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Author: meshtag

doc/image_processing/2D_Convolution_and_Correlation.rst

@@ -1,39 +1,93 @@
-Convolution forms a pillar in a large number of image processing applications. It is a mathematical 
-operation which is applied in a same manner over the entire considered region in an image. 
+Two dimensional convolution and correlation
+-------------------------------------------
+
+
+What is the role of convolution in image processing?
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+
+Convolution forms a base pillar in a large number of image processing applications. It is a 
+mathematical operation which is applied in a same manner over the entire considered region in an image. 
 Accessing all pixels in a similar manner creates the room for optimization in this algorithm. 
 We have thus targeted our efforts towards improving the spatial access of pixel elements during 
 2D convolution and correlation.
 
-There are many ways of performing 2D convolution, the naive one includes irregular access of source 
+
+---------------------------------
+
+
+What are we doing differently?
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+
+There are many ways of performing 2D convolution, the naive ones includes irregular access of source 
 image pixels. Due to this, a large number of cache misses occur which further in turn degrade the 
 performance. We have tried to improve this access pattern and hence reduce the number of overall 
 cache misses.
 
-What are we doing differently?
 
 Our strategy involves storing relevant 2D image data in a 1D buffer container. This container will 
 then be used for accessing required patches of pixels during convolution. These patches will be 
-accessed by a sliding window algorithm so that their dot product with 1D aligned kernel can be stored 
+accessed by a sliding window algorithm so that their dot product with 2D kernel can be stored 
 in destination image.
 
-But how much data should be stored inside the buffer?
+
+---------------------------------
+
+
+How much data should be stored inside the buffer?
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
 
 We store only that much data which will be required by kernel during its one single horizontal slide. 
-In other words, the size of buffer container will be kernel_size * img_width + boundary_offset. 
-Here boundary_offset is the addition in buffer size which will take place due to boundary 
+In other words, the size of buffer container will be 
+
+``buffer size = kernel_size * img_width + boundary_offset``
+
+where ``boundary_offset = (extrapolation_required == 1) ? ((kernel_size - 1) * kernel_size) : 0``
+
+Here, boundary_offset is the addition in buffer size which will take place due to boundary 
 extrapolation for corner pixels.
 
+
+----------------------------------
+
+
 What to do after forming the buffer container?
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
 
 Once our data is present inside the buffer container, a sliding window algorithm is applied for 
 calculating the dot product between kernel and image patch contained inside sliding window.
-
 Boundary extrapolation is taken care by changing elements of buffer container suitably.
 
+
+----------------------------------
+
+
 How do we take care of variable anchor point positions?
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
 
 While filling the buffer container, we will take care of this by starting/ending the filling process 
 at suitable points in source image.
-
 This will automatically take care of the problem and will achieve desired effect due to the followed 
 pattern of pixel assignment.
+
+
+-------------------------------------------------------
+
+
+What to do with spatial separable kernels?
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+
+We have written a custom algorithm which can detect whether a two dimensional kernel can be 
+spatially separated. It does separation as well as checking for separability 
+simultaneously. If at any step during checking, the kernel is deemed non-separable, the separation 
+algorithm stops, otherwise horizontal and vertical separated kernels are returned.
+The algorithm uses the fact that a separable 2D matrix is essentially a rank1 matrix. It further 
+uses basic linear algebra for separating the kernel.
+
+
+-------------------------------------------------------