Factor Graphs for Robot Perception
1. Paper overview
In one sentence: Conditioned sensor models turn SLAM into a sparse estimation problem, but adequate constraints and trustworthy local optimization remain essential. e-identitye-scopee-probleme-graphe-linearizatione-evaluation-scope
| At a glance | What to know |
|---|---|
| Research problem | Source description SLAM must estimate both a robot's trajectory and an initially unknown landmark map from uncertain sensor measurements. A bearing relates a pose to a landmark but usually cannot determine either alone. The opening example adds an absolute pose measurement to anchor the solution. The challenge is therefore to combine partial constraints while respecting their uncertainty and dependency structure. e-probleme-map |
| Core mechanism | |
| Reading caution | Source description The selected chapters contain a simulated graph and algorithmic explanations, with no benchmark results table, train/test protocol, uncertainty estimates or empirical ablation. Figure 2.1 uses ground-truth positions only to display connectivity. It reports approximately 100 time steps and 20 landmarks, not localization accuracy. e-structuree-evaluation-scope |
Core contributions
- Source description
Chapter 2 connects Gaussian-noise MAP estimation to weighted nonlinear least squares, whitening, QR or Cholesky factorization, and iterative solvers. Its contribution here is a unified explanation, rather than a newly trained architecture. e-least-squarese-linearizatione-directe-iteration
- Author claim
The bibliographic discussion motivates smoothing over the entire trajectory through sparse information structure and situates square root SAM relative to filtering. These are surveyed arguments; the selected chapters do not supply a controlled performance comparison. e-smoothinge-evaluation-scope
Figure 1.1. The same unknown scene must explain both robot motion and landmark observations. Original paper, p. 8 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Start with the lower row: x₁, x₂ and x₃ denote successive robot poses, and the solid arrows indicate motion. Now follow the dashed observation links. Two poses observe l₁, while only x₃ observes l₂. These lines describe which quantities participate in an observation; they do not supply measured distances or confidence intervals. Section 1.1 explains why both rows are unknown in SLAM. The next page adds an absolute position/orientation observation at x₁ to anchor the problem, although that observation is not drawn in this first schematic. Keep this scene in mind as the following two figures change its probabilistic representation. e-probleme-map
What it supports. SLAM couples localization and mapping because landmark observations depend on both robot and landmark state. The drawing introduces three poses and two landmarks as a teaching example. It establishes the structure of the estimation problem, not a successful reconstruction or a comparison of algorithms.
Where the evidence stops. The schematic is not drawn with a numerical coordinate scale, and its links do not prove that all states are identifiable. Anchoring the first pose addresses the absolute reference frame; other variables still need sufficient measurement information.
2. Motivation
2.1 The problem and the proposed response
SLAM must estimate both a robot's trajectory and an initially unknown landmark map from uncertain sensor measurements. A bearing relates a pose to a landmark but usually cannot determine either alone. The opening example adds an absolute pose measurement to anchor the solution. The challenge is therefore to combine partial constraints while respecting their uncertainty and dependency structure. e-probleme-map
2.2 What this reading follows
How can a robot estimate its path when the landmarks used to localize it are also unknown? Dellaert and Kaess answer by separating a model of how measurements arise from a computation that explains measurements already observed. This illustrated reading covers the title material, abstract and first two chapters of their 2017 monograph. It follows the same scene from a geometric sketch through a Bayesian network to a factor graph, then examines weak constraints and nonlinear solver updates. The payoff is an explicit route from uncertain sensor observations to pose and map estimates. The evidence consists of derivations and teaching examples, with later chapters and empirical applications outside the reviewed scope. e-identitye-scopee-probleme-graphe-linearizatione-evaluation-scope
3. Research context
We place the paper in the collection through its world–action interface. The catalog labels and the reading’s assessment are shown separately.
| Catalog dimension | Recorded classification |
|---|---|
| Major category | Foundational work |
| Architecture | Not applicable |
| Prediction paradigm | Not applicable |
| Quadrant | Not applicable |
3.1 Evidence-based assessment
Supports the recorded classification
The recorded foundational-work category and state-estimation emphasis are supported. The reviewed architecture is a probabilistic estimation graph, with known controls entering motion factors, rather than a joint future/action predictor or inverse-dynamics controller. Thus the recorded architecture, prediction paradigm and quadrant being Not applicable is appropriate. The planning connection is enabling state estimation, not a demonstrated planning algorithm. e-modelse-graphe-lm
This is the collection’s architectural analysis, not a new related-work survey. Benchmark comparisons and their protocols appear in Section 6.
