This question has Statement 1 and Statement 2. Of the four choices given after the statements, choose the one that best describes the two statements.
Statement1:The separability of orthogonal transformation matrix reduces the computational complexity from O(n) to 0(2n³)
Statement 2: Orthogonal transformation of Image can be thought of linear summation of set of orthogonal basis images where coefficients of linear summation are estimated by projecting image onto the orthogonal basis images.
a) Statement 1 is True and Statement 2 is False
b) Statement 1 is False and Statement 2 is True
c) Statement 1 is True and Statement 2 is True
d) Statement 1 is False and Statement 2 is False

Let's analyze the two statements one by one.

**Statement 1: The separability of orthogonal transformation matrix reduces the computational complexity from \( O(n) \) to \( O(2n^3) \)**

- The time complexity of applying an orthogonal transformation is generally not \( O(n) \), and in many cases, separability reduces computational complexity, often from a higher order to a lower order, but not to \( O(2n^3) \). Instead, separability of a transformation matrix can reduce a problem that would otherwise require \( O(n^4) \) operations to a problem requiring \( O(n^3) \) operations, or sometimes even lower, depending on the structure of the problem. Therefore, this statement seems to be **False**.

**Statement 2: Orthogonal transformation of Image can be thought of linear summation of set of orthogonal basis images where coefficients of linear summation are estimated by projecting image onto the orthogonal basis images.**

- This statement describes a common method in image processing. When an image undergoes an orthogonal transformation (like DCT, DFT, etc.), it can indeed be represented as a linear combination of orthogonal basis images, with the coefficients determined by projecting the image onto these basis functions. This is a correct description of how orthogonal transformations work. Thus, this statement is **True**.

Given the analysis above:
- Statement 1 is False.
- Statement 2 is True.

The correct choice is **(b) Statement 1 is False and Statement 2 is True**.

Would you like to explore more details about these transformations, or do you have any further questions? Here are some related questions you might consider:

1. What are some examples of orthogonal transformations in image processing?
2. How does separability reduce computational complexity in matrix operations?
3. Can you explain the concept of basis images in the context of image transformations?
4. What is the difference between orthogonal and non-orthogonal transformations?
5. How is the Fast Fourier Transform (FFT) an example of reducing computational complexity?

**Tip:** In linear algebra, the concept of orthogonal transformations plays a crucial role in simplifying problems, especially in areas like signal processing, where operations on large datasets need to be efficient.

Explore the concepts of orthogonal transformation and computational complexity reduction in matrix operations. Analyze the statements about orthogonal transformations' impact on computational efficiency, with detailed explanations and examples.

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