Constraint propagation using Projection rule

I’ve found this example of constraint propagation using projection rule enter image description here

We have

C = { x1 ≠ x2, x1 ≥ x2 }  < C; x1 ∈ {1,2,3}, x2 ∈ {1,2,3} > 

They say that applying propagation rule, does not give any simplification.

I’m not sure why this is the case. Shouldn’t we get?

< C; x1 ∈ {2,3}, x2 ∈ {1,2} > 

Other steps in the example, make sense that to me, e.g.

< C; x1 ∈ {2}, x2 ∈ {1,2,3} > 

produces

< C; x1 ∈ {2}, x2 ∈ {1} > 

How to use known Projection Matrices of cameras to generate new fundamental matrix located correctly in 3D space?

I have a few cameras which have been calibrated (using a checkerboard) so I know the fundamental matrix between an origin camera and each remaining camera

I wish to take pairs of camera with no fundamental matrix – but each has a projection matrix – and calculate the fundamental matrix with the view to reconstruct the stereo pair in 3D. To clarify – as all cameras have a projection matrix – I should be able to take any combination and generate a Fundamental matrix – so I can reconstruct and combine stereo pairs in 3D using as many pairs as possible

currently I use this formula: enter image description here

(decomposing Projection matrices from camera to get C)

but that didn’t quite work with stereo pairs that don’t include the origin. I noticed that to calculate the fundamental matrix that rotation was relative so I added the relative rotation formula when recalculating P from K/R/T (intrinsic camera params, Rotation matrix, Translation matrix)

enter image description here

but now each reconstruction appears on the camera position (I think) instead of all stereo pairs being mapped into same space to give a dense reconstruction

Does anyone know what I am doing wrong? thanks for any help

Graph of function, continuous projection

$ X$ and $ Y$ are topological spaces. $ f:X\rightarrow Y$ a map (we don’t suppose that $ f$ is continuous). Consider $ A=\{(x,f(x))\in X\times Y| x\in X\}$ . is $ \pi: A\rightarrow X$ , $ $ (x,f(x))\mapsto x$ $ a homeomorphism ? If not, is it enough to assume that $ A$ is closed subspace in $ X\times Y$ ? If $ X$ and $ Y$ are metrizable spaces, how to prove that $ \pi$ is a homemorphism using sequential continuity ? Suppose that by miracle $ \pi$ is homeomorphism but $ f$ is not continuous, is it possible such phenomenon ?

Projection of a polytope along 4 orthogonal axes

Consider the following problem:

Given an $ \mathcal{H}$ -polytope $ P$ in $ \mathbb{R}^d$ and $ 4$ orthogonal vectors $ v_1, …, v_4 \in \mathbb{R}^d$ , compute the projection of $ P$ to the subspace generated by $ v_1, …, v_4$ (and ouput it as an $ \mathcal{H}$ -polytope).

I know that the problem of computing projections along $ k$ orthogonal vectors in NP-hard (if $ k$ and $ d$ are part of the input), as shown in this paper. But does it help if $ k$ is a constant? Specifically, does it help if $ k \leq 4$ ? Do we have a polynomial algorithm in this case?

Estimating quality of projection

I asked this question on math.se, but didn’t receive an answer

Suppose we are given a vector $ v$ and vectors $ \mu_i$ :

$ v = \mu_1+\mu_2+…+\mu_m$ , where $ \mu_i \in R^n$ , all $ \mu_i$ are of unit length.

Oracle will give me $ k$ vectors $ \mu_{j_1}, \mu_{j_2},…\mu_{j_k}$ from the original set such that when I project $ v$ onto subspace spanned by these vectors the length of the projection is highest possible. In other words, from the set of all combinations of $ k$ vectors from $ [\mu_1,…\mu_n]$ the $ [\mu_{j_1}, \mu_{j_2},…\mu_{j_k}]$ give highest length of projection. Lets denote by $ v_{\text{proj}}$ projection of $ v$ onto $ [\mu_{j_1}, \mu_{j_2},…\mu_{j_k}]$

I want to estimate quality of projection before oracle gives me this $ k$ vectors. I want to give upper bound on $ ||v – v_{\text{proj}}|| $

As far as I understood it is very difficult to obtain these $ k$ vectors by myself. However, I know that for any two vectors $ \mu_i, \mu_j$ , $ ||\mu_i-\mu_j|| \leq \alpha$ , where $ \alpha$ is a given positive number.

