I posted some time ago a CGAL question that was kindly answered by pointing to the Polyhedron demo and the corefinement plugin. The basic idea being that one open polyhedron A is cut by another open polyhedron B, and I need the list of intersection half edges owned by A, or better, A minus the part of A in B.
The co-refinement demo does this, but I want to select, as a result, all parts of A not in B. This does not match the available predicates in the demo (A - B (leaves parts of B inside A) , B - A (leaves parts of B outside A), A inter B, A union B). I tried combining/modifying them to get what I want but I must be missing something. The information on the 'darts' seem to be mutually exclusive.
The picture below illustrates this : A as been cut by B (I have a hole with the shape of B) but some parts of B are still in A (the facets on the hole border).
(edit : sorry : not enough reputation to post an image here :-()
Any advices on how to write a predicate that would select only A with a hole, and leave out any face coming from B?
Thank you!
Related
When I try the bokeh segmentation effect using body-pix#1.0.0, It detects/segments the person (A) in front of the camera. If another person (B) is standing behind, away from A, B is being blurred out. If the person B comes very close to the contour of A, then person B is also getting detected. This is the preferred behaviour.
Now when I try with body-pix#2.0.0, both Person A and B are getting detected even though I am using segmentPerson API. Pls note, person B is standing much away from person A, still both are getting detected. The advantage I see with 2.0 is that the contour of the person detected is much more accurate and smoother than that in 1.0 which had a gap in the contour and the bokeh effect was missing around this gap. In 2.0, the contour is more accurate. But multiple people are getting detected. Is there any parameter I could tweak to restrict this to single person detection and use the smoother contour?
Thanks
For those who wants to know the answer. Source: https://github.com/tensorflow/tfjs/issues/2547
If you want to use BodyPix 2.0 to only blur just a subset of people (e.g. the large people), a quick way would be to use BodyPix 2.0's Multi-Person Segmentation API: https://github.com/tensorflow/tfjs-models/tree/master/body-pix#multi-person-segmentation.
This method returns an array of PersonSegmentation object. In your case it will be an array of two PersonSegmentation object: one for Person A and one for Person B.
You could then remove certain people (in your case Person B) from that array and pass the resulting array (with only 1 element: Person A) to the drawBokehEffect https://github.com/tensorflow/tfjs-models/tree/master/body-pix#bodypixdrawbokeheffect.
To automate this process for other cases (3 or more people):
Each PersonSegmentation object has a .pose field that contains the 2D coordinates (in image pixel space) of the person's 17 keypoints. They can be used to compute the smallest bounding box area for each person. The person bounding box area can then be used as a criteria to remove small people in the image.
I was trying to solve Google Code Jam problems and there is one of them that I don't understand. Here is the question (World Finals 2013 - problem C): https://code.google.com/codejam/contest/2437491/dashboard#s=p2&a=2
And here follows the problem analysis: https://code.google.com/codejam/contest/2437491/dashboard#s=a&a=2
I don't understand why we can use binary search. In order to use binary search the elements have to be sorted. In order words: for a given element e, we can't have any element less than e at its right side. But that is not the case in this problem. Let me give you an example:
Suppose we do what the analysis tells us to do: we start with a left bound angle of 90° and a right bound angle of 0°. Our first search will be at angle of 45°. Suppose we find that, for this angle, X < N. In this case, the analysis tells us to make our left bound 45°. At this point, we can have discarded a viable solution (at, let's say, 75°) and at the same time there can be no more solutions between 0° and 45°, leading us to say that there's no solution (wrongly).
I don't think Google's solution is wrong =P. But I can't figure out why we can use a binary search in this case. Anyone knows?
I don't understand why we can use binary search. In order to use
binary search the elements have to be sorted. In order words: for a
given element e, we can't have any element less than e at its right
side. But that is not the case in this problem.
A binary search works in this case because:
the values vary by at most 1
we only need to find one solution, not all of them
the first and last value straddle the desired value (X .. N .. 2N-X)
I don't quite follow your counter-example, but here's an example of a binary search on a sequence with the above constraints. Looking for 3:
1 2 1 1 2 3 2 3 4 5 4 4 3 3 4 5 4 4
[ ]
[ ]
[ ]
[ ]
*
I have read the problem and in the meantime thought about the solution. When I read the solution I have seen that they have mostly done the same as I would have, however, I did not thought about some minor optimizations they were using, as I was still digesting the task.