4. Problem formulation
4.1 Inputs and outputs
| Inputs | Outputs |
|---|---|
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4.2 Equations and their role
5. Method in detail
5.1 1. Fix the measurements, then ask which states explain them
The modeling direction starts with poses and landmarks and predicts what a sensor would observe. That direction is useful for constructing p(X,Z), and the Bayesian-network arrows make it explicit. Once the robot has measured Z, however, the unknowns are X. Section 1.6 drops the evidence normalizer and treats the measurement density as a likelihood function of those unknowns. A Gaussian error model for the sensor does not make this likelihood Gaussian in position: a bearing can leave many positions equally plausible. Figure 1.3 therefore replaces explicit observation variables with factors whose measured values are fixed parameters. The practical lesson is to distinguish the model's generative semantics from the estimator's task. Sampling plausible observations and recovering states from actual observations use the same probabilistic ingredients but perform different computations. e-bayese-samplinge-mape-graph
Figure 1.2. A generative model predicts observations from poses, landmarks and their local dependencies. Original paper, p. 11 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Read each arrow toward its child variable. Along the bottom, pose transitions point from x₁ to x₂ to x₃. For z₂, arrows arrive from x₁ and l₁, expressing the conditional measurement density p(z₂|x₁,l₁). The analogous pairs produce z₃ and z₄, while z₁ depends only on the first pose. Square nodes mark measurements that will be observed. Equations (1.4)–(1.7) also include priors for the first pose and both landmarks, although priors have no parent arrows to draw. Section 1.5 uses this dependency order to describe sampling a trajectory, landmarks and sensor readings from the generative model. e-bayese-modelse-samplinge-graph
What it supports. Local sensor models allow the full joint density to be assembled from manageable components. The four observation nodes make the distinction between unknown state and observed data explicit. Generating these observations is a modeling operation; the next step fixes their values and solves for the unknown states.
Where the evidence stops. Arrows encode conditional dependencies, not commands to execute or the order of an inference algorithm. The model assumes known data association; the source does not learn which landmark produced each observation.
5.2 2. Turn sensor disagreement into a common linear system
For the Gaussian-noise factors in Section 2.2, maximizing the product of factors is equivalent to minimizing the sum of squared Mahalanobis residuals. Each residual compares a predicted measurement h_i(X_i) with the observed z_i. At the current estimate, the Jacobian H_i describes how that prediction changes under a small state update. Whitening then applies the inverse covariance square root to both H_i and the prediction error. This is essential when measurements have different units or uncertainty: it makes the rows compatible without discarding their statistical weights. Stack the whitened terms into A and b and solve the local least-squares problem. The source's structural point is that a factor only contributes derivatives for its adjacent variables. Sparse graph connections thus become sparse blocks in the numerical system, while nonlinear geometry still requires relinearization. e-least-squarese-linearizatione-iteration
Figure 1.3. Conditioning removes observation variables while preserving their likelihood constraints. Original paper, p. 17 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Compare this crop with the Bayesian network. The square measurement nodes have disappeared because their observed values are now fixed parameters. The large circles still represent the five unknown pose and landmark variables. Small black dots represent nine factors: two pose-transition factors, three bearing likelihoods, the first-pose and two landmark priors, and the absolute-pose likelihood. Notice the two separate unary factors attached to x₁. A dot touches only the state variables used by its function; for example, the factor between x₃ and l₂ depends on that pair. Equations (1.20)–(1.28) verify these connections and the product they encode. e-bayese-graphe-linearizatione-map
What it supports. The graph displays the computational dependencies of the unnormalized posterior. Each factor can be evaluated locally, while all factors together constrain a shared set of states. This representation makes the later Jacobian's sparse block structure understandable without drawing the entire numerical matrix.
