I am trying to find the cylindrical projection of an image using the cylindrical projection equations. After finding the warpped values for the X and Y coordinates, I am trying to use the inbuilt MATLAB imwarp function to project the image onto the new coordinates, so that I can use the interpolation for smooth image. However, I get a different warpped image. Here is the output what I am getting:

What I expect the image to be:

Below is the cylindrical projection code:

clc; close all; clear;

% Inputs
fileName = 'checker.jpg';

% Focal lengths
fx = 50;
fy = 50;

% Read image
image = (imread(fileName));

% Get image size
[ydim, xdim, bypixs] = size(image);

% Camera intrinsics
K = [fx, 0, xdim/2; 0, fy, ydim/2; 0, 0, 1];

% Distortion coefficients [k1, k2, k3, p1, p2]
DC = [0, 0, 0, 0, 0];

% Get distrotion coefficients
fx = K(1,1);
fy = K(2,2);
k1 = DC(1);
k2 = DC(2);
k3 = DC(3);
p1 = DC(4);
p2 = DC(5);

% Get image size
[ydim, xdim, bypixs] = size(image);

% Get the center of image
xc = xdim/2;
yc = ydim/2;

% Create X and Y coordinates grid
[X,Y] = meshgrid(1:xdim, 1:ydim);

% Perform the cylindrical projection
theta = (X - xc) / fx;
h = (Y - yc) / fy;

% Cylindrical coordinates to Cartesian
xcap = sin(theta);
ycap = h;
zcap = cos(theta);

xyz_cap = cat(3, xcap, ycap, zcap);
xyz_cap = reshape(xyz_cap,[],3);

% Normalized coords
xyz_cap_norm = (K * xyz_cap')';
xn = xyz_cap_norm(:,1) ./ xyz_cap_norm(:,3);
yn = xyz_cap_norm(:,2) ./ xyz_cap_norm(:,3);

% Radial and tangential distortion 
r = xn.^2 + yn.^2;
xd_r = xn .* (1 + k1 * r.^2 + k2 * r.^4 + k3 * r.^6);
yd_r = yn .* (1 + k1 * r.^2 + k2 * r.^4 + k3 * r.^6);

xd_t = 2 * p1 * xn .* yn + p2 * (r.^2 + 2 * xn.^2);
yd_t = p1 * (r.^2 + 2 * yn.^2) + 2 * p2 * xn .* yn;

xd = xd_r + xd_t;
yd = yd_r + yd_t;

% Convert to 2D array
xd = reshape(ceil(xd),[ydim, xdim]);
yd = reshape(ceil(yd),[ydim, xdim]);

% Get projections
imageCylindrical = imwarp(image, cat(3,xd, yd), 'cubic');
imshow(imageCylindrical)

Please let me know what I am doing wrong using the imwarp function. Any help is appreciated!