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By: Yair Weiss and Edward H. Adelson. Presenting: Ady Ecker and Max Chvalevsky. EM for Motion Segmentation “Perceptually organized EM: A framework that combines information about form and motion” “A unified mixture framework for motion segmentation: incorporating spatial coherence and estimating the number of models”

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2 Contents Motion segmentation. Expectation Maximization. EM for motion segmentation. EM modifications for motion segmentation. Summery.

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Part 1: Motion Segmentation

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4 Motion segmentation problem Input: 1. Sequence of images. 2. Flow vector field – output of standard algorithm. Problem: Find a small number of moving objects in the sequence of images. vxvx vyvy v

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5 Segmentation Output Classification of each pixel in each image to its object. Full velocity field. flow data velocity field

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6 Segmentation goal

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7 Motion vs. static segmentation Combination of motion and spatial data. Object can contain parts with different static parameters (several colors). Object representation in an image can be non-continuous when: There are occlusions. Only parts of the object are captured...

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8 Difficulties Motion estimation. Integration versus segmentation dilemma. Smoothing inside the model while keeping models independent.

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9 Motion estimation - review Estimation cannot be done from local measurements only. We have to integrate them.

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10 Motion integration In reality we will not have clear distinction between corners and lines.

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11 Integration without segmentation When there are several motions, we might get false intersection points of velocity constraints at T-junctions.

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12 Integration without segmentation False corners (T-junctions) introduce false dominant directions (upwards).

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13 Contour ownership Most pixels inside the object don’t supply movement information. They move with the whole object.

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14 Smoothing We would like to smooth information inside objects, not between objects.

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15 Smoothness in layers

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16 Human segmentation Humans perform segmentation effortlessly. Segmentation may be illusive. Tendency to prefer (and tradeoff): Small number of models. Slow and smooth motion. The segmentation depends on factors such as contrast and speed, that effect our confidence in possible motions.

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17 Segmentation illusion – The split herringbone

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18 Segmentation Illusion - plaids

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Part 2: Expectation Maximization

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20 Clustering

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21 Clustering Problems Structure: Vectors in high-dimension space belong to (disjoint) groups (clusters, classes, populations). Given a vector, find its group (label). Examples: Medical diagnosis. Vector Quantization. Motion Segmentation.

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22 Clustering by distance to known centers

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23 Finding the centers from known clustering

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24 EM: Unknown clusters and centers Maximization step: Find the center (mean) of each class Start with random model parameters Expectation step: Classify each vector to the closest center

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25 Illustration

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26 EM Characteristics Simple to program. Separates the iterative stage to two independent simple stages. Convergence is guaranteed, to some local minimum. Speed and quality depend on: Number of clusters. Geometric Shape of the real clusters. Initial clustering.

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27 Soft EM Each point is given a probability (weight) to belong to each class. The E step: The probabilities of each point are updated according to the distances to the centers. The M step: Class centers are computed as a weighted average over all data points.

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28 Soft EM (cont.) Final E step: classify each point to the nearest (most probable) center. As a result: Points near a center of a cluster have high influence on the location of the center. Points near clusters boundaries have small influence on several centers. Convergence to local minima is avoided as each point can softly change its group.

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29 Perceptual Organization Neighboring or similar points are likely to be of the same class. Account for this in the computation of weights by prior probabilities.

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30 Example: Fitting 2 lines to data points Input: Data points that where generated by 2 lines with Gaussian noise. Output: The parameters of the 2 lines. The assignment of each point to its line. riri (x i,y i ) y=a 1 x+b 1 + vy=a 2 x+b 2 + v v~N(0,1)

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31 The E Step Compute residuals assuming known lines: Compute soft assignments:

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32 Least-Squares review In case of single line and normal i.i.d. errors, maximum likelihood estimation reduces to least-squares: The line parameters (a,b) are solutions to the system:

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33 The M Step In the weighted case we find Weighted least squares system is solved twice for (a 1,b 1 ) and (a 2,b 2 ).

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34 Illustrations

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Illustration

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36 Estimating the number of models In weighted scenario, additional models will not necessarily reduce the total error. The optimal number of models is a function of the parameter – how well we expect the model to fit the data. Algorithm: start with many models. redundant models will collapse.

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Illustration l=log(likelihood)

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Part 3: EM for Motion Segmentation

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39 Segmentation of image motion: Input Products of image sequence: Local flow – output of standard algorithm. Pixel intensities and color. Pixel coordinates. Static segmentation: Based on the same local data. Problematic as explained before.

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40 Segmentation output segmentation Models: ‘blue’ model‘red’ model

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41 Notations r - pixel. O r - flow vector at pixel r. k - model id. k - parameters of model k. v k (r) - velocity predicted by model k at location r. D k (r) = D(r, k ) - distance measure. - expected noise variance. g k (r) - probability that pixel ‘r’ is a member of model ‘k’.