Small values of $ \alpha$ will tell me that all $ \mu_i$ are close to each other and heading towards same direction. I would suspect then that projection will be good, and its length will be close to the length of original vector. How can I use this to give an upper bound $ ||v – v_{\text{proj}}|| $ ?

My attempts:

Without loss of generality lets assume that $ k$ optimal vectors are first $ k$ vectors in the list, i.e $ \mu_1,\mu_2,…\mu_k$ . Lets denote by $ P$ projection operator on the space spanned by $ \mu_1,\mu_2,…\mu_k$ .

$ \|v – v_{\text{proj}}\| = \|v – P(v)\| = \|v – P(\mu_1+\mu_2+…+\mu_m)\| = $

$ \|v – P(\mu_1) – P(\mu_2) – … – P(\mu_m)\| = $

$ \| v – \mu_1 – \mu_2 – … – \mu_k – P(\mu_{k+1}) – P(\mu_{k+2}) – … – P(\mu_m)\| = $

$ \|\mu_{k+1} – P(\mu_{k+1}) + \mu_{k+2} – P(\mu_{k+2}) + … + \mu_{m} – P(\mu_{m})\|$

$ \|v – v_{\text{proj}}\| \leq \|\mu_{k+1} – P(\mu_{k+1})\| + \|\mu_{k+2} – P(\mu_{k+2}) + … + \|\mu_{m} – P(\mu_{m})\|$

$ \|v – v_{\text{proj}}\| \leq (m-k)\alpha$

So in order to make $ \|v – v_{\text{proj}}\| \leq \epsilon$ , we need $ k \geq \frac{m\alpha – \epsilon}{\alpha}$

I am not satisfied with this result because $ k$ grows linearly with $ m$ . I want it to grow much slower, something like $ \log(m)$ . My goal is to show that under some constraints on $ \mu_i$ , we need only approximately $ \log(m)$ vectors to approximate $ v$ .

I think the bound can be improved substantially. First Cauchy inequality isn’t very tight and second, I used $ |\mu_{k+1} – P(\mu_{k+1})\| \leq \alpha$ which is also very loose.

I am open for additional constraints on $ \mu_1,…\mu_m$ to achieve logarithmic growth

Alex Ravsky on math.se has noted, that we also need a constraint on $ \alpha$ in order to achieve logarithmic growth. Assume that $ k$ $ \leq n$ , $ \mu_i$ is th $ i$ -th standard ort of the space $ \mathbb{R}^n$ , and $ \alpha = \sqrt{2}$ . Then $ \|v – v_{\text{proj}}\| = \sqrt{m-k}$

Projection of an invariant almost complex structure to a non integrable one

My apology in advance if my question is obvious or elementary

We identify elements of $ S^3$ with their quaternion representation $ x_1+x_2 i +x_3 j +x_4 k$ . We consider two independent vector fields $ S_1(a)=ja$ and $ S_2(a)=ka$ on $ S^3$ . On the other hand $ P: S^3\to S^2$ is a $ S^1$ -principal bundle with the obvious action of $ S^1$ on $ S^3$ . Then the span of $ S_1, S_2$ is the standard horizontal space associated to the standard connection of the principal bundle $ S^3 \to S^2$ . Then each horizontal space has an almost complex structure $ J$ . This is the standard structure associated to $ S_1, S_2$ coordinate.

Is this structure invariant under the action of $ S^1$ ? If yes, we can define a unique almost complex structure on $ S^2$ which is $ P$ related to the structure on total space. Now is this structure on $ S^2$ integrable?

As a similar question, is there an example of a principal bundle $ P\to X,$ such that $ P$ is a real manifold and $ X$ is a complex manifold and a connection admit an invariant almost complex structure which project to a non integrable structure?

