Pose-SLAM and rendering
Frank Dellaert
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247172dae8
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e52685d441
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{
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Efficent Marginals Computation\n",
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"\n",
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"GTSAM can very efficiently calculate marginals in Bayes trees. In this post, we illustrate the “shortcut” mechanism for **caching** the conditional distribution $P(S \\mid R)$ in a Bayes tree, allowing efficient other marginal queries. We assume familiarity with **Bayes trees** from [the previous post](#)."
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]
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},
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"cell_type": "markdown",
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"source": [
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"## Toy Example\n",
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"\n",
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"We create a small Bayes tree:\n",
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"\n",
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"\\begin{equation}\n",
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"P(a \\mid b) P(b,c \\mid r) P(f \\mid e) P(d,e \\mid r) P(r).\n",
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"\\end{equation}\n",
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"\n",
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"Below is some Python code (using GTSAM’s discrete wrappers) to define and build the corresponding Bayes tree. We'll use a discrete example, i.e., we'll create a `DiscreteBayesTree`.\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"from gtsam import DiscreteConditional, DiscreteBayesTree, DiscreteBayesTreeClique, DecisionTreeFactor"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Make discrete keys (key in elimination order, cardinality):\n",
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"keys = [(0, 2), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)]\n",
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"names = {0: 'a', 1: 'f', 2: 'b', 3: 'c', 4: 'd', 5: 'e', 6: 'r'}\n",
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"aKey, fKey, bKey, cKey, dKey, eKey, rKey = keys\n",
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"keyFormatter = lambda key: names[key]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"outputs": [],
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"source": [
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"# 1. Root Clique: P(r)\n",
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"cliqueR = DiscreteBayesTreeClique(DiscreteConditional(rKey, \"0.4/0.6\"))\n",
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"\n",
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"# 2. Child Clique 1: P(b, c | r)\n",
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"cliqueBC = DiscreteBayesTreeClique(\n",
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" DiscreteConditional(\n",
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" 2, DecisionTreeFactor([bKey, cKey, rKey], \"0.3 0.7 0.1 0.9 0.2 0.8 0.4 0.6\")\n",
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" )\n",
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")\n",
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"\n",
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"# 3. Child Clique 2: P(d, e | r)\n",
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"cliqueDE = DiscreteBayesTreeClique(\n",
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" DiscreteConditional(\n",
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" 2, DecisionTreeFactor([dKey, eKey, rKey], \"0.1 0.9 0.9 0.1 0.2 0.8 0.3 0.7\")\n",
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" )\n",
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")\n",
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"\n",
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"# 4. Leaf Clique from Child 1: P(a | b)\n",
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"cliqueA = DiscreteBayesTreeClique(DiscreteConditional(aKey, [bKey], \"1/3 3/1\"))\n",
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"\n",
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"# 5. Leaf Clique from Child 2: P(f | e)\n",
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"cliqueF = DiscreteBayesTreeClique(DiscreteConditional(fKey, [eKey], \"1/3 3/1\"))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Build the BayesTree:\n",
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"bayesTree = DiscreteBayesTree()\n",
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"\n",
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"# Insert root:\n",
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"bayesTree.insertRoot(cliqueR)\n",
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"\n",
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"# Attach child cliques to root:\n",
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"bayesTree.addClique(cliqueBC, cliqueR)\n",
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"bayesTree.addClique(cliqueDE, cliqueR)\n",
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"\n",
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"# Attach leaf cliques:\n",
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"bayesTree.addClique(cliqueA, cliqueBC)\n",
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"bayesTree.addClique(cliqueF, cliqueDE)\n",
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"\n",
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"# bayesTree.print(\"bayesTree\", keyFormatter)"
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]
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},
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{
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"execution_count": 5,
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"source": [
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"import graphviz\n",
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"graphviz.Source(bayesTree.dot(keyFormatter))"
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]
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},
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{
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"source": [
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"## Naive Computation of P(a)\n",
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"The marginal $P(a)$ can be computed by summing out the other variables in the tree:\n",
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"$$\n",
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"P(a) = \\sum_{b, c, d, e, f, r} P(a, b, c, d, e, f, r)\n",
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"$$\n",
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"\n",
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"Using the Bayes tree structure, we have\n",
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"\n",
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"$$\n",
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"P(a) = \\sum_{b, c, d, e, f, r} P(a \\mid b) P(b, c \\mid r) P(f \\mid e) P(d, e \\mid r) P(r) \n",
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"$$\n",
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"\n",
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"but we can ignore variables $e$ and $f$ not on the path from $a$ to the root $r$. Indeed, by associativity we have\n",
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"\n",
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"$$\n",
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"P(a) = \\sum_{r} \\Bigl\\{ \\sum_{e,f} P(f \\mid e) P(d, e \\mid r) \\Bigr\\} \\sum_{b, c, d} P(a \\mid b) P(b, c \\mid r) P(r)\n",
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"$$\n",
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"\n",
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"where the grouped terms sum to one for any value of $r$, and hence\n",
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"\n",
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"$$\n",
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"P(a) = \\sum_{r, b, c, d} P(a \\mid b) P(b, c \\mid r) P(r).\n",
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"$$"
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Memoization via Shortcuts\n",
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"\n",
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"In GTSAM, we compute this recursively\n",
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"\n",
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"#### Step 1\n",
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"We want to compute the marginal via\n",
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"$$\n",
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"P(a) = \\sum_{r, b} P(a \\mid b) P(b).\n",
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"$$\n",
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"where $P(b)$ is the separator of this clique.\n",
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"\n",
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"#### Step 2\n",