Solution:
Step1: They choose a median so that each of the line splits the set into half, therefore there will be two provinces having x mines, while the other two provinces will have N - x mines, respectively, because the two lines each split the set into half and
2 * x + 2 * (2 * N - x) = 2 * x + 4 * N - 2 * x = 4 * N.
If x = N, then we were lucky and accidentally found a solution.
Step2: They are taking advantage of the "fact" that no three lines are collinear. I believe they are wrong, as the task did not tell us this is the case and they have taken advantage of this "fact", because they assumed that the task is solvable, however, in the task they were clearly asking us to tell them if the task is impossible with the current input. I believe this part is smelly. However, the task is not necessarily solvable, not to mention the fact that there might be a solution even for the case when three mines are collinear.
Thus, somewhere in between X had to be exactly equal to N!
Not true either, as they have stated in the task that
You should output IMPOSSIBLE instead if there is no good placement of
borders.
Step 3: They are still using the "fact" described as un-true in the previous step.
So let us close the book and think ourselves. Their solution is not bad, but they assume something which is not necessarily true. I believe them that all their inputs contained mines corresponding to their assumption, but this is not necessarily the case, as the task did not clearly state this and I can easily create a solvable input having three collinear mines.
Their idea for median choice is correct, so we must follow this procedure, the problem gets more complicated if we do not do this step. Now, we could search for a solution by modifying the angle until we find a solution or reach the border of the period (this was my idea initially). However, we know which provinces have too much mines and which provinces do not have enough mines. Also, we know that the period is pi/2 or, in other terms 90 degrees, because if we move alpha by pi/2 into either positive (counter-clockwise) or negative (clockwise) direction, then we have the same problem, but each child gets a different province, which is irrelevant from our point of view, they will still be rivals, I guess, but this does not concern us.
Now, we try and see what happens if we rotate the lines by pi/4. We will see that some mines might have changed borders. We have either not reached a solution yet, or have gone too far and poor provinces became rich and rich provinces became poor. In either case we know in which half the solution should be, so we rotate back/forward by pi/8. Then, with the same logic, by pi/16, until we have found a solution or there is no solution.
Back to the question, we cannot arrive into the situation described by you, because if there was a valid solution at 75 degrees, then we would see that we have not rotated the lines enough by rotating only 45 degrees, because then based on the number of mines which have changed borders we would be able to determine the right angle-interval. Remember, that we have two rich provinces and two poor provinces. Each rich provinces have two poor bordering provinces and vice-versa. So, the poor provinces should gain mines and the rich provinces should lose mines. If, when rotating by 45 degrees we see that the poor provinces did not get enough mines, then we will choose to rotate more until we see they have gained enough mines. If they have gained too many mines, then we change direction.
I have graph like one in Figure 1 (the first image) and want to connect the red nodes to have cycle, but cycles do not have to be Hamiltonian like Figure 2 and Figure 3 (the last two images). The problem has much bigger search space than TSP since we can visit a node twice. Like the TSP, it is impossible to evaluate all the combinations in a large graph and I should try heuristic, but the problem is that, unlike the TSP, the length of cycles or tours is not fixed here. Because, visiting all the blue nodes is not mandatory and this cause having variable length including some of the blue nodes. How can I generate a possible "valid" combination every time for evaluation? I mean, a cycle can be {A, e, B, l, k, j, D, j, k, C, g, f, e} or {A, e, B, l, k, j, D, j, i, h , g, C, g, f, e}, but not {A, e, B, l, k, C, g, f, e} or {A, B, k, C, i, D}.
Update:
The final goal is to evaluate which cycle is optimal/near optimal considering length and risk (see below). So I am not only going to minimize the length but minimizing the risk as well. This cause not being able to evaluate risk of cycle unless you know all its nodes sequence. Hope this clarifies why I can not evaluate new cycle at the middle of its generating process.
We can:
generate and evaluate possible cycles one by one;
or generate all possible cycles and then do their evaluation.
Definition of the risk:
Assume cycle is a ring which connects primary node (one of the red nodes) to all other red nodes. In case of failure in any part (edge) of the ring, no red nodes should be disconnected form the primary node (this is desired). However there are some edges we have to pass twice (due to not having Hamiltonian cycle which connects all the red nodes) and in case of failure in those edges, some of red nodes may be totally disconnected. So risk of cycle is summation of the length of risky edges (we have twice in our ring/tour) multiplied by number of red nodes we lose in case of cutting each risky edge.
