how to error estimate of 0.005 calculated from the singular values in svd from the F-norm for an image

how to determine a reduced rank approximation of our image so that the relative error (measured in the Frobenius norm) is no more than 0.5% for a grayscale as well as RGB image

Answer 1

[Anshuman]

Hi karthik,

You can do the following steps to determine a reduced rank approximation of an image such that the relative error is no more than 0.5%:

1. Load the image into Matlab using the imread function.
2. Convert the image to a matrix using the double function. For grayscale images, this will result in a 2D matrix, while for RGB images, this will result in a 3D matrix.
3. Compute the Frobenius norm using the norm function.
4. Find the maximum rank of the reduced rank approximation. This can be done by computing the singular value decomposition (SVD) of the image matrix and examining the singular values. The rank can be set to the number of singular values that account for at least 99.5% of the total energy. You can use the svd function in Matlab to obtain the singular values.
5. Compute the reduced rank approximation of the image by truncating the SVD at the determined rank. This can be done by setting the smallest singular values to zero and reconstructing the image using the truncated SVD.
6. Compute the relative error (measured in the Frobenius norm) between the original image and the reduced rank approximation. This can be done by computing the Frobenius norm of the difference between the original and approximated matrices, and then dividing by the Frobenius norm of the original matrix.
7. If the relative error is greater than 0.5%, repeat steps 4-6 with a lower rank until the desired relative error is achieved.

Note that for RGB images, you will need to perform these steps separately for each color channel (i.e., red, green, and blue) and then combine the approximations to obtain the final reduced rank approximation of the image.

Hope it helps!