I'm trying to work my way through an image processing paper (Creating Full View Panoramic Image Mosaics and Environment Maps) and am stuck on how the Jacobian is calculated, especially terms ($J_{17}, J_{18}, J_{27}, J_{28}$) in Section 3. The math is done in homography coordinates.
$$\boldsymbol{x}=[x,y,1]^T$$ $$D= \begin{bmatrix} d_0 & d_1 & d_2 \\ d_3 & d_4 & d_5 \\ d_6 & d_7 & 0 \end{bmatrix}$$
The tilda $\sim$ indicates that the left/right side are equal to scale - the below matrix is element-wise divided by its final term. $$\boldsymbol{x}''=[x'',y'',1]^T\sim(I+D)x = \begin{bmatrix} (1+d_0)x + d_1y + d_2 \\ d_3x + (1+d_4)y + d_5 \\ d_6x + d_7y + 1 \end{bmatrix} $$ This yields: $$\boldsymbol{x''}= \begin{bmatrix} \frac{(1+d_0)x + d_1y + d_2}{d_6x + d_7y + 1} \\ \frac{d_3x + (1+d_4)y + d_5}{d_6x + d_7y + 1} \\ 1 \end{bmatrix} $$ Now taking the Jacobian w.r.t $d=[d_0,d_1,...,d_8]^T$ $$J_d(x)=\frac{\partial{x''}}{\partial{d}}= \begin{bmatrix} x & y & 1 & 0 & 0 & 0 & -x^2 & -xy \\ 0 & 0 & 0 & x & y & 1 & -xy & -y^2 \end{bmatrix} $$ I can see how the the terms in columns 1-6 could be calculated from $(I+D)x$ but not from $x''$, moreover the last 2 columns make absolutely no sense to me. Also its possible $\boldsymbol{x}''=[x'',y'']$ as it would explain $J$ having 2 vs. 3 rows. The paper's vague there.