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

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In summary, a gradient is a measure of how something is changing from one point to another, typically in a specific direction and with a certain magnitude. In mathematics, it refers to a vector that describes the slope of a surface at a particular point. In image processing, a gradient is often used to define edges and is typically linear. However, it is important to be aware of potential round-off errors when working with gradients. A basic understanding of calculus is helpful in understanding gradients, but it is not always necessary for practical applications.
  • #1
Jake
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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" [Broken] 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 :)
 
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  • #2
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.
 
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  • #3
Perhaps try

http://amath.colorado.edu/outreach/demos/hshi/2001Spr/snake/snake.html [Broken]
"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?
 
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  • #4
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 :-)
 
  • #5
Rach3 said:
Perhaps try

http://amath.colorado.edu/outreach/demos/hshi/2001Spr/snake/snake.html [Broken]
"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
 
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  • #6
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.
 
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1. What is a gradient?

A gradient is a mathematical concept that represents the rate of change of a function with respect to its variables. In simpler terms, it is a way to measure how much a function is changing at a certain point.

2. Why are gradients important in science?

Gradients are important in science because they allow us to understand and analyze complex systems and processes. By calculating gradients, we can determine the direction and speed of change in a system, which can help us make predictions and design experiments.

3. How are gradients calculated?

Gradients are calculated using differential calculus, specifically the derivative of a function. This involves finding the slope of the function at a specific point and determining how it changes as the variables change.

4. What is the significance of the direction of a gradient?

The direction of a gradient is significant because it tells us the direction in which a function is increasing or decreasing the most rapidly. This can help us understand the behavior and patterns of a system, and make predictions about its future behavior.

5. How can gradients be applied in real-world situations?

Gradients have numerous applications in various fields of science, including physics, biology, economics, and engineering. They can be used to analyze and model natural phenomena, optimize processes and systems, and make predictions about future outcomes.

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