Consider an 8 × 8 image f(x, y) given below. Divide the image into 2 x 2 blocks and apply 2 x 2 DCT to each block. Assume that F(u, v) denotes the 8 × 8 matrix containing the transform coefficients. Show the complete form of F(u, v). 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100
Consider an 8 × 8 image f(x, y) given below. Divide the image into 2 x 2 blocks and apply 2 x 2 DCT to each block. Assume that F(u, v) denotes the 8 × 8 matrix containing the transform coefficients. Show the complete form of F(u, v). 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100 100 -100
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. **Consider the Following Computational Problem:**
We are given an 8 × 8 image matrix denoted as \( f(x, y) \). The task involves dividing this image matrix into 2 × 2 blocks and then applying a 2 × 2 Discrete Cosine Transform (DCT) to each of these blocks. Assume that \( F(u, v) \) represents the 8 × 8 matrix containing the transform coefficients. You are asked to present the complete form of \( F(u, v) \).
**Original 8 × 8 Image Matrix \( f(x, y) \):**
\[
\begin{bmatrix}
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
\end{bmatrix}
\]
**Instruction:**
1. Divide this 8 × 8 matrix into sixteen 2 × 2 blocks.
2. Perform a 2 × 2 DCT on each block.
3. Populate \( F(u, v) \) with the DCT coefficients.
**Note:** Detailed computation of the DCT and compilation of the transform matrix \( F(u, v) \) are required for a comprehensive solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbffa73e6-1761-4c9e-97cb-0c164f013989%2F050bad2c-a2b6-467b-b26b-ae47be1eae3d%2F71tuzd7_processed.png&w=3840&q=75)
Transcribed Image Text:1. **Consider the Following Computational Problem:**
We are given an 8 × 8 image matrix denoted as \( f(x, y) \). The task involves dividing this image matrix into 2 × 2 blocks and then applying a 2 × 2 Discrete Cosine Transform (DCT) to each of these blocks. Assume that \( F(u, v) \) represents the 8 × 8 matrix containing the transform coefficients. You are asked to present the complete form of \( F(u, v) \).
**Original 8 × 8 Image Matrix \( f(x, y) \):**
\[
\begin{bmatrix}
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
100 & -100 & 100 & -100 & 100 & -100 & 100 & -100 \\
\end{bmatrix}
\]
**Instruction:**
1. Divide this 8 × 8 matrix into sixteen 2 × 2 blocks.
2. Perform a 2 × 2 DCT on each block.
3. Populate \( F(u, v) \) with the DCT coefficients.
**Note:** Detailed computation of the DCT and compilation of the transform matrix \( F(u, v) \) are required for a comprehensive solution.
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