SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\mathbf{I}$  is an identity matrix of size  $r\times r$  with  $r = \mathrm{rank}(\mathbf{X})$ . It restricts that the bases are all orthometric.  $\lambda ,\gamma ,\tau >0$  are hyper-parameters to balance different loss items. Problem (14) ultimately outputs an exact sparse subspace  $\mathbf{X}\in \mathbb{R}^{D\times r}$ , helping to identify the true model  $\mathcal{M}$  with  $\mathbf{X}\sim \Psi (\mathcal{M})$ . In the following, we will explore an efficient solution for this multi-objective problem.
2.3 Solution
Problem (14) involves  $\ell_0$  minimization, which is the holy grail of sparse approximation. However,  $\ell_0$  optimization is NP-hard, and the non-linear objective makes it even more difficult. A practical way is
to use  $\ell_1$  norm as the best convex approximation instead. Thus, Problem (14) can be relaxed into:
$$
\min  _ {\mathbf {X}, \mathbf {E}, r} \frac {1}{2} \| \mathbf {M} ^ {\top} \mathbf {X} - \mathbf {E} \| _ {F} ^ {2} + \lambda \| \mathbf {X} \| _ {1} + \gamma \| \mathbf {E} \| _ {2, 1} - \tau \operatorname {r a n k} (\mathbf {X}), \tag {15}
$$
$$
s. t. \left\| \mathbf {x} _ {i} \right\| _ {2} = 1, \quad \mathbf {x} _ {i} ^ {\top} \mathbf {x} _ {j} = 0, \forall i, j = 1, 2, \dots , r, i \neq j.
$$
Another troublesome issue is to maximize rank  $(\mathbf{X})$ . Consider that Problem (15) actually aims to obtain  $r$  orthometric bases  $[\hat{\mathbf{x}}_1, \hat{\mathbf{x}}_2, \dots, \hat{\mathbf{x}}_r]$  and a sparse outlier matrix  $\hat{\mathbf{E}} = [\hat{\mathbf{e}}_1, \hat{\mathbf{e}}_2, \dots, \hat{\mathbf{e}}_r]$  satisfying  $\mathcal{L}(\mathbf{M}, \hat{\mathbf{x}}_i, \hat{\mathbf{e}}_i) = \frac{1}{2} \| \mathbf{M}^\top \hat{\mathbf{x}}_i - \hat{\mathbf{e}}_i \|_2^2 + \lambda \| \hat{\mathbf{x}}_i \|_1 + \gamma \| \hat{\mathbf{e}}_i \|_1 < \tau, \forall i \in \{1, 2, \dots, r\}$ . Thus we decompose (15) into progressively estimating a new sparse bases orthometric to all given bases up to  $\mathcal{L}(\mathbf{M}, \hat{\mathbf{x}}_i, \hat{\mathbf{e}}_i) < \tau$  not holds. For any given bases  $\{\mathbf{B} = (\mathbf{x}_j) | j = 1, 2, \dots, i-1\}$  (those have been estimated), a new sparse basis  $\mathbf{x}_i (i > 1)$  can be estimated by solving
$$
\min  _ {\mathbf {x}, \mathbf {e}} \mathcal {L} (\mathbf {M}, \mathbf {x}, \mathbf {e}) = \frac {1}{2} \| \mathbf {M} ^ {\top} \mathbf {x} - \mathbf {e} \| _ {2} ^ {2} + \lambda \| \mathbf {x} \| _ {1} + \gamma \| \mathbf {e} \| _ {1}, \quad s. t. \| \mathbf {x} \| _ {2} = 1, \mathbf {x} ^ {\top} \mathbf {y} = 0, \forall \mathbf {y} \in \mathbf {B}. \tag {16}
$$
Since there exist two variables  $(\mathbf{x},\mathbf{e})$  to be estimated, it is impractical to optimize them directly with gradient descent method as used in [52, 13, 17]. A valid way is to alternatively optimize  $\mathbf{x}$  and  $\mathbf{e}$  in an iterative pipeline, which encourages us to compute the closed-form solution for  $(\mathbf{x},\mathbf{e})$  in each iteration. For such purpose, we first calculate the derivatives of objective w.r.t. variables, then make them equal to zero. In the  $k$ -th iteration and for given  $\mathbf{x}^{k-1}$ , we can easily obtain
