Question

如何在Matlab中获得代表带有复制的2d卷积的2d卷积矩阵。所以我想要的是那种：

T = getConvMtx(H, m, n);
res1 = T * im;
res2 = imfilter(im, H, 'replicate');

让res1和res2实际上是平等的。

Matlab对convmtx2的实现为您提供了一个假设零填充的卷积矩阵。

我将我的实施作为答案之一，尽管效率极低。这就是为什么我会为更好的方法提供价值，反馈和建议的原因。

Answer 1

以下是我设法获得所需结果的方法。但是，效率非常低，因此非常欢迎任何反馈和建议：

function T = getConvMtx(H,m,n)

vHalfKerSz = floor(size(H) / 2);

mInds = reshape(1:m*n, m, n);
mInds = padarray(mInds, vHalfKerSz, 'replicate');

Tcols = zeros(m*n*numel(H), 1);
Trows = zeros(m*n*numel(H), 1);
Tvals = zeros(m*n*numel(H), 1);

i = 0; p = 0;
for c = 1:n
    for r = 1:m
        p = p + 1;

        mKerInds = mInds(r:r+size(H,1)-1, c:c+size(H,2)-1);

        [U, ~, ic] = unique(mKerInds(:));

        for k = 1:length(U)
            i = i + 1;
            Tcols(i) = U(k);
            Trows(i) = p;
            Tvals(i) = sum(H(mKerInds == U(k)));
        end
    end
end

T = sparse(Trows(1:i), Tcols(1:i), Tvals(1:i), m*n, m*n);

end

以及一些示例用法：

n = 100;
im = rand(n);
h = fspecial('gaussian', 5, 1);

mConvMtx = getConvMtx(h, n, n);
im2 = reshape(mConvMtx * im(:), size(im));

im3 = imfilter(im, h, 'replicate');

% figure;imshow(im3);
% figure;imshow(im2);
sum(abs(im2(:) - im3(:)))  %will give a very small number due to precision issues

以下是按顺序排列的结果（im，im2，im3）：

im2 im3

Answer 2

我创建了一个用于创建图像过滤矩阵的函数（类似于MATLAB imfilter()的想法）：

function [ mK ] = CreateImageFilterMtx( mH, numRows, numCols, operationMode, boundaryMode )
%UNTITLED6 Summary of this function goes here
%   Detailed explanation goes here

OPERATION_MODE_CONVOLUTION = 1;
OPERATION_MODE_CORRELATION = 2;

BOUNDARY_MODE_ZEROS         = 1;
BOUNDARY_MODE_SYMMETRIC     = 2;
BOUNDARY_MODE_REPLICATE     = 3;
BOUNDARY_MODE_CIRCULAR      = 4;

switch(operationMode)
    case(OPERATION_MODE_CONVOLUTION)
        mH = mH(end:-1:1, end:-1:1);
    case(OPERATION_MODE_CORRELATION)
        % mH = mH; %<! Default Code is correlation
end

switch(boundaryMode)
    case(BOUNDARY_MODE_ZEROS)
        mK = CreateConvMtxZeros(mH, numRows, numCols);
    case(BOUNDARY_MODE_SYMMETRIC)
        mK = CreateConvMtxSymmetric(mH, numRows, numCols);
    case(BOUNDARY_MODE_REPLICATE)
        mK = CreateConvMtxReplicate(mH, numRows, numCols);
    case(BOUNDARY_MODE_CIRCULAR)
        mK = CreateConvMtxCircular(mH, numRows, numCols);
end


end


function [ mK ] = CreateConvMtxZeros( mH, numRows, numCols )
%UNTITLED6 Summary of this function goes here
%   Detailed explanation goes here

numElementsImage    = numRows * numCols;
numRowsKernel       = size(mH, 1);
numColsKernel       = size(mH, 2);
numElementsKernel   = numRowsKernel * numColsKernel;

vRows = reshape(repmat(1:numElementsImage, numElementsKernel, 1), numElementsImage * numElementsKernel, 1);
vCols = zeros(numElementsImage * numElementsKernel, 1);
vVals = zeros(numElementsImage * numElementsKernel, 1);

kernelRadiusV = floor(numRowsKernel / 2);
kernelRadiusH = floor(numColsKernel / 2);

pxIdx       = 0;
elmntIdx    = 0;

for jj = 1:numCols
    for ii = 1:numRows
        pxIdx = pxIdx + 1;
        for ll = -kernelRadiusH:kernelRadiusH
            for kk = -kernelRadiusV:kernelRadiusV
                elmntIdx = elmntIdx + 1;

                pxShift = (ll * numCols) + kk;

                if((ii + kk <= numRows) && (ii + kk >= 1) && (jj + ll <= numCols) && (jj + ll >= 1))
                    vCols(elmntIdx) = pxIdx + pxShift;
                    vVals(elmntIdx) = mH(kk + kernelRadiusV + 1, ll + kernelRadiusH + 1);
                else
                    vCols(elmntIdx) = pxIdx;
                    vVals(elmntIdx) = 0; % See the accumulation property of 'sparse()'.
                end
            end
        end
    end
end

mK = sparse(vRows, vCols, vVals, numElementsImage, numElementsImage);


end


function [ mK ] = CreateConvMtxSymmetric( mH, numRows, numCols )
%UNTITLED6 Summary of this function goes here
%   Detailed explanation goes here

numElementsImage    = numRows * numCols;
numRowsKernel       = size(mH, 1);
numColsKernel       = size(mH, 2);
numElementsKernel   = numRowsKernel * numColsKernel;

