Image gradient sampling grid

Hi, everyone. I am having a difficulty with my project in MATLAB. I really do not understand this concept of sampling grid and from the circle to R. Thank you for any help in advance! Here is the question:

Id is a discrete grayscale image of size N^2, thus an element of R N^N. This discrete image Id can be viewed as sampling on regular grid of a function I from the square [-1,1]^2 to R, with the grid spacing being h=2/(N-1) in both directions. How to compute I from Id?

Answer 1

[Image Analyst]

You never said what the badly-named "I" is, so I don't know how to compute it from Id.

All the rest of the question is just a bunch of mathematical mumbo jumbo that really doesn't amount to anything except to allow the author to impress himself. Trust me, you can safely ignore it.

[Anar Alshanbayeva]

Hi, thank you for such quick response. I don't understand the theory behind this mumbo jumbo though. Clarifying the I and Id, In fact Id is the discrete image, the matrix, with indices i,j between 1 and N, and I is the equivalent image expressed at coordinates x,y. Basically, from what I understood, they want me to make an Image I the borders of which will be (-1, 1) and which will represent the values from the Image Id (which is a grayscale image), but only converting the matrix numbers of Id image (which has values b/w 0-255) to a matrix of I image (which will have values b/w -1,1 and will be bordered in dimension as a square -1, 1). I don't know if this is my correct understanding .

[Image Analyst]

Not sure what that means. A digital image is an array with rows and columns. A real image is just a bunch of light (or whatever). Not sure if you want to interpolate a different number of pixels, or if you want some kind of spatial calibration system that keeps track of the real-world (x,y) given the (column, row) indexes. For example element (1,1) corresponds to x=-1 and y=-1, and element (480, 1) (last row of a 480x640 image) corresponds to x=-1, y=+1, and element (480,640) corresponds to x=1, y=1. If so, you'll have to keep track of the spatial calibration factor(s) separately whenever you want to convert from real world coordinates to indexes.

See my spatial calibration demo, attached.