A real example of 3D graph I am working on including 5 red nodes and 95 blue nodes is in below:
And here is link to Excel sheet containing adjacency matrix of the above graph (the first five nodes are red and the rests are blue).
Upon a bit more reflection, I decided it's probably better to just rewrite my solution, as the fact that you can use red nodes twice, makes my original idea of mapping out the paths between red nodes inefficient. However, it isn't completely wasted, as the blue nodes between red nodes is important.
You can actually solve this using a modified version of BFS, as more-less a backtracking algorithm. For each unique branch the following information is stored, most of which simply allows for faster rejection at the cost of more space, only the first two items are actually required:
The full current path. (list with just the starting red node)
The remaining red nodes. (initially all red nodes)
The last red node. (initially the start red node)
The set of blue nodes since last red node. (initially empty)
The set of nodes with a count of 1. (initially empty)
The set of nodes with a count of 2. (initially empty)
The algorithm starts with a single node then expands adjacent nodes using BFS or DFS, this repeats until the result is a valid tour or is the node to be expanded is rejected. So the basic psudoish code (current path and remaining red points) looks something like below. Where rn is the set of red nodes, t is the list of valid tours, p/p2 is a path of nodes, r/r2 is a set of red nodes, v is the node to be expanded, and a is a possible node to expand to.
function PATHS2HOME(G,rn)
create a queue Q
create a list t
p = empty list
v ← rn.pop()
r ← rn
add v to p
Q.enqueue((p,r))
while Q is not empty
p, r ← Q.dequeue()
if r is empty and the first and last elements of p are the same:
add p to t
else
v ← last element of p
for all vertices a in G.adjacentVertices(v) do
if canExpand(p,a)
p2 ← copy(p)
r2 ← copy(r)
add a to the end of p2
if isRedNode(a) and a in r2
remove a from r2
Q.enqueue( (p2,r2) )
return t
The following conditions prevent expansion of a node. May not be a complete list.
Red nodes:
If it is in the set of nodes that have a count of 2. This is because the red node would have been used more than twice.
If it is equal to the last red node. This prevents "odd" tours when a red node is adjacent to three other blue nodes. Thus say the red node A, was adjacent to blue nodes b, c and d. Then you would end a tour where part of the tour looks like b-A-c-A-d.
Blue nodes:
If it is in the set of nodes that have a count of 2. This is because the red node would have been used more than twice.
If it is in the set of blue nodes since last red node. This is because it would cause a cycle of blue nodes between red nodes.
Possible optimizations:
You could map out the paths between red nodes, use that to build something of a suffix tree, that shows red nodes that can be reached given the following path Like. The benefit here is that you avoid expanding a node if the path that expansion leads to red nodes that have already been visited twice. Thus this is only a useful check once at least 1 red node has been visited twice.
Use a parallel version of the algorithm. A single thread could be accessing the queue, and there is no interaction between elements in the queue. Though I suspect there are probably better ways. It may be possible to cut the runtime down to seconds instead of hundreds of seconds. Although that depends on the level of parallelization, and efficiency. You could also apply this to the previous algorithm. Actually the reasons for which I switched to using this algorithm are pretty much negated by
You could use a stack instead of a queue. The main benefit here is by using a depth-first approach, the size of the queue should remain fairly small.
Consider the DFA :
What will be δ(A,01) equal to ?
options:
A) {D}
B) {C,D}
C) {B,C,D}
D) {A,B,C,D}
The correct answer is option B) but I don't get how. Please some one explain me the steps to solve it and also in general how do we solve sit for any DFA and any transition?
Thanks.
B) Option is not Correct answer! for this transition graph.
In Transition Graph(TG) symbol ε means NULL-move (ε-move). There is two NULL-moves in TG.
One: (A) --- `ε` ---->(B)
Second: (A) --- `ε` ---->(C)
A ε-move means without consume any symbol you can change state. In your diagram from A to B, or A to C.
What will be δ(A,01) equal to ?
Question asked "what is path from state A if input is 01". (as I understand because there is only one final state)
01 can be processed in either of two ways.
(A) -- ε --->(B) -- 0 --> (B) -- 1 --> (D)
(A) -- ε --->(C) -- 0 --> (B) -- 1 --> (D)
Also, there is no other way to process string 01 even if you don't wants to reach final state.
[ANSWER]
So there is misprinting in question (or either you did).
