Constraints based on Magnitudes

Unless properly taken care, the phase based algebraic constraints can have problems with the phase wrap around. We now present a rank-three constraint based on magnitudes of the vector Fourier coefficients. We start with Equation 4. This equation can be rewritten as

$$U^l[k] = (m_{l11} U^0[k] + m_{l12} V^0[k]) \exp({\frac{j2\pi\lambda_l k}{N}})$$

$$V^l[k] = (m_{l21} U^0[k] + m_{l22} V^0[k]) \exp({\frac{j2\pi\lambda_l k}{N}})$$

Writing in terms of the real and imaginary components of the complex numbers

$$U^l_R[k] + j U^l_I[k] = ((m_{l11} U^0_R[k] + m_{l12} V^0_R[k]) +$$

$$j ( m_{l11} U^0_I[k] + m_{l12} V^0_I[k]) ) e^{ \frac{j 2 \pi \lambda_l k}{N}}$$

Taking the square of the magnitudes of both sides, we get

$$\vert U^l[k]\vert^2 = (U^l_R[k])^2 + (U^l_I[k])^2$$

$$= (m_{l11} U^0_R[k] + m_{l12} V^0_R[k])^2 +$$

$$( m_{l11} U^0_I[k] + m_{l12} V^0_I[k]) ) ^2$$

$$= m_{l11}^2 (U^0_R[k]) ^ 2 + m_{l12}^2 ( V^0_R[k] )^2 +$$

$$2 (m_{l11} U^0_R[k])(m_{l12} V^0_R[k]) +$$

$$m_{l11}^2 (U^0_I[k]) ^ 2 + m_{l12}^2 ( V^0_I[k] )^2 +$$

$$2 (m_{l11} U^0_I[k])(m_{l12} V^0_I[k])$$

$$= m_{l11}^2 [(U^0_R[k])^2 + (U^0_I[k]) ^ 2 ] +$$

$$m_{l12}^2 [( V^0_R[k] )^2 + ( V^0_I[k] )^2 ] +$$

$$2 m_{l11} m_{l12} [ U^0_R[k] V^0_R[k] + U^0_I[k] V^0_I[k] ]$$ (14)

Similarly,

$$\vert V^l[k]\vert^2 = (V^l_R[k])^2 + (V^l_I[k])^2$$

$$= m_{l21}^2 [(U^0_R[k])^2 + (U^0_I[k]) ^ 2 ] +$$

$$m_{l22}^2 [( V^0_R[k] )^2 + ( V^0_I[k] )^2 ] +$$

$$2 m_{l21} m_{l22} [ U^0_R[k] V^0_R[k] + U^0_I[k] V^0_I[k] ]$$ (15)

Its evident from equations 14 and 15 that the magnitude of the components of the Fourier domain representation in any view can be expressed in terms of the components in a reference view.

This result can also be expressed in the following manner. Given $M$ views, we can construct a ( $2M+1) \times (N-1)$ matrix as follows. The first row consists of the sum of products $( U^0_R[k] V^0_R[k] + U^0_I[k] V^0_I[k])$, $0$ being the reference view. Every view contributes two rows to this matrix (except the reference view, which contributes 3 rows) the magnitudes of $U$ in one row and the magnitudes of $V$ in the other. Let $\Theta ^{\prime \prime \prime }$ =

$${\tiny \left [ \begin{array}{ccc} (U^0_R[1] V^0_R[1] + U^0_... ...... & ((V^M_R[G])^2 + (V^M_I[G]) ^ 2) \end{array} \right ] }$$ (16)

(using $G$ for $(N - 1)\;$)

From equations 14 and 15, one can conclude that the rank of $\Theta^{''}$ is 3, irrespective of the number of views. Therefore, the constraint,

$$rank(\Theta^{\prime \prime \prime}) = 3$$ (17)

is a necessary condition for recognition in multiple views related by affine image-to-image homographies. This observation is consistent with the notion that the various views of a shape lie in a lower dimensional linear subspace. We can also say that the squares of the magnitudes of the Fourier Domain representation of a contour can be used as a signature of the boundary. These are, naturally, view independent as they can be computed from a single view.

Figure 1: Four affine transformed views of a dinosaur

(a)

(b)

(c)

(d)