Combine convolve filter in matlab - matlab

Question

Is there a way to take the low pass and high pass filters in the following code and combine them into a single kernel and apply one conv2() function?
note: length(lfilter) = 21, length(hfilter) = 81.
what we are basically doing in the last step is saying to remove the large objects from the image (after already removing the very small objects with a Gaussian blur).
properties (Constant)
minStar = 2; % min star radius
maxStar = 8; % max star radius
threshold = 12;
end
function filter2(this)
normalize = #(x) x/sum(x);
lfilter = normalize(exp(-((-ceil(5*this.minStar):ceil(5*this.minStar))/(2*this.minStar)).^2));
hfilter = normalize(exp(-((-ceil(5*this.maxStar):ceil(5*this.maxStar))/(2*this.maxStar)).^2));
this.low = conv2(lfilter',lfilter,this.raw,'same');
this.high = conv2(hfilter',hfilter,this.raw,'same');
this.filtered = this.low - this.high;
this.foreground = this.filtered > this.threshold;
end

Answer 1

Since the convolution operator is associative:
conv( a, conv(b,c) ) == conv( conv(a,b), c )
you should be able to combine the two kernels into one just by convolving them with each other.
In your case something like this should work:
new_kernel = conv2(lfilter',lfilter, conv2(hfilter',hfilter), 'same');
Convolution is commutative as well so the order in which you perform the convolutions shouldn't matter.
EDIT: as I explain in the comment below, the asker's method of performing four 1D convolutions ends up being faster than a single 2D convolution.

Answer 2

I just got the answer in the matlab forums. http://www.mathworks.com/matlabcentral/answers/169713-combine-convolution-filters-bandpass-into-a-single-kernel
The gist is that you have to use padding to fill in both sides of the shorter filter, and then you can just combine the vectors.
Convolution is a linear operation so yes, you can combine the two filtering operations into one. Just make the filters the same size and add/subtract them. For example:
lfilter = normalize(exp(-((-ceil(5*minStar):ceil(5*minmax))/(2*this.minStar)).^2));
hfilter = normalize(exp(-((-ceil(5*maxStar):ceil(5*minmax))/(2*this.maxStar)).^2));
padlength = (length(hfilter) - length(lfilter))/2;
lfilter = padarray(lfilter, [0 padlength]);
lhfilter = lfilter - hfilter;
this.filtered = conv2(lhfilter',lhfilter,this.raw,'same');