Where the evidence stops. The nine-factor count includes landmark priors retained in this example; the text notes that SLAM often omits them when no prior map exists. The graph specifies dependencies, not factor strengths or a guarantee of unique recovery.
5.3 3. Separate solving an approximation from trusting its update
A linear solver finds the best update for the current approximation; it does not establish that this update improves the original nonlinear objective. Section 2.4 describes Cholesky and QR routes to the local solution. Section 2.5 then distinguishes how nonlinear methods use it. Gauss–Newton takes the resulting step and can diverge if the approximation is poor. Algorithm 2.1 instead adds diagonal-scaled damping, evaluates the actual nonlinear cost, and accepts only a strict decrease. Rejected LM steps leave the estimate unchanged and increase damping. The dogleg illustration presents another approach: combine separately computed descent and Gauss–Newton steps within a trust region, retaining their directions when reconsidering an update. The reader should distinguish this objective-based feedback from physical robot feedback. These chapters optimize a state estimate; they provide no action-selection or execution loop. e-directe-iteratione-lme-dogleg
Figure 2.1. Graph connectivity reveals where a simulated map has little information. Original paper, p. 23 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Follow the cyan pose chain from the lower left through the bend to the right. Odometry factors form its backbone, while observation factors fan out to blue landmark squares. The source describes approximately 100 time steps and about 20 landmarks. It places the variables at ground-truth positions solely to make the graph interpretable. Compare the many links meeting some landmarks with the isolated link to the lower-right landmark. Section 2.1 specifically uses that landmark to explain underconstraint: if its sole observation measures bearing, multiple positions along the allowed direction remain compatible with the observation. No error bars or estimated trajectory are plotted. e-structuree-modelse-least-squarese-evaluation-scope
What it supports. Connectivity can reveal an observability problem before any solver is run. A singly observed landmark may remain undetermined even though it belongs to the same connected graph as well-observed landmarks. This is a qualitative diagnostic in the source, not an ablation measuring the effect of removing observations.
Where the evidence stops. More links alone do not quantify uncertainty; geometry, covariances and priors matter too. Ground-truth placement must not be mistaken for an inferred map. The excerpt supplies neither numerical reconstruction error nor an exact simulation configuration.
Figure 2.2. Dogleg limits the proposed state update while retaining the two computed step directions. Original paper, p. 32 ↗
Excerpt from the authors’ paper; cropped without altering the figure or table.
How to read it. Begin at the current estimate, where the three bold update arrows originate. In the illustrated case, the gradient-descent endpoint lies inside the circular trust region and the Gauss–Newton endpoint lies outside. Follow the thin segment connecting those endpoints: its intersection with the boundary determines the dogleg endpoint. The bold dogleg arrow shows the resulting displacement, whereas the two-segment construction explains its name. This agrees with the caption and Section 2.5.4. If the proposed update is rejected, the source says the two computed directions can be recombined within a revised region without a new matrix factorization at the unchanged estimate. e-doglege-dogleg-notatione-lme-linearization
What it supports. Dogleg addresses the cost of reconsidering an unsafe nonlinear update. The source contrasts its reusable directions with LM's need to refactor after changing damping. This is a mechanism and computational argument; Figure 2.2 contains no convergence curve, measured runtime or accuracy comparison.
Where the evidence stops. Below Eq. (2.28), the printed L(Δ)=AᵀAΔ−Aᵀb is vector-valued, inconsistent with a scalar predicted cost reduction. Earlier squared-residual equations provide the verifiable objective. The diagram is preserved faithfully; the gain-ratio definition needs clarification before literal implementation.
5.4 Training and inference
During training
No neural training stages, learned latent representation, frozen modules or training split are specified in these chapters. Sensor noise may be assumed or learned in principle, but the presented algorithms optimize state estimates for supplied observations; no noise-learning procedure is given. e-bayese-modelse-iteration
During inference
Start from an initial estimate, compute an update and repeat until convergence. Steepest descent requires a step-size choice and can converge slowly. Gauss–Newton solves the linearized normal equations and can diverge when the quadratic approximation is poor. e-iteration
Algorithm 2.1 initializes LM damping at 10⁻⁴ and uses diagonal-scaled damping. It accepts only a strict decrease in the nonlinear cost, divides damping by 10 after acceptance, and multiplies it by 10 after rejection. Powell's dogleg combines separately computed descent and Gauss–Newton steps inside an explicit trust region, reusing their directions after rejection. e-lme-dogleg
This feedback checks the estimator's objective. The selected algorithm returns state estimates; no action extraction, planning rollout or physical execution loop is specified. e-modelse-lm
5.5 Implementation flow
- Specify local probabilistic models
Build a pose Markov chain, landmark priors and conditional sensor densities. Measurement functions predict observations from nearby state variables; zero-mean Gaussian noise supplies covariance weighting. Data association is assumed to have been performed beforehand. Known controls may parameterize a motion model; controls are not estimated as an action policy. e-bayese-models
- Condition on the observations
Fix Z and optimize over the unknown states X. Each conditional becomes a factor connected only to the variables it depends on. In the toy example, five variable nodes and nine factor nodes encode the unnormalized posterior, with observed measurements stored as factor parameters. e-mape-graph
- Linearize and whiten
At the current estimate, approximate each measurement function with its Jacobian. Premultiply both Jacobian and prediction error by the inverse covariance square root. Stacking these blocks produces A and b; the graph's local dependencies determine the sparse block pattern. The derivation assumes vector-valued state updates. e-linearization
- Solve the local system
Cholesky factors the information matrix AᵀA into RᵀR and uses triangular solves. QR factors A directly, avoiding explicit formation of AᵀA. Both yield a square root representation. The text presents QR as more numerically stable, while emphasizing that sparse factorization cost depends on structure. e-direct
6. Experiments & results
This selected-chapter reading follows Dellaert and Kaess from a probabilistic SLAM model to square root smoothing and mapping. Given noisy observations, known data associations and suitable priors, the estimator jointly adjusts robot poses and landmarks. Conditioning turns a generative Bayesian network into a factor graph; Gaussian measurement errors then lead to nonlinear least squares, repeated linearization and matrix factorization. The central lesson is how local sensor constraints become a global estimation problem. The reviewed chapters provide derivations and diagrams, not an empirical evaluation of learned world models or robot control.
This is an illustrated selected-chapter monograph review. Chapters 1–2 provide five explanatory figures but no quantitative benchmark table, empirical ablation, training evaluation or control-execution result. All five figures are included as mechanism visuals; Figure 2.1 supplies a qualitative observability diagnostic. Figure 2.2 explains solver geometry rather than measured performance. Later chapters and appendices are outside the authorized reading scope, so their contents cannot fill those gaps. The absence of experimental visuals here is not a claim that the complete monograph lacks applications or results. e-scopee-structuree-doglege-evaluation-scope
6.1 Read the original evidence
No results visual is included in this edition. The source-specific visual limitations are recorded above.
6.2 Results and evaluation conditions
No quantitative results are included in this reading.
6.3 Ablations and diagnostic examples
No ablation or diagnostic visual is included in this edition. The source-specific visual limitations are recorded above.
7. Analysis & limitations
7.1 What the evidence leaves open
The selected chapters contain a simulated graph and algorithmic explanations, with no benchmark results table, train/test protocol, uncertainty estimates or empirical ablation. Figure 2.1 uses ground-truth positions only to display connectivity. It reports approximately 100 time steps and 20 landmarks, not localization accuracy. e-structuree-evaluation-scope
Known data association and adequate constraints are substantive assumptions. A lone bearing can leave a landmark undetermined. A good initialization matters, and the local linearization does not establish a general global-convergence guarantee. The vector-space update also leaves manifold handling to an unread chapter. e-modelse-structuree-linearizatione-iteration
A source inconsistency affects dogleg reproduction: below Eq. (2.28), L(Δ)=AᵀAΔ−Aᵀb is vector-valued, although the gain-ratio denominator requires a scalar predicted cost reduction. The earlier squared-residual objective supplies a consistent reference; this report does not silently repair the printed definition. e-dogleg-notation
7.2 Questions for discussion
- When does an additional measurement add a new constraint direction rather than repeat an existing one?