Uniqueness of projection under spectral norm

I am considering $ $ \min_{M\in \mathcal{M}} \|X – M\|:=x \neq 0, $ $ where $ X$ , $ M$ are $ m\times n$ matrices, $ \|\cdot\|$ is spectral norm and $ \mathcal{M}$ is a matrix subspace. I wonder to what extent the solutions are unique, i.e. what can we say about the set $ $ \mathcal{M}^* = \{M|\|X – M\|=x\}. $ $ Currently, I only know it is a convex set. For $ M_1, M_2 \in \mathcal{M}^*$ , let $ M = p M_1 + (1-p) M_2$ , $ p \in (0,1)$ . For any vector $ v$ , using Cauchy’s inequality, we have $ $ v^T(X – M)^T(X – M)v \leq x^2, $ $ thus $ M \in \mathcal{M}^*$ . Moreover, the equality holds only when $ $ (X – M_1)v = (X – M_2)v = u,~~~~s.t.~u^T u = x^2. $ $ I wonder whether there are more properties I could say about this set, e.g. its dimension/the properties of the common eigenvectors/the condition under which the solution is unique.

Projection on space of permutations of lower diagonal matrices

Let $ G$ be the group of all $ n\times n$ permutation matrices and let $ V$ be the vector space of all $ n$ -dimensional lower diagonal matrices. Then I define the set $ $ X = \{P\cdot L\cdot P^T \mid \forall P\in G, L\in V\}, $ $ where “$ \cdot$ ” is matrix multiplication. That is, $ X$ is the set of all lower diagonal matrices with possible rows and colums that are permuted at the same time.

I am interested in the orthogonal projection of a general matrix $ M\in\mathbb{R}^{n\times n}$ on the set $ X\subset \mathbb{R}^n$ . Does anyone know if there is a more efficient method to do it than projecting on each space $ $ V_P = \{P\cdot L\cdot P^T \mid \forall L\in V\} $ $ for every individual $ P\in G$ and then taking the shortest one?

Thank you in advance!

On the projection onto the intersection of line segment and convex set

Let $ \mathcal{H}$ be a real Hilbert space. For given $ a \neq b \in \mathcal{H}$ , we denote the line segment by $ $ \left[ a , b \right] := \left\lbrace \left( 1 – t \right) a + t b \ \colon \ t \in \left[ 0 , 1 \right] \right\rbrace . $ $ Consider a convex set $ C$ and assume that we can compute its projection $ P_{C} \left( x \right)$ for every $ x \in \mathcal{H}$ .

Can we deduce the projection formula for the set $ \left[ a , b \right] \cap C$ ?

It is clear that $ P_{\left[ a , b \right]} \left( x \right) = x$ for $ x \in \left[ a , b \right]$ . When $ x \notin \left[ a , b \right]$ , we can compute its projection onto $ \left[ a , b \right]$ notice that \begin{align} \left\lVert x – a \right\rVert & \leq \left\lVert x – y \right\rVert + \left\lVert y – a \right\rVert , \qquad \forall y \in \left[ a , b \right] , \ \left\lVert x – b \right\rVert & \leq \left\lVert x – y \right\rVert + \left\lVert y – b \right\rVert , \qquad \forall y \in \left[ a , b \right] . \end{align} Taking the infimum on bothsides we can also get an explicit formula for the projection, in particular $ $ P_{\left[ a , b \right]} \left( x \right) = \begin{cases} x & \textrm{ if } x \in \left[ a , b \right] , \ a & \textrm{ if } x \notin \left[ a , b \right] \textrm{ and } \left\lVert x – a \right\rVert \leq \left\lVert x – b \right\rVert , \ b & \textrm{ if } x \notin \left[ a , b \right] \textrm{ and } \left\lVert x – a \right\rVert \geq \left\lVert x – b \right\rVert . \end{cases} $ $

I’m not sure about the projection of the intersection. I guess it would be the composition of two projection in a correct order but I don’t know how to proceed.