Two Dimensional Transformations and Viewing Questions and Answers

Q: In general pivot point rotation what is the correct sequence of transformations ?

The sequence is Translation, rotation, translation.Basic rotation formulas turn a point about the origin. To rotate about any other pivot point (xr, yr), the object is first translated so that the pivot moves to the origin, then rotated about the origin, then translated back so the pivot returns to its original position. The composite matrix is \(T(x_r, y_r) \cdot R(\theta) \cdot T(-x_r, -y_r)\).Scaling plays no part in a rotation, and rotating twice around a translation does not keep the pivot fixed, so the other three sequences are wrong.

Q: The transformation matrix \(\begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) is used to obtain the reflection about the.......... .

The matrix gives the reflection about the diagonal line Y = -x.Apply it to a point in homogeneous coordinates:\[ \begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \begin{bmatrix} -y \\ -x \\ 1 \end{bmatrix} \]The point (x, y) goes to (-y, -x), which is its mirror image in the line y = -x. For example (2, 0) goes to (0, -2).Reflection about y = x swaps the coordinates without changing signs (the same matrix with +1s). Reflection about the y-axis has diagonal (-1, 1, 1), and about the line y = 0 (the x-axis) has diagonal (1, -1, 1).

Q: A transformation that distorts the shape of an object such that the transformed shape appears as if the object was composed of internal layers that had be caused to slide over each other is called as ......... .

The transformation is shear.A shear shifts each point in proportion to its distance from a fixed axis, for example \(x' = x + sh_x \cdot y\). The effect is as if the object were made of thin layers sliding over one another, turning a square into a parallelogram.Reflection mirrors an object and translation moves it, both without distortion. Dither is a technique for simulating extra colours or shades with patterns of pixels, not a geometric transformation.

In general pivot point rotation what is the correct sequence of transformations ?

A.Rotation, translation, rotation
B.Rotation, scaling, rotation
C.Scaling, rotation, translation
D.Translation, rotation, translation

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: D

The sequence is Translation, rotation, translation.

Basic rotation formulas turn a point about the origin. To rotate about any other pivot point (xr, yr), the object is first translated so that the pivot moves to the origin, then rotated about the origin, then translated back so the pivot returns to its original position. The composite matrix is \(T(x_r, y_r) \cdot R(\theta) \cdot T(-x_r, -y_r)\).

Scaling plays no part in a rotation, and rotating twice around a translation does not keep the pivot fixed, so the other three sequences are wrong.

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The transformation matrix \(\begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) is used to obtain the reflection about the.......... .

A.Y-axis, flipping x-coordinates
B.line Y = 0
C.Diagonal line Y = -x
D.Diagonal line Y = x

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: C

The matrix gives the reflection about the diagonal line Y = -x.

Apply it to a point in homogeneous coordinates:

\[ \begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \begin{bmatrix} -y \\ -x \\ 1 \end{bmatrix} \]

The point (x, y) goes to (-y, -x), which is its mirror image in the line y = -x. For example (2, 0) goes to (0, -2).

Reflection about y = x swaps the coordinates without changing signs (the same matrix with +1s). Reflection about the y-axis has diagonal (-1, 1, 1), and about the line y = 0 (the x-axis) has diagonal (1, -1, 1).

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A transformation that distorts the shape of an object such that the transformed shape appears as if the object was composed of internal layers that had be caused to slide over each other is called as ......... .

A.Reflection
B.Shear
C.Dither
D.Translation

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: B

The transformation is shear.

A shear shifts each point in proportion to its distance from a fixed axis, for example \(x' = x + sh_x \cdot y\). The effect is as if the object were made of thin layers sliding over one another, turning a square into a parallelogram.

Reflection mirrors an object and translation moves it, both without distortion. Dither is a technique for simulating extra colours or shades with patterns of pixels, not a geometric transformation.

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The aspect ratio of an image is defined as

A.The ratio of width to its height measured in unit length.
B.The ratio of height to width measured in number of pixels.
C.The ratio of depth to width measured in unit length.
D.The ratio of width to depth measured in number of pixels.

Exam Year(s)

UGC NET - 2012,

Answer: A

The aspect ratio of an image is the ratio of its width to its height, measured in the same unit of length, such as 4 : 3 or 16 : 9. The statement about the ratio of height to width, or those with depth, do not define the aspect ratio: depth is a third dimension that a flat image does not have, and the height-to-width ratio is its inverse. The keyed answer is the first option, the ratio of width to its height measured in unit length.

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The transform which possesses the "multi-resolution" property is

A.Fourier transform
B.Short-time-Fourier transform
C.Wavelet transform
D.Karhunen-Loere transform

Exam Year(s)

UGC NET - 2012,

Answer: C

The wavelet transform has the multi-resolution property: it analyses a signal at different scales, using wide windows for low frequencies and narrow windows for high frequencies. The Fourier transform has no localisation in time, the short-time Fourier transform has a window of fixed size and so a fixed resolution, and the Karhunen-Loeve transform is a data dependent transform used for decorrelation. The keyed answer is the wavelet transform.

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If the Fourier transform of the function \(f(x, y)\) is \(F(m, n)\), then the Fourier transform of the function \(f(2x, 2y)\) is :

A.\(\frac{1}{4}F\left(\frac{m}{2}, \frac{n}{2}\right)\)
B.\(\frac{1}{4}F(2m, 2n)\)
C.\(\frac{1}{4}F(m, n)\)
D.\(\frac{1}{4}F\left(\frac{m}{4}, \frac{n}{4}\right)\)

Exam Year(s)

UGC NET - 2012,

Answer: A

For the two dimensional Fourier transform, scaling the function in space scales its transform in the opposite way: if f(x, y) has the transform F(m, n), then f(ax, by) has the transform

\[ \frac{1}{|ab|} F\left(\frac{m}{a}, \frac{n}{b}\right) \]

With a = b = 2 this gives \(\frac{1}{4}F(m/2, n/2)\). The option with F(2m, 2n) has scaled the wrong way, the one with F(m, n) has no scaling, and the one with m/4, n/4 has used a factor of 4 for the arguments. The keyed answer is the first option.

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An example of a dictionary-based coding technique is

A.Run-length coding
B.Huffman coding
C.Predictive coding
D.LZW coding

Exam Year(s)

UGC NET - 2012,

Answer: D

LZW coding is a dictionary based coding technique: it builds a dictionary of strings as it reads the data, and replaces repeated strings with the codes of the dictionary entries. Run-length coding replaces runs of the same symbol, Huffman coding assigns variable length codes by symbol frequency, and predictive coding encodes the difference from a predicted value, so none of them is dictionary based. The keyed answer is LZW coding.

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You are given four images represented as

\(I_1 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\)

\(I_2 = \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}\)

\(I_3 = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\)

\(I_4 = \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}\)

The value of entropy is maximum for image

A.\(I_1\)
B.\(I_2\)
C.\(I_3\)
D.\(I_4\)

Exam Year(s)

UGC NET - 2012,

Answer: C

The entropy of a binary image depends on the proportion of 1s and 0s, and is highest when they are equally likely. For \(I_1\) all four pixels are 0, so the entropy is 0. For \(I_2\) one pixel in four is 1, and for \(I_4\) three in four are 1, so both give

\[ H = -\tfrac{1}{4}\log_2\tfrac{1}{4} \]\[ \quad - \tfrac{3}{4}\log_2\tfrac{3}{4} \approx 0.81 \]

For \(I_3\) two pixels are 1 and two are 0, so p = 1/2 and the entropy is 1 bit, the maximum. The keyed answer is \(I_3\).

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