What is a Gradient and How is it Calculated in Image Processing?

[Jake]

I was wondering if anyone could help me with this rather arbitrary math problem.

I need a description of what a gradient is in mostly non-math terms. I know a gradient is a sort of slope, but I don't understand where you would draw the line of what is and what is not a gradient. Meaning which slopes are and are not a gradient. Is this even calculable? I know http://en.wikipedia.org/wiki/Gradient" on it has some equations but I don't understand any of that.

I am using this for an image processing project where the maximum pixel intensity value is 765 and minimum is 0. So for example is a sequence of pixels with the values 10,15,20,25 a gradient? I'd think so, but is 9,16,21,24 also a gradient? Where do you draw the line between gradient and non-gradient?

Thanks a lot :)

 

[gnomedt]

A gradient, in the mathematical sense, is a measure of how much a surface is changing. If you could imagine a small stretch of wavy ocean, you could plug in the x and y of any point on that section of ocean and you'd "get out" an arrow (a vector). That arrow will point in the steepest direction -- for example, for a climber standing at the foot of Everest, the gradient of the land at the climbers position is a vector that points UP at Everest. because from where he's standing, that's the steepest way he could go. It's hard to describe, but it's a vector Calculus topic. Loosely, though, a gradient is how something is changing from one place to another, over space. That means a lot of things can have gradients -- height, color, temperature, anything that can change from place to place.

In your sense, a gradient is used in the color sense -- one color fading to another. It's arbitrary, and I don't see any reason why you should need to "define" a gradient. Indeed, if I interpret your numbers right, both of your examples are gradients, because the color changes over space. The former gradient might look smoother though.

 

[Rach3]

Perhaps try

http://amath.colorado.edu/outreach/demos/hshi/2001Spr/snake/snake.html
"Visualizing calculus: The use of the gradient in image processing"

A gradient is not a sequence of numbers, or a path. It is a vector at a single point, that describes the how the scalar function slopes there - in which direction it slopes, and how steeply it slopes. That's what a vector describes - a direction and a magnitude. This is of course multivariable calculus, and it will be of no use to you if you avoid the math. What is your background in calculus?

 

[Jake]

Ok, that clears it up. It didn't make sense that gradients could not be arbitrarily defined, although it would make things a bit simpler :|

Thanks a lot :-)

 

[Jake]

Perhaps try

http://amath.colorado.edu/outreach/demos/hshi/2001Spr/snake/snake.html
"Visualizing calculus: The use of the gradient in image processing"

A gradient is not a sequence of numbers, or a path. It is a vector at a single point, that describes the how the scalar function slopes there - in which direction it slopes, and how steeply it slopes. That's what a vector describes - a direction and a magnitude. This is of course multivariable calculus, and it will be of no use to you if you avoid the math. What is your background in calculus?

None at all. However I don't think I need it. I am simply using gradients as one method of defining an edge within an image. The problem was figuring out where to draw the line of a single gradient and an edge. Since a gradient can be completely arbitrary, it looks like that can't be the only clue in finding edges.

Thanks though, the link is helpful

 

[gnomedt]

Well no. A mathematical gradient is different than the very specific type of gradient that you're talking about. In an image, a gradient is linear, so you could determine the beginning and end of it. It's an adopted use of the word "gradient." But since you were asking on a Math forum (and not a Photoshop forum), I assumed you wanted a definition of a gradient in general, and every image has a mathematical gradient.

One warning: watch out for round-off errors. That could be a big pain.