This set of Digital Image Processing Questions and Answers for Entrance exams focuses on “Laplacian in Frequency Domain”.

1. The expression [∂^{2} f(x,y)/∂x^{2} +∂^{2} f(x,y)/∂y^{2}] is considered as _________ where f(x, y) is an input image.

a) Laplacian of f(x, y)

b) Gradient of f(x, y)

c) All of the mentioned

d) None of the mentioned

View Answer

Explanation: The Laplacian for an image f(x, y) is defined as: ∇

^{2}f=∂

^{2}f/∂x

^{2}+ ∂

^{2}f/∂y

^{2}.

2. If the Laplacian in frequency domain is: where is the Fourier transform operator and F(u, v) is Fourier transformed function of f(x, y), then what is -(u^{2}+ v^{2}) is considered as?

a) Laplacian operation

b) Filtering operation

c) Shift operation

d) None of the mentioned

View Answer

Explanation: The Laplacian in frequency domain is simply implemented by using filter:

H(u, v)= -(u

^{2}+ v

^{2}).

3. The Laplacian in frequency domain is simply implemented by using filter __________

a) H(u, v)= -(u^{2}– v^{2})

b) H(u, v)= -(1)

c) H(u, v)= -(u^{2}+ v^{2})

d) none of the mentioned

View Answer

Explanation: Laplacian in frequency domain is: I[(∂

^{2}f(x,y))/∂x

^{2}+(∂

^{2}f(x,y))/∂y

^{2}]= -(u

^{2}+v

^{2})F(u,v), where ℑ is the Fourier transform operator and F(u, v) is Fourier transformed function of f(x, y) and -(u

^{2}+ v

^{2}) is the filter.

4. Assuming that the origin of F(u, v), Fourier transformed function of f(x, y) an input image, has been correlated by performing the operation f(x, y)(-1)x+y prior to taking the transform of the image. If F and f are of same size, then what does the given operation is/are supposed to do?

a) Resize the transform

b) Rotate the transform

c) Shifts the center transform

d) All of the mentioned

View Answer

Explanation: The given operation f(x, y)(-1)x+y shifts the center transform so that (u, v)=(0,0) is at point (M/2, N/2) for F and f of same size M*N.

5. Assuming that the origin of F(u, v), Fourier transformed function of f(x, y) an input image, has been correlated by performing the operation f(x, y)(-1)x+y prior to taking the transform of the image. If F and f are of same size M*N, where does the point (u, v) =(0,0) shifts?

a) (M -1, N -1)

b) (M/2, N/2)

c) (M+1, N+1)

d) (0, 0)

View Answer

Explanation: The given operation f(x, y)(-1)x+y shifts the center transform so that (u, v)=(0, 0) is at point (M/2, N/2) for F and f of same size M*N.

6. Assuming that the origin of F(u, v), Fourier transformed function of f(x, y) an input image, has been correlated by performing the operation f(x, y)(-1)x+y prior to taking the transform of the image. If F and f are of same size M*N, then which of the following is an expression for H(u, v), the filter used for implementing Laplacian in frequency domain?

a) H(u, v)= -(u^{2}+ v^{2})

b) H(u, v)= -(u^{2}– v^{2})

c) H(u, v)= -[(u – M/2)^{2}+ (v – N/2)^{2}].

d) H(u, v)= -[(u – M/2)^{2}– (v – N/2)^{2}].

View Answer

Explanation: The given operation f(x, y)(-1)x+y shifts the center transform so that (u, v)=(0, 0) is at point (M/2, N/2) and hence the filter is: H(u, v)= -[(u – M/2)

^{2}+ (v – N/2)

^{2}].

7. Computing the Fourier transform of the Laplacian result in spatial domain is equivalent to multiplying the F(u, v), Fourier transformed function of f(x, y) an input image, and H(u, v), the filter used for implementing Laplacian in frequency domain. This dual relationship is expressed as _________

a) Fourier transform pair notation

b) Laplacian

c) Gradient

d) None of the mentioned

View Answer

Explanation: The Fourier transform of the Laplacian result in spatial domain is equivalent to multiplying the F(u, v) and H(u, v). This dual relationship is expressed as Fourier transform pair notation given by: ∇

^{2}f(x,y)-[(u – M/2)

^{2}+ (v – N/2)

^{2}]F(u,v), for an image of size M *N.

8. Computing the Fourier transform of the Laplacian result in spatial domain is equivalent to multiplying the F(u, v), Fourier transformed function of f(x, y) an input image of size M*N, and H(u, v), the filter used for implementing Laplacian in frequency domain. This dual relationship is expressed as Fourier transform pair notation given by_____________

a) ∇^{2} f(x,y)↔[(u –M/2)^{2}+ (v –N/2)^{2}]F(u,v)

b) ∇^{2} f(x,y)↔-[(u+M/2)^{2}– (v+N/2)^{2}]F(u,v)

c) ∇^{2} f(x,y)↔-[(u –M/2)^{2}+ (v –N/2)^{2}]F(u,v)

d) ∇^{2} f(x,y)↔[(u+M/2)^{2}– (v+N/2)^{2}]F(u,v)

View Answer

Explanation: The Fourier transform of the Laplacian result in spatial domain is equivalent to multiplying the F(u, v) and H(u, v). This dual relationship is expressed as Fourier transform pair notation given by:∇

^{2}f(x,y)↔-[(u – M/2)

^{2}+ (v – N/2)

^{2}]F(u,v), for an image of size M*N.

9. An enhanced image can be obtained as: g(x,y)=f(x,y)-∇^{2} f(x,y), where Laplacian is being subtracted from f(x, y) the input image. What does this conclude?

a) That the center spike would be negative

b) That the immediate neighbors of center spike would be positive.

c) All of the mentioned

d) None of the mentioned

View Answer

Explanation: For the above given enhanced image the Laplacian subtraction suggest that the center coefficient of Laplacian mask is negative and so the center spike is negative with its immediate neighbors being positive.

10. An enhanced image can be obtained as: g(x,y)=f(x,y)-∇^{2} f(x,y), where Laplacian is being subtracted from f(x, y) the input image of size M*Non which an operation f(x, y)(-1)x+yis applied.Unlike enhancing in spatial domain with one single mask, it is possible to perform the same in frequency domain using one filter. Which of the following is/are the required filter(s)?

a) H(u, v)= -[1 + u^{2}+ v^{2}].

b) H(u, v)= -[(u – M/2)2+ (v– N/2)2].

c) H(u, v)= [1 + (u – M/2)2+ (v – N/2)2].

d) All of the mentioned

View Answer

Explanation: The filter H(u, v)= [1 + (u – M/2)2+ (v – N/2)2] is used to perform the same enhancement in frequency domain like in spatial domain.

11. Why is scaling of Laplacian filtered images necessary?

a) Because it contain high positive values

b) Because it contain high negative value

c) Because it contain both positive and negative values

d) None of the mentioned

View Answer

Explanation: A Laplacian filtered image contain both positive and negative values of comparable magnitudes. So, scaling is necessary.

12. Which of the following fact is true for the masks that includes diagonal neighbors than the masks that doesn’t?

a) Mask that excludes diagonal neighbors has more sharpness than the masks that doesn’t

b) Mask that includes diagonal neighbors has more sharpness than the masks that doesn’t

c) Both masks have same sharpness result

d) None of the mentioned

View Answer

Explanation: Including diagonal neighbor pixels enhances sharpness of the image. So, Mask that includes diagonal neighbors has more sharpness than the masks that doesn’t.

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