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"To compute the separator marginal, we use the **shortcut** $P(b|r)$:\n",
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"$$\n",
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"P(b) = \\sum_{r} P(b \\mid r) P(r).\n",
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"$$\n",
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"In general, a shortcut $P(S|R)$ directly conditions this clique's separator $S$ on the root clique $R$, even if there are many other cliques in-between. That is why it is called a *shortcut*.\n",
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"\n",
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"#### Step 3 (optional)\n",
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"If the shortcut was already computed, then we are done! If not, we compute it recursively:\n",
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"$$\n",
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"P(S\\mid R) = \\sum_{F_p,\\,S_p \\setminus S}P(F_p \\mid S_p) P(S_p \\mid R).\n",
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"$$\n",
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"Above $P(F_p \\mid S_p)$ is the parent clique, and by the running intersection property we know that the seprator $S$ is a subset of the parent clique's variables.\n",
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"Note that the recursion is because we might not have $P(S_p \\mid R)$ yet, so it might have to be computed in turn, etc. The recursion ends at nodes below the root, and **after we have obtained $P(S\\mid R)$ we cache it**.\n",
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"\n",
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"In our example, the computation is simply\n",
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"$$\n",
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"P(b|r) = \\sum_{c} P(b, c \\mid r),\n",
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"$$\n",
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"because this the parent separator is already the root, so $P(S_p \\mid R)$ is omitted. This is also the end of the recursion.\n"
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]
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0\n",
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"Marginal P(a):\n",
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" Discrete Conditional\n",
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" P( 0 ):\n",
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" Choice(0) \n",
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" 0 Leaf 0.51\n",
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" 1 Leaf 0.49\n",
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"\n",
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"\n",
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"3\n"
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]
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}
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],
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"source": [
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"# Marginal of the leaf variable 'a':\n",
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"print(bayesTree.numCachedSeparatorMarginals())\n",
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"marg_a = bayesTree.marginalFactor(aKey[0])\n",
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"print(\"Marginal P(a):\\n\", marg_a)\n",
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"print(bayesTree.numCachedSeparatorMarginals())\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"3\n",
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"Marginal P(b):\n",
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" Discrete Conditional\n",
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" P( 2 ):\n",
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" Choice(2) \n",
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" 0 Leaf 0.48\n",
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" 1 Leaf 0.52\n",
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"\n",
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"\n",
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"3\n"
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]
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}
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],
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"source": [
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"\n",
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"# Marginal of the internal variable 'b':\n",
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"print(bayesTree.numCachedSeparatorMarginals())\n",
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"marg_b = bayesTree.marginalFactor(bKey[0])\n",
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"print(\"Marginal P(b):\\n\", marg_b)\n",
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"print(bayesTree.numCachedSeparatorMarginals())\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {},
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{
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"name": "stdout",
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"text": [
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"3\n",
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"Joint P(a, f):\n",
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" DiscreteBayesNet\n",
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" \n",
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"size: 2\n",
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"conditional 0: P( 0 | 1 ):\n",
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" Choice(1) \n",
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" 0 Choice(0) \n",
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" 0 0 Leaf 0.51758893\n",
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" 0 1 Leaf 0.48241107\n",
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" 1 Choice(0) \n",
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" 1 0 Leaf 0.50222672\n",
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" 1 1 Leaf 0.49777328\n",
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"\n",
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"conditional 1: P( 1 ):\n",
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" Choice(1) \n",
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" 0 Leaf 0.506\n",
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" 1 Leaf 0.494\n",
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"\n",
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"\n",
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"3\n"
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]
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}
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],
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"source": [
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"\n",
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"# Joint of leaf variables 'a' and 'f': P(a, f)\n",
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"# This effectively needs to gather info from two different branches\n",
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"print(bayesTree.numCachedSeparatorMarginals())\n",
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"marg_af = bayesTree.jointBayesNet(aKey[0], fKey[0])\n",
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"print(\"Joint P(a, f):\\n\", marg_af)\n",
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"print(bayesTree.numCachedSeparatorMarginals())\n"
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]
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},
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{
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@ -1,102 +0,0 @@
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"""
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GTSAM Copyright 2010-2018, Georgia Tech Research Corporation,
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Atlanta, Georgia 30332-0415
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All Rights Reserved
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Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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See LICENSE for the license information
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Simple robotics example using odometry measurements and bearing-range (laser) measurements
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Author: Alex Cunningham (C++), Kevin Deng & Frank Dellaert (Python)
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"""
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# pylint: disable=invalid-name, E1101
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from __future__ import print_function
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import math
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import gtsam
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import gtsam.utils.plot as gtsam_plot
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import matplotlib.pyplot as plt
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def main():
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"""Main runner."""