$$
\mathbf {e} ^ {k} = \mathbf {M} ^ {\top} \mathbf {x} ^ {k - 1} - \gamma \operatorname {s g n} (\mathbf {e} ^ {k}) = \mathcal {T} _ {\gamma} (\mathbf {M} ^ {\top} \mathbf {x} ^ {k - 1}), \tag {17}
$$
where  $\operatorname{sgn}(\cdot)$  is a sign function.  $\mathbf{e}^k$  is in fact solved by a standard threshold shrinkage operation  $\mathcal{T}_{\gamma}$  introduced in the Lemma of [47]. (Refer to Append.B for details)
Given  $\mathbf{e}^k$  Update  $\mathbf{x}^k$ . Due to the coexistence of quadratic and sparse constraints, it is hard to directly optimize. Fortunately, [7] provides efficient solution for such linear inverse problem with theoretical and practical verification. We exploit its basic idea into our dual sparsity problem, that is to build at each iteration a regularization of the linearized differentiable function, then obtain
$$
\mathbf {x} ^ {k} = \arg \min  _ {\mathbf {x}} \left\{\frac {L}{2} \| \mathbf {x} - \left(\mathbf {x} ^ {k - 1} - \frac {1}{L} \nabla f \left(\mathbf {x} ^ {k - 1}\right)\right) \| _ {2} ^ {2} + \lambda \| \mathbf {x} \| _ {1} \right\}, \tag {18}
$$
where the smallest Lipschitz constraint of the gradient  $\nabla f(\mathbf{x})$  can be calculated by  $L(f) = \lambda_{\max}(\mathbf{MM}^{\top})$ , and  $\lambda_{\max}(\cdot)$  indicates the maximum eigenvalue of a matrix, and we denote
$$
\mathbf {q} _ {L} \left(\mathbf {x} ^ {k - 1}\right) = \mathbf {x} ^ {k - 1} - \frac {1}{L} \nabla f \left(\mathbf {x} ^ {k - 1}\right), \tag {19}
$$
where  $\nabla f(\mathbf{x}^{k - 1}) = \mathbf{M}(\mathbf{M}^{\top}\mathbf{x}^{k - 1} - \mathbf{e}^{k})$ . In this case, Problem (18) is essentially similar to solving  $\mathbf{e}$ , and we can obtain the update formula of  $\mathbf{x}$  with  $\mathbf{x}^k = \mathcal{T}_{\frac{\lambda}{L}}(\mathbf{q}_L(\mathbf{x}^{k - 1}))$ . (Proof is in Append.C)
The above solution for  $\mathbf{x}$  actually reduces to a subgradient method [26], which converges at a rate no worse than  $\mathcal{O}(1 / k)$ . However, it was proven in [34, 35] that there exists a gradient method with an  $\mathcal{O}(1 / k^2)$  complexity which is an 'optimal' first order method for smooth problems [7]. The core strategy is that the iterative shrinkage operation performs at the point  $\tilde{\mathbf{x}}^{k - 1}$  which uses a specific linear combination of previous two points  $(\mathbf{x}^{k - 1},\mathbf{x}^{k - 2})$  as  $\tilde{\mathbf{x}}^{k - 1} = \mathbf{x}^{k - 1} + \left(\frac{t_{k - 1} - 1}{t_k}\right)(\mathbf{x}^{k - 1} - \mathbf{x}^{k - 2})$  with  $t_k = \frac{1}{2} (1 + \sqrt{1 + 4t_{k - 1}^2})$ . Then updating  $\mathbf{x}$  with convergence rate  $\mathcal{O}(1 / k^2)$  is (See Fig. 2)
$$
\mathbf {x} ^ {k} = \mathcal {T} _ {\frac {\lambda}{L}} \left(\mathbf {q} _ {L} \left(\tilde {\mathbf {x}} ^ {k - 1}\right)\right). \tag {20}
$$
Next, we further consider constraints of Problem (16). It can be easily solved by a projection operation [17, 14] in each iteration, including a sphere projection  $\mathbf{x} \gets \frac{\mathbf{x}}{\|\mathbf{x}\|_2}$ , and an orthogonal projection  $\mathbf{x} \gets (\mathbf{I} - \mathbf{BB}^\top)\mathbf{x}$ , because the projector of orthogonal complement is  $\mathbf{I} - \mathbf{BB}^\top$ .
Eqs. (17) and (20) provide iterative optimization for searching sparse solution of  $\mathbf{x}$  and  $\mathbf{e}$ , thus we term the whole algorithm as Dual Sparsity Pursuit (DSP). Repeatedly using DSP, we can successfully address the general unknown model fitting task, and the whole process is concluded in Append.D. Note that, in two-view problem, each model has at least  $r = 1$  basis (e.g.  $\mathbf{F}$ ) and at most  $r = 3$  bases (e.g.  $\mathbf{H}$ ), thus we denote  $R$  as the max number of bases to be estimated, with  $R = 3$ .
Model Reasoning Analysis. A critical problem is how to reason out the model type and estimate the parameters under a general case. Several methods [44, 41] have been early proposed to use a selection strategy [33, 31, 32], but it is not always clear due their ambiguities [37]. This would be well mitigated in our DSP. To be specific, for 2D point set fitting, we can directly distinguish the correct model from  $\{L, P, E\}$  through the sparsity of estimated  $\mathbf{x}$ , as shown in Fig. 2. As for two-view models, we can also distinguish them via the sparsity and rank of estimated  $\mathbf{X}$ , referring to the model embedding  $\Psi(\cdot)$ . The following remark may deliver good understandi
Figure 2: The optimization procedure of our DSP for 2D line fitting.  $\mathbf{x} = [a,b,c,d,e]^{\top}$  and  $\mathbf{x}' = [a',b',c',d',e']^{\top}$  denote the solutions w/o and with acceleration, respectively.
Remark 1 If the true model is  $\mathbf{F}$ , the solution for  $(\mathbf{x}_2, \mathbf{e}_2)$  would result in a larger fitting error than  $(\mathbf{x}_1, \mathbf{e}_1)$ , i.e.,  $\|\mathbf{M}^\top \mathbf{x}_1\|_2 \ll \|\mathbf{M}^\top \mathbf{x}_2\|_2$ . In other words, the number of inliers detected by  $(\mathbf{x}_2, \mathbf{e}_2)$  would be much smaller than  $(\mathbf{x}_1, \mathbf{e}_1)$ . Since the second basis is essentially to find a homography or affine structure. On the contrary, if the true model is  $\mathbf{H}$  or  $\mathbf{A}$ , the detected inliers between these two solutions would have small difference, as the orthometric and sparse bases indeed exist.
Implementation Details: (1) Parameter setting. In DSP,  $\lambda$  and  $\gamma$  are two hyper-parameters. Based on [44], we set  $\lambda = 0.005\log (4N)\times$ $[1,1,0.5,1,1,0.5,0.5,0.5,0.5,0.1]^{\top}$  as default (for 2D model, we set  $\lambda =$ $\log (2N)\times [0.01,0.1,0.1,1,1]^{\top})$  . In addition, we set  $\gamma = 0.06$  at the beginning, then update it with  $0.98\gamma$  for each twenty iterations, and constrain  $\gamma_{\mathrm{min}} = 0.02$  . Moreover, we set the max iteration as  $2k$  , and stop if  $\varepsilon = \| \mathbf{x}_k - \mathbf{x}_{k - 1}\| _2\leq 1e - 6$  . As for  $\tau$  , it controls the number of estimated basis, i.e.,  $r$  . We set  $\xi = \mathcal{L}(\mathbf{M},\mathbf{x}_i,\mathbf{e}_i) / \mathcal{L}(\mathbf{M},\mathbf{x}_{i - 1},\mathbf{e}_{i - 1})$  and describe its distribution on all real data as in Fig. 3. Based on the best  $\xi$  , we set  $\tau = 1.2\mathcal{L}(\mathbf{M},\mathbf{x}_{i - 1},\mathbf{e}_{i - 1})$  during the estimation of  $\mathbf{x}_i$  . (2)
Figure 3: Distribution of  $\xi$  on  $\mathbf{H}$  data and  $\mathbf{F}$  data.
Synthesized Data. We synthesize 300 image pairs for each model, which consist of different outlier ratio  $(\mathrm{OR} = 20\%, 50\%, 80\%)$  and noise level  $(\mathrm{NL} = 0.2, 0.5, 0.8, 1.0, 1.5$  in pixel). (3) Real Image Data. 8 public datasets [8, 2] are used, and we divide them into two groups including Fund: kusvod2, CPC, TUM, KITTI, T&T; and Homo: homogr, EVD, Hpatch. (4) Comparisons. 6 methods are used for evaluation, including three SAC methods RNASAC [19], USAC [38], and MAGSAC++ [6]; one globally optimized method EAS [18]; and two deep methods OANet [50], SuperGlue [40]. (5) Metrics. All methods applied the normalized 4-point and 8-point algorithms for H and F estimation (which are applied as post-process [18] in EAS, DSP and two deep methods with 100 iterations). To measure the accuracy, we use Geometrical Error (GE, in pixel), which is defined as the re-projection error and Sampson distance [4, 5] for H and F estimation, respectively. We defined the failed case as that the GE is larger than 5 pixels, or the model is wrongly identified. To avoid the failed cases for a dataset, we use the medium value  $E_{med}$  of GE as measurement. (More details are in Append.E).
Model Reasoning and Robust Fitting Test:
(1) Test on Synthetic Data. In the simulation test, an advanced robust estimator EAS [18] is used to estimate all possible models first. Then, three widely used model selection criteria, i.e., AIC [1], BIC [42] and GRIC [25], are equipped. The accuracy  $(\%)$  of model selection are reported in Tab. 1. AIC can achieve good performance for homography and affine models, but not for epipolar cases, particularly for high OR. BIC shows poor performance in fundamental model selection, since it prefers under-fitting in theory. GRIC fully considers the properties of vision problem performing well in most cases. As for our DSP, it can identify the
Table 1: Model reasoning results on syn. data w.r.t different OR . Each value  $x$  indicates that  $x\%$  models are correctly selected.
OR	Data Model	AIC	BIC	GRIC	DSP (ours)
20%	F,100	100	62	100	100
H,100	100	98	100	100
A,100	100	96	100	100
50%	F,100	100	0	100	100
H,100	100	95	100	100
A,100	99	97	98	100
80%	F,100	85	0	93	100
H,100	95	96	98	99
A,100	96	92	95	98
all	97.2	70.7	98.4	99.7
geometric models with the best accuracy even in case of high outlier ratio.
Table 2: Results of model reasoning and fitting on real image pairs. All datasets are divided into Fund and Homo. based on the model type. Method with * means first estimating all possible models, then using GRIC to select the "best" one. Bold and underline indicate the best and second, respectively.
Data\Methods	RANSAC*	USAC*	MAGSAC++*	EAS*	OANet*	SuperGlue*	DSP (ours)
Fund.	E_med	0.8478	0.6545	0.6260	0.6900	0.7579	0.8926	0.6017
Time(s)	0.5666	0.0239	0.3918	0.2037	0.0152	0.0721	0.0551
FR	0.2925	0.1992	0.2077	0.2142	0.2632	0.2709	0.1136
Homo.	E_med	1.0428	0.8744	0.8978	0.8472	0.8480	0.9392	0.8227
Time(s)	2.010	0.0550	1.3419	0.4463	0.0342	0.0716	0.2794
FR	0.3021	0.2604	0.1424	0.0972	0.0903	0.1181	0.066
All	E_med	0.8794	0.6711	0.6383	0.7091	0.7670	0.9026	0.6249
Time(s)	0.6638	0.0260	0.4558	0.22	0.0130	0.0721	0.0703
FR	0.2932	0.2043	0.2033	0.2064	0.2515	0.2606	0.1123
(2) Test on Real Image Data. In this part, those datasets are divided into Fund and Homo based on their model type. In fact, during real applications, it is more urgent to make sure whether the true model is F or H [41, 33], thus our used real image data are sufficient to verify the performance for real cases. In addition, considering the competitive performance in Tab. 1, we only use GRIC as comparison in model selection stage. Six representative estimators are used for comparison. That is, first using those robust estimators to obtain all possible models, then using GRIC to select the best one. Qualitative results on two representative image pairs are shown in Fig. 4, which obviously reveal the mechanism of our DSP as we assumed in Remark 1. In addition, our algebra results are somewhat coarse, but they can be well refined by a post-processing [18]. Quantitative results are evaluated in Tab. 2, including  $E_{med}$ , Run Time (Time) and failure rate (FR). From this ta
ble, we find that the other estimators typically achieve poor model selection even if using the best criterion GRIC. Three SAC methods get poor model identification, particularly for Homo data, since many image pairs are from extreme view change with high outlier ratios. OANet and SuperGlue similarly achieve low model reasoning accuracy, since some datasets are from complicated scenarios, such as EVD, T&T and CPC, which are not seen by these two deep methods in their training. Their model fitting performances directly affect the model reasoning accuracy. Overall speaking and comparing with GRIC, our DSP reasons out the true model directly from its solutions with much higher accuracy. As for running time, OANet and USAC almost achieve the best efficiency. Because OANet is a deep model with GPU acceleration (SuperGlue is slower since it takes descriptors as input to generate matches), and USAC integrates the local optimization and fast model verification strategy in its universal framework, but sacrifices the accuracy to some extent.
**Known Model Estimation Test:** Known model estimation is the main focus of current researches, thus we also test on those real image pairs by giving the priori of model type, and provide extensive comparisons. The statistic results are shown in Fig. 5, which only contains four challenging datasets, while others are in *Append.E.3*. Statistic results show that the performance of SAC-based methods would heavily degrade with more time consuming and less accuracy on these challenging datasets. This is because they intrinsically have to sample an outlier-free subset to best fit the given model, this is much difficult for these datasets. OANet and SuperGlue achieve poor fitting accuracy due to the same reason particularly for EVD datasets. As for EAS and our DSP, they commonly optimize from global formulation, thus showing similar top accuracy. However, our DSP performs better comparing with EAS, due to the specific modeling to restrain noise and outliers. And it is much faster, because the use of an acceleration strategy that makes the convergence rate closed to  $\mathcal{O}(1 / k^2)$ .
Figure 4: Detected inliers of DSP for real data:  $\mathbf{F}(\text { left })$  and  $\mathbf{H}(\text { right })$ . Top to bottom: input data, the first  $(\mathbf{x}_1, \mathbf{e}_1)$ , second basis  $(\mathbf{x}_2, \mathbf{e}_2)$ , and refined results. For visibility, at most 200 random selected matches are shown (blue = true positive, green = false negative, red = false positive).
Figure 5: Qualitative statistic results of Geometric Error (top) and Run Time (bottom) for known model estimation. A point on the curve with coordinate  $(x,y)$  denotes that there are  $100 \times x$  percent of image pairs which have GE or RT no more than y. The lower the better.