vRows = reshape(repmat(1:numElementsImage, numElementsKernel, 1), numElementsImage * numElementsKernel, 1);
vCols = zeros(numElementsImage * numElementsKernel, 1);
vVals = zeros(numElementsImage * numElementsKernel, 1);

kernelRadiusV = floor(numRowsKernel / 2);
kernelRadiusH = floor(numColsKernel / 2);

pxIdx       = 0;
elmntIdx    = 0;

for jj = 1:numCols
    for ii = 1:numRows
        pxIdx = pxIdx + 1;
        for ll = -kernelRadiusH:kernelRadiusH
            for kk = -kernelRadiusV:kernelRadiusV
                elmntIdx = elmntIdx + 1;

                pxShift = (ll * numCols) + kk;

                if(ii + kk > numRows)
                    pxShift = pxShift - (2 * (ii + kk - numRows) - 1);
                end

                if(ii + kk < 1)
                    pxShift = pxShift + (2 * (1 -(ii + kk)) - 1);
                end

                if(jj + ll > numCols)
                    pxShift = pxShift - ((2 * (jj + ll - numCols) - 1) * numCols);
                end

                if(jj + ll < 1)
                    pxShift = pxShift + ((2 * (1 - (jj + ll)) - 1) * numCols);
                end

                vCols(elmntIdx) = pxIdx + pxShift;
                vVals(elmntIdx) = mH(kk + kernelRadiusV + 1, ll + kernelRadiusH + 1);

            end
        end
    end
end

mK = sparse(vRows, vCols, vVals, numElementsImage, numElementsImage);


end


function [ mK ] = CreateConvMtxReplicate( mH, numRows, numCols )
%UNTITLED6 Summary of this function goes here
%   Detailed explanation goes here

numElementsImage    = numRows * numCols;
numRowsKernel       = size(mH, 1);
numColsKernel       = size(mH, 2);
numElementsKernel   = numRowsKernel * numColsKernel;

vRows = reshape(repmat(1:numElementsImage, numElementsKernel, 1), numElementsImage * numElementsKernel, 1);
vCols = zeros(numElementsImage * numElementsKernel, 1);
vVals = zeros(numElementsImage * numElementsKernel, 1);

kernelRadiusV = floor(numRowsKernel / 2);
kernelRadiusH = floor(numColsKernel / 2);

pxIdx       = 0;
elmntIdx    = 0;

for jj = 1:numCols
    for ii = 1:numRows
        pxIdx = pxIdx + 1;
        for ll = -kernelRadiusH:kernelRadiusH
            for kk = -kernelRadiusV:kernelRadiusV
                elmntIdx = elmntIdx + 1;

                pxShift = (ll * numCols) + kk;

                if(ii + kk > numRows)
                    pxShift = pxShift - (ii + kk - numRows);
                end

                if(ii + kk < 1)
                    pxShift = pxShift + (1 -(ii + kk));
                end

                if(jj + ll > numCols)
                    pxShift = pxShift - ((jj + ll - numCols) * numCols);
                end

                if(jj + ll < 1)
                    pxShift = pxShift + ((1 - (jj + ll)) * numCols);
                end

                vCols(elmntIdx) = pxIdx + pxShift;
                vVals(elmntIdx) = mH(kk + kernelRadiusV + 1, ll + kernelRadiusH + 1);

            end
        end
    end
end

mK = sparse(vRows, vCols, vVals, numElementsImage, numElementsImage);


end


function [ mK ] = CreateConvMtxCircular( mH, numRows, numCols )
%UNTITLED6 Summary of this function goes here
%   Detailed explanation goes here

numElementsImage    = numRows * numCols;
numRowsKernel       = size(mH, 1);
numColsKernel       = size(mH, 2);
numElementsKernel   = numRowsKernel * numColsKernel;

vRows = reshape(repmat(1:numElementsImage, numElementsKernel, 1), numElementsImage * numElementsKernel, 1);
vCols = zeros(numElementsImage * numElementsKernel, 1);
vVals = zeros(numElementsImage * numElementsKernel, 1);

kernelRadiusV = floor(numRowsKernel / 2);
kernelRadiusH = floor(numColsKernel / 2);

pxIdx       = 0;
elmntIdx    = 0;

for jj = 1:numCols
    for ii = 1:numRows
        pxIdx = pxIdx + 1;
        for ll = -kernelRadiusH:kernelRadiusH
            for kk = -kernelRadiusV:kernelRadiusV
                elmntIdx = elmntIdx + 1;

                pxShift = (ll * numCols) + kk;

                if(ii + kk > numRows)
                    pxShift = pxShift - numRows;
                end

                if(ii + kk < 1)
                    pxShift = pxShift + numRows;
                end

                if(jj + ll > numCols)
                    pxShift = pxShift - (numCols * numCols);
                end

                if(jj + ll < 1)
                    pxShift = pxShift + (numCols * numCols);
                end

                vCols(elmntIdx) = pxIdx + pxShift;
                vVals(elmntIdx) = mH(kk + kernelRadiusV + 1, ll + kernelRadiusH + 1);

            end
        end
    end
end

mK = sparse(vRows, vCols, vVals, numElementsImage, numElementsImage);


end

该代码已针对MATLAB imfilter()进行了验证。
在我编写它时，它并未进行优化（不是以矢量化方式编写），以使人们可以轻松地掌握正在发生的事情。

我的StackOverflow Q2080835 GitHub Repository中提供了完整的代码。

Matlab - 带复制的2D卷积矩阵

2 个答案: