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Morphological filtering is a theory developed in the 1960s for the analysis and processing of discrete images. It defines a series of operators which transform an image by probing it with a predefined shape element. The way this shape element intersects the neighborhood of a pixel determines the result of the operation. This article presents the most important morphological operators. It also explores the problem of image segmentation using algorithms working on the image morphology.
Eroding and dilating images using morphological filters
Erosion and dilation are the most fundamental morphological operators. Therefore, we will present them in this first recipe.
The fundamental instrument in mathematical morphology is the structuring element. A structuring element is simply defined as a configuration of pixels (a shape) on which an origin is defined (also called anchor point). Applying a morphological filter consists of probing each pixel of the image using this structuring element. When the origin of the structuring element is aligned with a given pixel, its intersection with the image defines a set of pixels on which a particular morphological operation is applied. In principle, the structuring element can be of any shape, but most often, a simple shape such as a square, circle, or diamond with the origin at the center is used (mainly for efficiency reasons).
Getting ready
As morphological filters usually work on binary images, we will use a binary image produced through thresholding. However, since in morphology, the convention is to have foreground objects represented by high (white) pixel values and background by low (black) pixel values, we have negated the image.
How to do it…
Erosion and dilation are implemented in OpenCV as simple functions which are cv::erode and cv::dilate. Their use is straightforward:
// Read input image
cv::Mat image= cv::imread(“binary.bmp”);
// Erode the image
cv::Mat eroded; // the destination image
cv::erode(image,eroded,cv::Mat());
// Display the eroded image
cv::namedWindow(“Eroded Image”);”);
cv::imshow(“Eroded Image”,eroded);
// Dilate the image
cv::Mat dilated; // the destination image
cv::dilate(image,dilated,cv::Mat());
// Display the dilated image
cv::namedWindow(“Dilated Image”);
cv::imshow(“Dilated Image”,dilated);
The two images produced by these function calls are seen in the following screenshot. Erosion is shown first:
Followed by the dilation result:
How it works…
As with all other morphological filters, the two filters of this recipe operate on the set of pixels (or neighborhood) around each pixel, as defined by the structuring element. Recall that when applied to a given pixel, the anchor point of the structuring element is aligned with this pixel location, and all pixels intersecting the structuring element are included in the current set. Erosion replaces the current pixel with the minimum pixel value found in the defined pixel set. Dilation is the complementary operator, and it replaces the current pixel with the maximum pixel value found in the defined pixel set. Since the input binary image contains only black (0) and white (255) pixels, each pixel is replaced by either a white or black pixel.
A good way to picture the effect of these two operators is to think in terms of background (black) and foreground (white) objects. With erosion, if the structuring element when placed at a given pixel location touches the background (that is, one of the pixels in the intersecting set is black), then this pixel will be sent to background. While in the case of dilation, if the structuring element on a background pixel touches a foreground object, then this pixel will be assigned a white value. This explains why in the eroded image, the size of the objects has been reduced. Observe how some of the very small objects (that can be considered as “noisy” background pixels) have also been completely eliminated. Similarly, the dilated objects are now larger and some of the “holes” inside of them have been filled.
By default, OpenCV uses a 3×3 square structuring element. This default structuring element is obtained when an empty matrix (that is cv::Mat()) is specified as the third argument in the function call, as it was done in the preceding example. You can also specify a structuring element of the size (and shape) you want by providing a matrix in which the non-zero element defines the structuring element. In the following example, a 7×7 structuring element is applied:
cv::Mat element(7,7,CV_8U,cv::Scalar(1));
cv::erode(image,eroded,element);
The effect is obviously much more destructive in this case as seen here:
Another way to obtain the same result is to repetitively apply the same structuring element on an image. The two functions have an optional parameter to specify the number of repetitions:
// Erode the image 3 times.
cv::erode(image,eroded,cv::Mat(),cv::Point(-1,-1),3);
The origin argument cv::Point(-1,-1) means that the origin is at the center of the matrix (default), and it can be defined anywhere on the structuring element. The image obtained will be identical to the one we obtained with the 7×7 structuring element. Indeed, eroding an image twice is like eroding an image with a structuring element dilated with itself. This also applies to dilation.
Finally, since the notion of background/foreground is arbitrary, we can make the following observation (which is a fundamental property of the erosion/dilation operators). Eroding foreground objects with a structuring element can be seen as a dilation of the background part of the image. Or more formally:
- The erosion of an image is equivalent to the complement of the dilation of the complement image.
- The dilation of an image is equivalent to the complement of the erosion of the complement image.
There’s more…
It is important to note that even if we applied our morphological filters on binary images here, these can also be applied on gray-level images with the same definitions.
Also note that the OpenCV morphological functions support in-place processing. This means you can use the input image as the destination image. So you can write:
cv::erode(image,image,cv::Mat());
OpenCV creates the required temporary image for you for this to work properly.