Image Gradient Video Lecture - Civil Engineering (CE)

So now let's dive into understanding what an image gradient is.
To help us define an image gradient, let's take the example of
this sample image again, where we're interested in is at each and
every point we want to compute the gradient.
And with respect to the neighborhood that we're in traversing and
again the both dimensions of the image.
A Gradient basically is defined as a measure of change in an Image function.
So that at this point, what are the changes of the image function with
respect to how things change in the x direction or
in this case, the columns or the y direction, the rows of an image?
We'd like to be able to compute this both in discontinuous and
the continuous form.
So how do we define it mathematically?
So basically, the differential of the function F,
which represents an image, with respect to x.
And then differential of the function image F, with respect to y.
And of course, both dimensions of this is represented by the del F,
is the gradient of the image F.
In essence, this is the way to measure the change in image function F,
in both x and y.
Let's look at that with a couple of simple examples.
Let's take a simple ramp image here, starts with a dark value,
let's say zero, to 255 where the pure white is.
At this point, I want to be able to measure the change of this image function,
as respect to how it is in this neighborhood here.
Very obvious if you take the same function here, you notice that all of
the change in this direction is for the x direction, none for the y.
So that of course, this is a very simple case where you can just represent
the del of F is nothing else at this point to be this function here.
The counter, of course, is when there is a pixel and actually it's now, or
this image here as a ramp here.
And of course, going from two fifth, 0 to 255,
from this point onwards, and, and in this case, we can
simply look at that there is not going to be any change in the x direction.
That seems to be fixed, and just in the y direction.
Combining these two,
we can look at if the changes are in both directions, then they will be,
of course the gradient direction would be at an angle, which is there.
In essence, what that basically means now is that the gradient point
in the direction of the most rapid increase in intensity.
>From here, the intensity is increasing rapidly in this direction, less so
in this or this direction, so this is my most dominant angle.
Let's look at this a little bit more carefully.
How would we compute the gradient direction?
Just take the inverse of the tangent, I mean,
this is my vector, del F with x and y.
If I know the y and
the x changes, I can basically now do the inverse tangent to get data.
The magnitude of this vector, which would of course be the modulus of
this value here, which would be the square plus square, square root.
So I have this thing, it'll give me the magnitude of the vector.
So the angle of the vector and
the magnitude of the vector are the ones we're looking for, and
that would basically define the gradient at any point in an image.
So while we're at it one thing I want everybody to think about is how does this,
how does this relate to edge direction.
When I have a point here and I have the gradient direction here the theta and
the magnitude.
Where is the edge?
Hopefully you will have the answer in a bit.