- How should a reconstruction distinguish lower residual cost from improved ground-truth state accuracy?
- Which dogleg implementation choices remain unresolved by the inconsistent local-cost definition?
8. Reproducibility audit
8.1 Requirements and known gaps
A minimal educational reconstruction needs explicitly chosen poses, landmarks, sensor geometry, covariance matrices, associations, priors and initialization. The displayed simulated graph does not specify exact coordinates, noise values or a random seed. These choices must be disclosed as reconstruction choices. e-modelse-structure
Algorithm 2.1 specifies damping and acceptance rules but leaves convergence numerical thresholds unspecified. Record stopping tolerances and derivative checks. Sparse solver implementations are named in Section 2.4, but these chapters do not fix software versions or provide hardware or wall-clock measurements. e-directe-iteratione-lm
Proposed checks should first isolate observability by adding a second bearing to a singly observed landmark, then compare Gauss–Newton with LM under identical initialization perturbations. Measure objective decrease, estimation error and rejected steps separately; neither check has been run here. e-structuree-lme-iteration
8.2 Proposed reproduction checks
The following checks are proposals motivated by the paper. They have not been run as part of this reading.
Check 1: Does another bearing remove an unconstrained landmark direction?
Reader-proposed check, not performed: reconstruct the lower-right landmark situation in Figure 2.1 with fixed, known robot poses and no prior on the tested landmark. Generate noise-free bearings from the stated measurement model while retaining fixed positive covariance weights. Compare a single observation with a second observation from a separated pose whose ray intersects the first. Keep the landmark truth and all other modeling choices fixed. Plot the landmark cost surface, examine the landmark Jacobian's singular values and repeat optimization from several initial landmark positions. A single bearing should leave a direction of equal fit; the additional independent bearing should remove that local ambiguity. If both configurations appear equally constrained, inspect for an unintended prior or a hidden fixed coordinate. Report chosen geometry and tolerances because the original drawing supplies no numeric configuration. e-modelse-structuree-least-squarese-linearization
Check 2: Does LM's acceptance rule help when initialization deteriorates?
Reader-proposed check, not performed: create a fully constrained synthetic SLAM instance using the reviewed Gaussian factors, fixed data associations and an absolute reference. Compare Gauss–Newton with Algorithm 2.1's diagonal-scaled LM using identical residuals, Jacobians, linear solver and stopping criteria. Apply the same sequence of initial-state perturbations to both methods. Record nonlinear objective values, ground-truth pose/map error, linear solves and LM accept/reject decisions. The discriminating observation is whether LM rejects cost-increasing proposals and achieves a lower final objective in cases where Gauss–Newton diverges; an advantage is not assumed for every initialization. Verify the source's initial damping and factor-of-ten updates, and declare numerical convergence tolerances explicitly. Separate improved objective behavior from improved state accuracy and do not present this as reproducing a published benchmark. e-probleme-modelse-iteratione-lm
8.3 Reading coverage
Visual audit: The title/author block, contents, abstract, all five original figures, the Gaussian-to-least-squares derivation, direct-solver equations, nonlinear update equations, Algorithm 2.1 and the closing Chapter 2 discussion were visually inspected on the declared pages. Every final crop was separately inspected. Bayesian-network arrows and factor connections agree with the corresponding equations; the dogleg endpoint geometry agrees with its caption and prose. The printed dogleg local-cost inconsistency is disclosed. PDF pp. 7 and 19–21 were read as supplied text but not rendered in this visual pass; no retained technical claim relies exclusively on those pages. PDF pp. 35–144 and the appendices remain outside the selected-chapter reading. No quantitative table or empirical ablation is present in the reviewed chapters.
PDF pages inspected for this edition: 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34. Appendix coverage: not read.
Original figures and tables remain the work of the source’s authors. Extractions preserve their scientific content; any HTML wrapper layout is disclosed with each figure. The surrounding reading notes are our own.
