how can I get the basic function of the cubic B-spline [N i,d (t)]? thank you.

how can I get the basic function of the B-spline [N i,d (t)]?

Answer 1

[Aditya]

Hi farouk,

To compute the basic function of a B-spline, denoted as ( N_{i,d}(t) ), you need to follow the recursive definition of B-spline basis functions. The B-spline basis functions are defined over a knot vector, and the degree of the B-spline is determined by ( d ).Here is a sample code :

function N = bsplineBasis(i, d, t, knots)
    % bsplineBasis computes the B-spline basis function N_{i,d}(t)
    % i: index of the basis function
    % d: degree of the B-spline
    % t: the parameter value
    % knots: knot vector

    if d == 0
        % Zeroth-degree basis function
        if knots(i) <= t && t < knots(i+1)
            N = 1;
        else
            N = 0;
        end
    else
        % Higher-degree basis function
        % Compute the left term
        if knots(i+d) == knots(i)
            left_term = 0;
        else
            left_term = ((t - knots(i)) / (knots(i+d) - knots(i))) * bsplineBasis(i, d-1, t, knots);
        end
        
        % Compute the right term
        if knots(i+d+1) == knots(i+1)
            right_term = 0;
        else
            right_term = ((knots(i+d+1) - t) / (knots(i+d+1) - knots(i+1))) * bsplineBasis(i+1, d-1, t, knots);
        end
        
        N = left_term + right_term;
    end
end

Following is the documentation link which will help you :

https://www.mathworks.com/help/curvefit/bspline.html