
Equation (1) shows that multiplying f(x, y) by the indicated exponential term and taking
the transform of the product results in a shift of the origin of the frequency plane to the point (u_o,
v_o).

Consider the equation (1) with u_o = v_o = N/2 or
                                              exp[j2Π(u_ox + v_oy)/N] = e^{jΠ(x + y)}
                                                                                   = (-1)^{(x + y)}
and
                                          f(x, y)(-1)^x+y = F(u – N/2, v – N/2)

Thus the origin of the Fourier transform of f(x, y) can be moved to the center of its
corresponding N x N frequency square simply by multiplying f(x, y) by (-1)^x+y . In the one
variable case this shift reduces to multiplication of f(x) by the term (-1)^x. Note from equation (2)
that a shift in f(x, y) does not affect the magnitude of its Fourier transform as,

 

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13.State and prove separability property of 2D-DFT.

13.State and prove separability property of 2D-DFT.

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 The separability property of 2D-DFT states that, the discrete Fourier transform pair can
be expressed in the separable forms. i.e. ,
                                                                                                                                                             (1)

 For u, v = 0, 1, 2 . . . , N – 1, and


                                                                                                                                                             (2)

For x, y = 0, 1, 2 . . . , N – 1

The principal advantage of the separability property is that F(u,v) or f(x,y) can be
obtained in two steps by successive applications of the 1-D Fourier transform or its inverse. This
advantage becomes evident if equation (1) is expressed in the form


                                                                                                                                                         (3)

Where,

(4)

 For each value of x, the expression inside the brackets in eq(4) is a 1-D transform, with
frequency values v = 0, 1, . . . , N-1. Therefore the 2-D function f(x, v) is obtained by taking a
transform along each row of f(x, y) and multiplying the result by N. The desired result, F(u, v), is
then obtained by taking a transform along each column of F(x, v), as indicated by eq(3)

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Free IBPS Bank PO and Clerk coaching, study material, practice tests

Free IBPS Bank PO and Clerk coaching, study material, practice tests

   December 04, 2013       legend

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Hi everyone coaching genie started free IBPS PO and clerk online coaching.I think It is one of the best online coaching service available free. You can visit the website here coachinggenie.blogspot.com. They arranged questions in practice test or mock test style.Infact you will get ibps po and clerk free study material and coaching, great!!

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