You can learn how to remove NULL-move from transition graph. Also HOW TO WRITE REGULAR EXPRESSION FOR A DFA
If you remove null-moves from TG you will get three ways to accept 01.
EQUIVALENT TRANSITION GRAPH WITHOUT NULL-MOVE
Note there is three start-states in Graph.
Three ways:
{ABD}
{CBD}
{BBD}
In all option state-(B) has to be come.
Also, you have written Consider the DFA : is wrong. The TG is not deterministic because there is there is non-deterministic move δ(A,ε) and next states are either B or C.
Given:
I have no idea what the accepted language is.
From looking at it you can get several end results:
1.) bb
2.) ab(a,b)
3.) bbab(a, b)
4.) bbaaa
How to write regular expression for a DFA
In any automata, the purpose of state is like memory element. A state stores some information in automate like ON-OFF fan switch.
A Deterministic-Finite-Automata(DFA) called finite automata because finite amount of memory present in the form of states. For any Regular Language(RL) a DFA is always possible.
Let's see what information stored in the DFA (refer my colorful figure).
(note: In my explanation any number means zero or more times and Λ is null symbol)
State-1: is START state and information stored in it is even number of a has been come. And ZERO b.
Regular Expression(RE) for this state is = (aa)*.
State-4: Odd number of a has been come. And ZERO b.
Regular Expression for this state is = (aa)*a.
Figure: a BLUE states = EVEN number of a, and RED states = ODD number of a has been come.
NOTICE: Once first b has been come, move can't back to state-1 and state-4.
State-5: comes after Yellow b. Yellow b means b after odd numbers of a.
Once you gets b after odd numbers of a(at state-5) every thing is acceptable because there is self a loop for (b,a) at state-5.
You can write for state-5 : Yellow-b followed-by any string of a, b that is = Yellow-b (a + b)*
State-6: Just to differentiate whether odd a or even.
State-2: comes after even a then b then any number of b. = (aa)* bb*
State-3: comes after state-2 then first a then there is a loop via state-6.
We can write for state-3 comes = state-2 a (aa)* = (aa)*bb* a (aa)*
Because in our DFA, we have three final states so language accepted by DFA is union (+ in RE) of three RL (or three RE).
So the language accepted by the DFA is corresponding to three accepting states-2,3,5, And we can write like:
State-2 + state-3 + state-5
(aa)*bb* + (aa)*bb* a (aa)* + Yellow-b (a + b)*
I forgot to explain how Yellow-b comes?
ANSWER: Yellow-b is a b after state-4 or state-3. And we can write like:
Yellow-b = ( state-4 + state-3 ) b = ( (aa)*a + (aa)*bb* a (aa)* ) b
[ANSWER]
(aa)*bb* + (aa)*bb* a (aa)* + ( (aa)*a + (aa)*bb* a (aa)* ) b (a + b)*
English Description of Language: DFA accepts union of three languages
EVEN NUMBERs OF a's, FOLLOWED BY ONE OR MORE b's,
EVEN NUMBERs OF a's, FOLLOWED BY ONE OR MORE b's, FOLLOWED BY ODD NUMBERs OF a's.
A PREFIX STRING OF a AND b WITH ODD NUMBER OF a's, FOLLOWED BY b, FOLLOWED BY ANY STRING OF a AND b AND Λ.
English Description is complex but this the only way to describe the language. You can improve it by first convert given DFA into minimized DFA then write RE and description.
Also, there is a Derivative Method to find RE from a given Transition Graph using Arden's Theorem. I have explained here how to write a regular expression for a DFA using Arden's theorem. The transition graph must first be converted into a standard form without the null-move and single start state. But I prefer to learn Theory of computation by analysis instead of using the Mathematical derivation approach.
I guess this question isn't relevant anymore :) and it's probably better to guide you through it then just stating the answer, but I think I got a basic expression that covers it (it's probably minimizable), so i'll just write it down for future searchers
(aa)*b(b)* // for stoping at 2
U
(aa)*b(b)*a(aa)* // for stoping at 3
U
(aa)*b(b)*a(aa)*b((a)*(b)*)* // for stoping at 5 via 3
U
a(aa)*b((a)*(b)*)* // for stoping at 5 via 4
The examples (1 - 4) that you give there are not the language accepted by the DFA. They are merely strings that belong to the language that the DFA accepts. Therefore, they all fall in the same language.
If you want to figure out the regular expression that defines that DFA, you will need to do something called k-path induction, and you can read up on it here.