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# Create noise models
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PRIOR_NOISE = gtsam.noiseModel.Diagonal.Sigmas(gtsam.Point3(0.3, 0.3, 0.1))
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ODOMETRY_NOISE = gtsam.noiseModel.Diagonal.Sigmas(
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gtsam.Point3(0.2, 0.2, 0.1))
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# 1. Create a factor graph container and add factors to it
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graph = gtsam.NonlinearFactorGraph()
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# 2a. Add a prior on the first pose, setting it to the origin
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# A prior factor consists of a mean and a noise ODOMETRY_NOISE (covariance matrix)
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graph.add(gtsam.PriorFactorPose2(1, gtsam.Pose2(0, 0, 0), PRIOR_NOISE))
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# 2b. Add odometry factors
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# Create odometry (Between) factors between consecutive poses
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graph.add(
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gtsam.BetweenFactorPose2(1, 2, gtsam.Pose2(2, 0, 0), ODOMETRY_NOISE))
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graph.add(
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gtsam.BetweenFactorPose2(2, 3, gtsam.Pose2(2, 0, math.pi / 2),
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ODOMETRY_NOISE))
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graph.add(
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gtsam.BetweenFactorPose2(3, 4, gtsam.Pose2(2, 0, math.pi / 2),
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ODOMETRY_NOISE))
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graph.add(
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gtsam.BetweenFactorPose2(4, 5, gtsam.Pose2(2, 0, math.pi / 2),
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||||
ODOMETRY_NOISE))
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||||
|
||||
# 2c. Add the loop closure constraint
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# This factor encodes the fact that we have returned to the same pose. In real
|
||||
# systems, these constraints may be identified in many ways, such as appearance-based
|
||||
# techniques with camera images. We will use another Between Factor to enforce this constraint:
|
||||
graph.add(
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gtsam.BetweenFactorPose2(5, 2, gtsam.Pose2(2, 0, math.pi / 2),
|
||||
ODOMETRY_NOISE))
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print("\nFactor Graph:\n{}".format(graph)) # print
|
||||
|
||||
# 3. Create the data structure to hold the initial_estimate estimate to the
|
||||
# solution. For illustrative purposes, these have been deliberately set to incorrect values
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||||
initial_estimate = gtsam.Values()
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||||
initial_estimate.insert(1, gtsam.Pose2(0.5, 0.0, 0.2))
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||||
initial_estimate.insert(2, gtsam.Pose2(2.3, 0.1, -0.2))
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||||
initial_estimate.insert(3, gtsam.Pose2(4.1, 0.1, math.pi / 2))
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||||
initial_estimate.insert(4, gtsam.Pose2(4.0, 2.0, math.pi))
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||||
initial_estimate.insert(5, gtsam.Pose2(2.1, 2.1, -math.pi / 2))
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||||
print("\nInitial Estimate:\n{}".format(initial_estimate)) # print
|
||||
|
||||
# 4. Optimize the initial values using a Gauss-Newton nonlinear optimizer
|
||||
# The optimizer accepts an optional set of configuration parameters,
|
||||
# controlling things like convergence criteria, the type of linear
|
||||
# system solver to use, and the amount of information displayed during
|
||||
# optimization. We will set a few parameters as a demonstration.
|
||||
parameters = gtsam.GaussNewtonParams()
|
||||
|
||||
# Stop iterating once the change in error between steps is less than this value
|
||||
parameters.setRelativeErrorTol(1e-5)
|
||||
# Do not perform more than N iteration steps
|
||||
parameters.setMaxIterations(100)
|
||||
# Create the optimizer ...
|
||||
optimizer = gtsam.GaussNewtonOptimizer(graph, initial_estimate, parameters)
|
||||
# ... and optimize
|
||||
result = optimizer.optimize()
|
||||
print("Final Result:\n{}".format(result))
|
||||
|
||||
# 5. Calculate and print marginal covariances for all variables
|
||||
marginals = gtsam.Marginals(graph, result)
|
||||
for i in range(1, 6):
|
||||
print("X{} covariance:\n{}\n".format(i,
|
||||
marginals.marginalCovariance(i)))
|
||||
|
||||
for i in range(1, 6):
|
||||
gtsam_plot.plot_pose2(0, result.atPose2(i), 0.5,
|
||||
marginals.marginalCovariance(i))
|
||||
|
||||
plt.axis('equal')
|
||||
plt.show()
|
||||
|
||||
|
||||
if __name__ == "__main__":
|
||||
main()
|
||||
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Reference in New Issue