Fit image datapoints to curve

I'm trying to control a robot from an ultrasonic greyscale image. So far I've used Sobel filtering to get a binary image, and Hough transform to detect line segments.

Now I want to create a control output to make the midpoints of the detected Houghlines aline most accurately with the drawn arc. The robot can move in the X and Y direction as well as rotate. The reference values I get from the code beneath contains [x-coordinate of midpoint, y-coordinate of midpoint, angle of the line segment, distance from midpoint to arc]. Does anyone know how to obtain the optimal rotation and movement to aline the points with the arc?

       I = imread('Weld_ultrasound.png');
        I = rgb2gray(I);
        BW2 = edge(I,'Sobel','nothinning');
        [H,T,R] = hough(BW2,'Theta',-90:1:89);
        P = houghpeaks(H,20,'threshold',ceil(0.2*max(H(:))));
        lines = houghlines(BW2,T,R,P,'FillGap',5,'MinLength',250);
     function ref = Houghlines2ref(lines)
     min_dist = 950;                % Set minimum distance
     max_dist = 1050;             % Set maximum distance
     ref = [];                              % [angle,distance]

     for i = 1:length(lines)
       mid = (lines(i).point1+lines(i).point2)/2;
       if sqrt(mid(1)^2+mid(2)^2) <= min_dist       
        ref = [ref; [mid,lines(i).theta, (sqrt(mid(1)^2+mid(2)^2) - min_dist)]];
       elseif sqrt(mid(1)^2+mid(2)^2) >= max_dist
        ref = [ref; [mid,lines(i).theta,(sqrt(mid(1)^2+mid(2)^2) - max_dist)]];
       end
     end

    I = imread('Weld_Ultrasound.png');
    I = rgb2gray(I);
    figure, imshow(I), hold on
    if isempty(ref) == 1
        return
    else    
        for k = 1:length(ref(:,1))
            p0 = [ref(k,1),ref(k,2)];
            [X,Y] = pol2cart(deg2rad(ref(k,3)),-ref(k,4));
            xy = [p0; p0+[X,Y]];
            plot(p0(1),p0(2),'rs')
            plot(xy(:,1),xy(:,2),'LineWidth',1,'Color','green'); % Plot referance vectors
        end
    end
    t = linspace(0,0.5*pi,128);     % Plot arc
    x = [0 1000*cos(t) 0];
    y = [0 1000*sin(t) 0];
    hold on
    plot( x, y, 'r' )
    end

Thanks

Original image;

Answer 1

[Matt J]

You would use a 2D version of an Iterative Closest Point (ICP) algorithm. This FEX contribution appears to have one:

https://www.mathworks.com/matlabcentral/fileexchange/12627-iterative-closest-point-method