How can I use of norm(a,b) in matlab if a, b are double type?

Question

I must to use angle = atan2(norm(cross(a,b)),dot(a,b)), for calculating the angle between two vectors a,b and these are double type and norm is undefined for this type. How do I resolve this problem? I need to calculate the angle between two vectors this way.

Answer 1

As @rayryeng has successfully answered this question, I would like to turn my post into a more general one by sharing my experience in debugging in Matlab. I hope anyone who somehow managed to find this post get more or less thinking about the habits a good programmer should have.

The question goes like: "How would I do if I get errors?"

1. Here's an excellent article by Eric in which he lists the rule-of-thumbs when you encounter a bug and wish to get rid of it. It's originally been cited by Stackoverflow, and that's the reason I read it.

2. If you still get no clue / idea how you can play with your code, see how this person does:

Pin-point the buggy line

0. (The number should start with 0) Make sure before running a script, you clear out any previously stored variables, including the notorious i and j's (you should never see them in any workspace). If any one is needed for the buggy code to run, save('buggy.mat','importantvar') before clear and load('buggy.mat') after clear.

   By doing so, you can isolate your buggy code from anything else, which could have bad influences. For example, in a previously called script, there is a line

   double = [2,4,6]; % you should never name a variable `double`
   

   and in the next script, you have

   >> e = str2num('uint8(200)')
   e =
     200
   >> double(e)
   Index exceeds matrix dimensions. 
   >> 
   >> f = single(2.36)
   f =
       2.3600
   >> double(f)
   Subscript indices must either be real positive integers or
   logicals. 
   >> 
   

   The reason is double is no longer an inbuild function, but a user-defined variable. Too bad to pick up a name carelessly!

   ....anyway, let's clear the workspace and get rid of double.

   >> clear
   

1. Read the error message, thoroughly.

   Now let's begin with OP's problem. The original code (trimmed) goes like this -

   img = imread('peppers.png');
   a = img(300,200,:);
   b = img(200,300,:);
   d = norm(cross(a,b));
   

   .... hence the error

   Undefined function 'norm' for input arguments of type 'uint8'.
   Error in untitled (line 6)
   d = norm(cross(a,b)); 
   

   Most beginners are only interested in the first line of the error message, which by it alone usually doesn't provide any useful help, or only in the red color, which leads to the famous question "my code does not work!"

   But think twice. You still have another 2 lines unread! Error in untitled (line 6) says I'm running a script named untitled and the (first) error lies in line 6, and the code in that line is d = norm(cross(a,b));.

   Now, at least you know a little more about your code - "My code d = norm(cross(a,b)); doesn't work!"

   Although most likely we may also vote this kind of question to get closed, it's still much much better than a simply "It does not work!".

2. Now we can pin-point the buggy line

   try
   %     this line will raise an error
       d = norm(cross(a,b));
   catch err
       disp(err.message)
   end
   

Look into the functions

First, make sure the inner function cross works as expected -

    >> cross(a,b)
    ans(:,:,1) =
        0
    ans(:,:,2) =
      255
    ans(:,:,3) =
        0
    >> 

Good. So now we can even narrow down the error to the outer norm.

One more thing to mention. You can always find Mathworks' documentation for any in-build function, by typing "matlab function", such as "matlab norm" in Google (or any other search engine) and clicking on the first result. If you prefer, you can also type in Matlab command window doc _function_ such as doc norm and read the doc in Matlab. It's of course a pleasure of us on Stackoverflow to give you the reference by doing the same thing, but it takes a longer time because a human is, in this aspect, always slower than a search engine.

The error reads Undefined function 'norm' for input arguments of type 'uint8'.. So the input for norm should not be uint8, unsigned 8-bit integer. But what should it be?

% why `norm` "does not work"? 
% this line runs perfectly well
norm(cross([1,2,3], [4,5,6]))
% so what is working? 
class([1,2,3]) % so `norm` works for `double`

One thing we can do now is convert a and b to double precision. Let's try it now.

% try fixing 'uint8' error
a2 = double(a);
b2 = double(b);
whos a b % now they are double, which `norm` should work for
try
%     this line will raise an error
    d = norm(cross(a2,b2));
catch err
    disp(err.message)
end

Now the error becomes Input must be 2-D.. What's wrong with the input?

% what is "must be 2-D" error? 
size(a2) % a2 is 3-D
disp(b2) % b2 is also 3-D

This gives output in command window

ans =
     1     1     3
(:,:,1) =
   255
(:,:,2) =
   150
(:,:,3) =
     0

In OP's problem, he/she is trying to calculate something about color difference (to the best of my knowledge) which involves the angle between two color vectors in RGB space. So the vectors are needed. With imread, each pixel of the image is stored as 3 elements in the matrix, first 2 dimension being its physical position, the 3 dimension being RGB channel components. Hence pixel(200,300) with color rgb[255,150,0] is stored by us in variable b wihch is a 3-D vector.

By understanding what we need and what Matlab can do, we can combine these two points into one. We need the norm of the cross product of a and b, while the useful information (the 3 component values) is stored in the 3rd dimension. Matlab can calculate the norm of the cross product of a vector with all its information in the 1st dimension. (Here, "dimension" refers to that of the Matlab variable; a vector with 3 elements in its 1st dimension is physically a 3-D vector).

After thinking twice, we are now able to debug our code - just put all 3 elements into the 1st dimension.

% so we want the 3 elements in the 3rd dimension become in the 1st dim
a3 = squeeze(a2);
b3 = reshape(b2,numel(b2),[]);
try
    d = norm(cross(a3,b3));
catch err
    disp(err.message)
end
d

Bonus: If by default Matlab treats a 3-D vector as a "1-D array", then most probably the cross function has not been working correctly. Let's make a check -

>> clear
>> a = [1,2,3]
a =
     1     2     3
>> b=[4,5,6]
b =
     4     5     6
>> cross(a,b)
ans =
    -3     6    -3
>> 

The result should be the same as the one we can get by calculating by hand.

Now if we put the components into the 3rd dimension of the variable -

>> clear
>> a(1,1,:)=[1,2,3]
a(:,:,1) =
     1
a(:,:,2) =
     2
a(:,:,3) =
     3
>> b(1,1,:)=[4,5,6]
b(:,:,1) =
     4
b(:,:,2) =
     5
b(:,:,3) =
     6
>> cross(a,b)
ans(:,:,1) =
    -3
ans(:,:,2) =
     6
ans(:,:,3) =
    -3
>> 

.... seems OK. cross also puts the result in the 3rd dimension. In fact, Mathworks' documentation says

If A and B are vectors, then they must have a length of 3.

If A and B are matrices or multidimensional arrays, then they must have the same size. In this case, the cross function treats A and B as collections of three-element vectors. The function calculates the cross product of corresponding vectors along the first array dimension whose size equals 3.

At last, one thing is always correct to anyone who wants to do something with programming - be cautious and prudent when writing your code.