Text reading scope & known omissions
- Title and contents, PDF pp. 1–5
- Abstract, printed p. 1 / PDF p. 6
- Chapter 1: Introduction, Sections 1.1–1.10, printed pp. 2–16 / PDF pp. 7–21
- Chapter 2: Smoothing and Mapping, Sections 2.1–2.6 including all four nonlinear solver subsections and Algorithm 2.1, printed pp. 17–29 / PDF pp. 22–34
Outside the original text pass
- PDF pages 35–144 (Chapters 3–7, bibliography and appendices) are omitted from the selected reading text. The full raw PDF and full extraction are retained separately; availability does not establish reading.
- Later chapters on sparsity, elimination ordering, incremental smoothing, manifold optimization and applications are outside this reading scope. Do not infer their methods/results from the TOC.
- Text extraction does not reconstruct figure images or equation/table layout; direct visual reading remains required.
- The low-text PDF page 136 is the Appendices divider, not a missing page. It is outside the selected reading scope.
- All five supplied text chunks were read completely. The extraction limitation was addressed by inspecting the original figures and all PDF pages supporting retained technical claims; the illustrated edition lists the exact visual scope. This is not a full-monograph review.
- Identity/version: the title and both authors match the catalog. The inspected title page identifies Foundations and Trends in Robotics, Vol. 6, No. 1–2 (2017), printed pp. 1–139, DOI 10.1561/2300000043. No separate revision identifier is shown; no other edition was inspected or substituted.
- No code, external cited works or supplementary resources were inspected, and no experiments were reproduced.
The visual audit above records the subsequent illustrated pass.
8.4 Traceable evidence
e-identityPDF p. 1, title/author/publication block
Exact title: Factor Graphs for Robot Perception. Authors: Frank Dellaert, Georgia Institute of Technology; Michael Kaess, Carnegie Mellon University. The block identifies the 2017 journal volume, printed page range and DOI.
Go to primary source ↓e-scopePDF pp. 2–5, Contents; PDF p. 6, Abstract
The contents separate Introduction and Smoothing and Mapping from later sparsity, ordering, incremental, manifold and application chapters and two appendices. The abstract identifies the work as a review; contents headings do not establish reading of those later chapters.
Go to primary source ↓e-problemPDF pp. 8–9 / printed pp. 3–4, Figure 1.1 and Sections 1.1–1.2
Three successive robot poses observe two landmarks; an absolute first-pose measurement anchors the example. SLAM estimates an unknown map jointly with robot localization under measurement uncertainty.
Go to primary source ↓e-bayesPDF pp. 10–11 / printed pp. 5–6, Section 1.3, Figure 1.2, Eqs. (1.3)–(1.7)
The joint density factors into pose transitions, landmark priors, an absolute pose observation and three bearing observation densities. Arrows point from parents to generated variables; square nodes denote observed measurements. Sensor noise can be assumed or learned in principle.
Go to primary source ↓e-modelsPDF pp. 12–13 / printed pp. 7–8, Section 1.4, Eqs. (1.8)–(1.14)
Data association is assumed available from preprocessing. Gaussian noise and Mahalanobis error specify measurement factors; motion factors can use odometry or known controls. Heavier-tailed measurement densities are mentioned, and three-dimensional manifolds are deferred.
Go to primary source ↓e-samplingPDF pp. 13–14 / printed pp. 8–9, Section 1.5
Ancestral sampling generates parents before their children. The toy example samples poses, landmarks and then observations, illustrating a generative model rather than an inference or control algorithm.
Go to primary source ↓e-mapPDF pp. 14–16 / printed pp. 9–11, Section 1.6, Eqs. (1.15)–(1.19)
MAP maximizes the posterior over X at fixed Z. The normalizer can be omitted. A Gaussian measurement distribution need not yield a Gaussian or individually informative likelihood over states; insufficient constraints can prevent a unique maximizer.
Go to primary source ↓e-graphPDF pp. 16–18 / printed pp. 11–13, Section 1.7, Figure 1.3, Eqs. (1.20)–(1.28)
The conditioned toy graph has five state-variable nodes and nine explicit factors. Every factor touches only its argument variables; known measurements become parameters and their variable nodes are omitted. The global function is the product of local factors.
