Sublabel-Accurate Relaxation of Nonconvex Energies

Thomas MÃ¶llenhoff, Emanuel Laude, Michael Moeller, Jan Lellmann, Daniel Cremers

**Concept Graph & Resume using Claude 3 Opus | Chat GPT4o | Llama 3:**

graph LR
classDef optimization fill:#f9d4d4, font-weight:bold, font-size:14px
classDef lifting fill:#d4f9d4, font-weight:bold, font-size:14px
classDef approximation fill:#d4d4f9, font-weight:bold, font-size:14px
classDef results fill:#f9f9d4, font-weight:bold, font-size:14px
A[Sublabel-Accurate Relaxation of

Nonconvex Energies] --> B[Variational computer vision minimizes

data and regularizer. 1] A --> C[Challenging non-convex optimization problems. 2] C --> D[Gradient descent finds local optima,

unintuitive parameters. 3] C --> E[Ishikawa 2003: globally optimizing

graph cut, grid/label bias. 4] C --> F[Pock et al. 2008: continuous setting,

label bias remains. 5] A --> G[Stereo matching: more labels needed,

becomes intractable. 6] A --> H[Proposes piecewise convex approximation,

fewer labels. 7] A --> I[Related work: continuous MRFs,

specific regularizers. 8] A --> J[Contributions: continuous sub-label accurate,

tightest convex relaxation. 9] A --> K[Convexification and functional lifting explained. 10] K --> L[Traditional lifting samples labels,

takes convex envelope. 11] K --> M[Proposed: costs between labels,

closer approximation, convex. 12] K --> N[Non-convex energy folded

in higher dimensions. 13] K --> O[Reformulated energy defined

along convex combinations. 14] K --> P[Tightest convex envelope

found analytically. 15] K --> Q[C characterized by

convex conjugates epigraphs. 16] A --> R[Implementation: projecting onto epigraphs,

piecewise linear/quadratic. 17] A --> S[Total variation regularizer

lifted to higher dimensions. 18] A --> T[Convex-concave saddle point problem,

primal-dual GPU algorithm. 19] A --> U[Solution obtained by back-projecting. 20] A --> V[Convex model: sub-label accurate

matches direct optimization. 21] V --> W[Transitions between direct

optimization and lifting. 22] A --> X[Exact solution with 10 labels,

traditional needs more. 23] A --> Y[Stereo: traditional has label bias,

many labels. 24] A --> Z[Proposed: smooth results, few labels,

less runtime/memory. 25] Z --> AA[Improvement with equal labels,

reasonable with 2. 26] A --> AB[First continuous sub-label accurate

relaxation for non-convex energies. 27] A --> AC[Fewer labels than traditional,

improves runtime/memory. 28] A --> AD[Generalizes to piecewise

convex approximations. 29] A --> AE[Code available online. 30] class B,C,D,E,F optimization class G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U lifting class V,W,X,Y,Z,AA,AB,AC,AD approximation class AE results

Nonconvex Energies] --> B[Variational computer vision minimizes

data and regularizer. 1] A --> C[Challenging non-convex optimization problems. 2] C --> D[Gradient descent finds local optima,

unintuitive parameters. 3] C --> E[Ishikawa 2003: globally optimizing

graph cut, grid/label bias. 4] C --> F[Pock et al. 2008: continuous setting,

label bias remains. 5] A --> G[Stereo matching: more labels needed,

becomes intractable. 6] A --> H[Proposes piecewise convex approximation,

fewer labels. 7] A --> I[Related work: continuous MRFs,

specific regularizers. 8] A --> J[Contributions: continuous sub-label accurate,

tightest convex relaxation. 9] A --> K[Convexification and functional lifting explained. 10] K --> L[Traditional lifting samples labels,

takes convex envelope. 11] K --> M[Proposed: costs between labels,

closer approximation, convex. 12] K --> N[Non-convex energy folded

in higher dimensions. 13] K --> O[Reformulated energy defined

along convex combinations. 14] K --> P[Tightest convex envelope

found analytically. 15] K --> Q[C characterized by

convex conjugates epigraphs. 16] A --> R[Implementation: projecting onto epigraphs,