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42 Segmentation output Segmented O: Model parameters: blue red r V red (r) O(r) V blue (r)

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43 The E Step Purpose: determine statistic classification of every pixel to models. k (r) - prior probability granted to model ‘k’. For classical EM, k (r) are equal for all ‘k’.

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44 The E Step (cont) Alternative representation: Soft decision enables slow convergence to better minimum instead of finding local minima.

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45 Distance measure functionality Correct physical interpretation of motion data. If possible – enable analytic solution.

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46 Distance measures (1) Optic flow constraint: – window centered at ‘r’. v k (r) – velocity of ‘k’ at location ‘r’. Quadratic. Provides closed MLE solution for the M-step.

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47 Distance measures (2) Deviation from constant intensity: – window centered at ‘r’. Good for high speed motion. Resolved by successive linearizations.

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48 The M step Purpose: layer optimization (according the soft classification of pixels). Produces weighted ‘average’ of the model. ‘Average’ depends on definition of D. Constrained by J (slow & smooth motion).

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49 J (cost) definition For loosely constrained (typical for image segmentation): For highly constrained : (#degrees of freedom < #owned pixels). 0

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50 EM: Unknown clusters and centers Estimation step: Classify each vector to the closest center Maximization step: Find the center (mean) of each class Start with random model parameters

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51 Natural image processing without segmentation a)Frame of a movie taken from driving car. b)Flow data along the dotted line. c)Smooth global approximation of motion (along the line).

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52 EM natural image processing a)The same picture. b)Rigid-like model segmentation. c)EM result.

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53 Textureless Regions Homogeneous regions have no clear layer preference (stay gray in ownership plots). Wrong segmentation decisions for “similar” motions (squares example). Probabilistic resolution of ambiguities (bars example).

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54 Illustration BARS : Probabilistic solution, vertical vs. horizontal. 2 squares moving diagonally right No segmentation for vertical lines. No segmentation for background. Motion directions identified correctly. Hand : Noisy segmentation

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55 Energy formulation of EM E-step: optimization with respect to g k (r). First term prefers hard decision. Second term (entropy) prefers no decision. M-step: optimization with respect to (embedded in D).

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Part 4: EM Modifications for Motion Segmentation

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57 Proposed modifications POEM – Perceptually Organized EM Combines local & neighbor motion with static analysis. Regional grouping. Color & intensity data. Contour ownership. Outlier detection T-junction points. Statistical outliers.

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58 POEM algorithm idea Determine the segmentation based on: Local pixel flow (standard EM). Neighbor pixel segmentation. Static data (optionally). Reason: neighboring pixels have higher probability to belong the same object. Similar pixels have even higher probability to belong the same object.

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59 PO Window of influence Neighbor votes:

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60 POE step Basic equation: estimation:

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61 POE step alternative representation The solution is computationally intensive.

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62 M step in POEM The M step is unchanged:

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63 POEM Energy formulation PO represented by the additional (last) term

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64 Contour ownership Implemented by modification of PO function: Step 1: preliminary segmentation & border detection. Step 2: contour ownership determination – by relative depth, consistent with T-junctions. Step 3: combining in voting procedure. Equation modification: window of influence between pixels (w(r,s)) gives additional weight to pixels on segment’s borders.

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65 Results of POEM Advantages : Resolves regions without texture by propagating information from borders. More robust to noise.

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66 Illustration 2 squares moving diagonally right Partially correct segmentation for vertical lines. Moving hand: Smooth solution

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67 Bars results Input Classical EM segmentation POEM segmentation without contour ownership POEM segmentation with contour ownership

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68 Flow outliers Segmentation into k layers and additional layer of “outliers” Probability to be outlier – function of: Prior – e.g. for T-junction. Likelihood – likelihood to be outlier. Outliers don’t participate directly in PO. Outliers layer is not smooth nor slow.

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Part 5: Summery

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70 Advantages The system showed relatively good results: For natural images. For artificial, challenging images. Simple - has few parameters. Universal. Modular - enables improvements: Utilizing additional data (mostly static). Optimizing parameters. Using advanced convergence methods. Altering priors to fit non-symmetric biological phenomena.

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71 Drawbacks Some images weren’t resolved completely: Edge deviations in ‘hand’ image. The system includes input-dependant . No process to determine value of was proposed. The ‘optic flow constraint’ measure is appropriate only for instantaneous motion. Other distance measures – much more difficult to solve.

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72 Conclusions The system tackles with ambiguity & noise. It estimates the degree of ambiguity. It assumes slow & smooth motions. The system is capable to explain its input by segmentation into separate layers of motion. It exploits static data to improve the segmentation.

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