In the previous blogs, we discussed different smoothing filters. Before moving forward, let’s first discuss Image Gradients which will be useful in edge detection, robust feature and texture matching. So, let’s first recall what a gradient is.

In mathematics, the term gradient of a function means how a function is changing wrt. its arguments or independent variables. The gradient term is more frequently used for multi-variable functions. For a single variable function, we refer to this as the slope.

The gradient of an N-variable function at each point is an N-D vector with the components given by the derivatives in the N-directions. e.g. for a 3-variable function (f(x,y,z)), the gradient, if it exists, is given by

Thus, the gradient provides two pieces of information – magnitude and direction. The direction of the gradient tells us the direction of greatest increase while the magnitude represents the rate of increase in that direction.

Because gradients are defined only for continuous functions and Image is a 2-d discrete function (F(x,y)). Thus we need to approximate the gradients and we do this using Finite differences. In this instead of h approaching 0 we assume h to be a fixed (non-zero) value.

Three forms of finite differences are commonly used: forward, backward and central.

But for calculating Image Gradients, we use the central difference to approximate gradients in x and y directions. Below example shows how to calculate the central difference in the x-direction for 200.

In the next blog, we will discuss how to derive different kernels such as Sobel, Prewitt, etc from this central difference formulae and then using convolution to approximate the image gradients.

Thus, at each image point, the gradient vector points in the direction of largest possible intensity increase, and the magnitude corresponds to the rate of change in that direction. Thus for an image f(x,y), the gradient direction and magnitude is given by

Thus in simple words, image gradient in x-direction measures the horizontal change in intensity while the gradient in y measures the vertical change in intensity.

Since edges are an abrupt change in the intensity values thus the largest gradient values will occur across an edge (neglecting noise) in an image. Thus, the x-gradient will find the vertical edges while y-gradient will highlight the horizontal edges as shown below.

Thus, we may conclude that edges are perpendicular to the gradient direction(largest). That’s why gradients are used in edge detection.

In the next blog, we will discuss different edge detection filters. Hope you enjoy reading.

If you have any doubt/suggestion please feel free to ask and I will do my best to help or improve myself. Good-bye until next time.