piecewise linear/quadratic. 17] A --> S[Total variation regularizer

lifted to higher dimensions. 18] A --> T[Convex-concave saddle point problem,

primal-dual GPU algorithm. 19] A --> U[Solution obtained by back-projecting. 20] A --> V[Convex model: sub-label accurate

matches direct optimization. 21] V --> W[Transitions between direct

optimization and lifting. 22] A --> X[Exact solution with 10 labels,

traditional needs more. 23] A --> Y[Stereo: traditional has label bias,

many labels. 24] A --> Z[Proposed: smooth results, few labels,

less runtime/memory. 25] Z --> AA[Improvement with equal labels,

reasonable with 2. 26] A --> AB[First continuous sub-label accurate

relaxation for non-convex energies. 27] A --> AC[Fewer labels than traditional,

improves runtime/memory. 28] A --> AD[Generalizes to piecewise

convex approximations. 29] A --> AE[Code available online. 30] class B,C,D,E,F optimization class G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U lifting class V,W,X,Y,Z,AA,AB,AC,AD approximation class AE results

**Resume: **

**1.-** Variational approaches in computer vision minimize energies with a data term and total variation regularizer.

**2.-** These are challenging non-convex optimization problems.

**3.-** Classical methods like gradient descent only find local optima and have unintuitive parameters.

**4.-** Ishikawa (2003) proposed globally optimizing by considering the product space and finding a graph cut, but this has grid/label bias.

**5.-** Pock et al. (2008) generalized this to a continuous setting, eliminating grid bias but still having label bias due to range discretization.

**6.-** Label discretization issues illustrated for stereo matching - more labels needed for smooth results but becomes intractable.

**7.-** This work proposes solving with fewer labels by using a piecewise convex approximation instead of piecewise linear.

**8.-** Some related work on MRFs with continuous state spaces, but focused on discrete setting or specific regularizers.

**9.-** Key contributions - first spatially continuous fully sub-label accurate formulation, proof of tightest convex relaxation, unifies lifting and direct optimization.

**10.-** Idea of convexification and functional lifting explained - transform energy to higher dimensional space.

**11.-** Traditional multi-labeling samples lifted energy at labels and takes convex envelope - linear relaxation.

**12.-** Proposed approach assigns costs between labels too before convex envelope - closer approximation, still convex.

**13.-** Non-convex energy folded along basis vectors in higher dimensional space.

**14.-** Reformulated energy defined as original along convex combinations of basis vectors, infinity elsewhere.

**15.-** Tightest convex envelope found analytically, is support function of convex set C.

**16.-** C characterized by epigraphs of convex conjugates of non-convex energy pieces.

**17.-** Implementation involves projecting onto epigraphs, done for piecewise linear and quadratic so far.

**18.-** Total variation regularizer also lifted to higher dimensional space, same as Pock et al. 2008.

**19.-** Results in a convex-concave saddle point problem, solved by primal-dual algorithm on GPU.

**20.-** Solution obtained by back-projecting from higher to lower dimensional space.

**21.-** Evaluated on convex model, sub-label accurate finds same solution as direct optimization with little overhead.

**22.-** Provides transition between direct optimization and functional lifting.

**23.-** Finds exact solution with 10 labels while traditional needs many more labels and memory.

**24.-** For stereo, traditional has visible label bias even with many labels.

**25.-** Proposed method gives smooth results with few labels and less runtime/memory.

**26.-** Clear improvement with equal label counts, reasonable results with just 2 labels.

**27.-** Concludes it's first spatially continuous sub-label accurate relaxation for certain non-convex energies.

**28.-** Uses far fewer labels than traditional lifting while improving runtime and memory.

**29.-** Generalizes from piecewise linear to piecewise convex approximations.

**30.-** Code is available online.

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