Logical Operations
[Arithmetic And Logic]
The Logical Operations category contains logical operations between two images or between an image and a constant as union, intersection, complementary. More...
Classes | |
| class | SoBitShiftProcessing |
 SoBitShiftProcessing engine More... | |
| class | SoInvertImageProcessing |
| SoInvertImageProcessing engine calculates the reverse of an image. More... | |
| class | SoLogicalImageProcessing |
| SoLogicalImageProcessing engine performs logical operations between two image. More... | |
| class | SoLogicalNotProcessing |
| SoLogicalNotProcessing engine More... | |
| class | SoLogicalValueProcessing |
| SoLogicalValueProcessing engine More... | |
Detailed Description
The Logical Operations category contains logical operations between two images or between an image and a constant as union, intersection, complementary.
Logical operations are usually performed on binary images, but may also be applied to grayscale images. In a binary image, a pixel/voxel with value 1 has the value true and a pixel/voxel with value 0 a value false. Each logical operation is linked to a boolean operation. In a binary image I, all pixels/voxels with value 1 belong to the set A, and all 0-value pixels to the complement set of A or the background.
Grayscale Images
When a logical operation is applied to a grayscale image, the logical operations for binary images are applied to each bit plane. Let us consider the two images 
and 
, and perform a logical AND operation.
Suppose that for pixel (n,m), intensities and are respectively equal to 
and 
. What will be the value of the result 
? If we consider the binary coding of the two numbers and apply the operator to each of the eight bits as shown on the Figure 1.
Figure 1
The result of this logical AND may appear rather strange and for this reason logical operations applied to grayscale images have to be handled with care.
Masking and Bit-Plane Combination
remains unchanged if 
- turns to 0 if
The two main applications of logical operations on grayscale images are for masking and for combining bit planes.
To mask an image, a logical AND is performed between images I and M where every pixel with value 1 is coded with 1 (short value) and every pixel with value 0 is coded with 0 (short value). In masking a grayscale image I with a binary image M:
To combine bit planes, if two images and have grayscale values in the range 0 to 15 (4-bit images), one has to shift the image by 4 bits on the left. A logical OR between image and image shifted by 4 bits (
) gives an 8-bit image made of the two images.
Color Images
![$I_1(i,j)=[R_1(i,j);G_1(i,j);B_1(i,j);A_1(i,j)]$](form_52.png)
,![$I_2(i,j)=[R_2(i,j);G_2(i,j);B_2(i,j);A_2(i,j)]$](form_53.png)
and![$O(i,j)=[R_O(i,j);G_O(i,j);B_O(i,j);A_O(i,j)]$](form_54.png)
Let us consider the two color images and . Suppose we want to perform 
where 
is a boolean operator.
When a logical operation is applied to color image, it operates on each color channel. A color image pixel/voxel is a quadruplet of values coded on one unsigned byte each:
then we will have the following relations:
For example, let us suppose that 
(opaque magenta) and 
(opaque light blue) performing an AND operation, then we should get :
When performing a logical operation between an image and a fixed value: 
, the following formulas apply:
![$O(i,j)=\left[\begin{array}{c} R_O(i,j)\\ G_O(i,j)\\ B_O(i,j)\\ A_O(i,j) \end{array}\right]=\left[\begin{array}{c} {R_1(i,j)}\otimes{C}\\ {G_1(i,j)}\otimes{C}\\ {B_1(i,j)}\otimes{C}\\ {A_1(i,j)}\otimes{C} \end{array}\right].$](form_60.png)
For example, let us suppose that (opaque magenta) and 
performing a XOR operation then we should get:
![$O(i,j)=\left[\begin{array}{c} R_O(i,j)\\ G_O(i,j)\\ B_O(i,j)\\ A_O(i,j) \end{array}\right]=\left[\begin{array}{c} {255}\otimes{7}\\ {0}\otimes{7}\\ {255}\otimes{7}\\ {255}\otimes{7} \end{array}\right]=\left[\begin{array}{c} {248}\\ {7}\\ {248}\\ {248} \end{array}\right].$](form_62.png)