Go to primary source ↓e-structurePDF pp. 22–23 / printed pp. 17–18, Section 2.1, Figure 2.1 and its discussion
The larger simulated graph depicts about 100 time steps and about 20 landmarks at ground-truth positions. Odometry forms a chain and observation factors connect poses to landmarks. The lower-right landmark has only one observation; if bearing-only, its location is not uniquely determined.
Go to primary source ↓e-least-squaresPDF p. 24 / printed p. 19, Section 2.2, Eqs. (2.1)–(2.4)
For factors with Gaussian error form, negative log transforms MAP into a sum of covariance-weighted squared residuals. Individual measurements can constrain only a subset of the involved state dimensions.
Go to primary source ↓e-linearizationPDF pp. 25–26 / printed pp. 20–21, Section 2.3, Eqs. (2.5)–(2.13)
Taylor expansion supplies Jacobians and state updates in a vector space. Inverse covariance square roots whiten Jacobians and prediction errors, giving the stacked least-squares problem in A and b. The sparse block structure mirrors the factor graph.
Go to primary source ↓e-directPDF pp. 26–28 / printed pp. 21–23, Section 2.4, Eqs. (2.14)–(2.22)
For full-column-rank A, normal equations can be solved through Cholesky; QR instead factors A and transforms b. Both lead to triangular solves. Dense operation counts differ, while sparse costs depend on structure. CHOLMOD and SuiteSparseQR are named without a fixed experimental configuration.
Go to primary source ↓e-iterationPDF pp. 29–30 / printed pp. 24–25, Sections 2.5–2.5.2, Eqs. (2.23)–(2.25)
Nonlinear optimization repeatedly updates an initial estimate. A small update is an example stopping criterion without a numerical tolerance. Steepest descent needs step-size selection; Gauss–Newton can converge rapidly near a suitable solution but can diverge with a poor quadratic fit.
Go to primary source ↓e-lmPDF pp. 30–31 / printed pp. 25–26, Section 2.5.3, Eqs. (2.26)–(2.27), Algorithm 2.1
The source distinguishes identity damping from diagonal-scaled damping. Algorithm 2.1 uses the latter, begins at λ=10⁻⁴, accepts g(Xᵗ+Δ)<g(Xᵗ), divides λ by 10 on acceptance, multiplies it by 10 otherwise and returns the latest state estimate.
Go to primary source ↓e-doglegPDF pp. 31–33 / printed pp. 26–28, Section 2.5.4, Figure 2.2 and Eq. (2.28)
Dogleg separately computes descent and Gauss–Newton steps and combines them within a trust region. Directions can be reused after rejection without refactoring, unlike changing LM damping. The source requires full-rank Jacobians for GN/dogleg and suggests LM for poorly constrained cases.
Go to primary source ↓e-dogleg-notationPDF p. 32 / printed p. 27, definition immediately below Eq. (2.28); PDF pp. 26 and 29 / printed pp. 21 and 24, Eqs. (2.13) and (2.23) and local objective
The PDF visibly prints L(Δ)=AᵀAΔ−Aᵀb below a gain ratio with denominator L(0)−L(Δ). Earlier pages give a scalar squared-residual local objective. The later printed expression has the dimensions of a state vector, creating a dimensional inconsistency for the proposed scalar ratio.
Go to primary source ↓e-smoothingPDF pp. 33–34 / printed pp. 28–29, Section 2.6
The survey contrasts smoothing over the full trajectory with filtering and attributes square root SAM to earlier work. It motivates sparse information structure and sparse QR/Cholesky solutions, qualifying compactness by typical mapping scenarios.
Go to primary source ↓e-evaluation-scopePDF pp. 22–34 / printed pp. 17–29, Chapter 2, especially Figure 2.1, Section 2.4 and Algorithm 2.1
The selected chapter provides a schematic simulated SLAM graph, algebraic cost arguments and solver descriptions. It contains no quantitative benchmark table or empirical ablation, and specifies no complete dataset split or numerical experimental configuration.
Go to primary source ↓8.5 Primary sources
Factor Graphs for Robot Perception ↗
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