Applications: (1) Multi-model Fitting. We first consider the multi-class multimodel fitting problem. Unlike multimodel fitting of single type, it would be more challenging if the data contain multiple models with unknown model type and model number. To this end, a public benchmark AdelaideRFM is used for evaluation. Quantitative results are re
ported in Tab. 3, where Misclassification Error (ME, %) is selected to characterize the performance. Six algorithms including T-link [29], RCMSA [36], RPA [30], MCT [31], MLink [32] and RFMSCAN [23], are used for comparison. The table shows that our DSP can achieve the best accuracy with low computational burden.
Table 3: Results of Multi-model fitting. The mean and median values of ME (\%), and Running Time (RT, s) are listed. Bold and underline indicate the best and second.
Tlink	RCMSA	RPA	MCT	MLink	RFM-SCAN	Ours
ME_ave	27.65	10.05	5.28	11.36	6.04	2.63	1.64
ME_med	27.88	6.087	4.35	1.21	4.22	1.20	0
RT(s)	1.95	2.16	10.84	6.44	9.72	0.01	0.18
(2) Loop Closure Detection. We next exploit 2 sequences from KITTI vision suite [20] (i.e., K00, K02) to evaluate our method in the loop closure detection (LCD) task, i.e., recognizing re-observations during the navigation of a robot. These two sequences have enough loop closures. Because DELG [10] is considered state-of-the-art
for LCD-related tasks such as image retrieval and landmark recognition, we choose it to simultaneously extract global and local features of images to perform LCD in a hierarchical way. Specifically, given a query image, the similarity of global features under L2-metric is first used to select its candidate frame, followed by performing model fitting between them based on local features. Only when sufficient matches are preserved would a loop-closing event be triggered. Results are reported in Tab. 4, which shows that our DSP achieves the best, since our dual sparsity constraints can well address navigation of the degraded cases such as only going ahead or making a turn.
Table 4: Results of LCD. Metric: the recall at  $100\%$  precision. Bold indicates the best.
RSAC	USAC	MSAC++	EAS	OANet	Ours
K00	0.9112	0.9118	0.9105	0.9086	0.7910	0.9162
K02	0.7632	0.7796	0.7757	0.7664	0.6441	0.7882
In this paper, we proposed a unified modeling and fast optimization paradigm named DSP for robust model fitting with unknown model type and heavy outliers. It enables to use as few parameters as possible to explain the inlier structure, thus achieving true model identification and parameter estimation. We introduced an alternating optimization strategy together with proximal approximation method to accurately estimate the sparse model parameters and outlier entries. In addition, our proposed method is validated to exactly disambiguate the true model on both synthetic and real data. Limitation: Our proposed method can only perform for single model, but real data often contains multiple different models, such as point cloud analysis. In the future, we will integrate the multi-model fitting problem into our modeling, thus achieving a four-fold task in a unified optimization.
The proposed method enjoys great potential to improve a wide range of industrial applications including image registration and retrieval, camera pose estimation and localisation, 3D reconstruction, etc. To be specific, the proposed method achieves significant accuracy gains of model reasoning and fitting on challenging matching benchmarks, which strongly indicates that it will directly benefit many related fields including robotics and autonomous driving, where the geometric model estimation is the foundation. However, our method does have some unwelcome repercussions like many other machine learning techniques. The accurate and robust model reasoning and estimation can be illegally used for a person or property without permission. It may even be weaponized to guide a UAV to perform self-location then carry out a terrorism attack. But these negative impacts are more related to the fields of application rather than the technology itself. And we believe, under strict supervision, that our work will bring more benefits than harms to society.
Funding in direct support of this work: NSFC grant 62276192.
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We provide necessary proof, pseudo code, additional analysis of our method, and more details about the implementation, the used datasets and additional experimental results.
A Notaions