@Ken No. If it were consistent the error would go to 0 asymptotically.

Here do we assume area of X = 1?

What is the difference of curvation and second derivative?

Should the red line be biased + consistent?

arkit also uses blendshapes for their emoji ar faces :)

What does ai stand for? Is it the position of knots or something we need to compute?

this image is so beautiful..

this is a great diagram, it helped me understand

Here's a link to an interactive app to play around with B-splines. Regarding the question on locality of a B-spline, your intuition is right that locality decreases when increasing the degree. And using the definition above, the bases B_{i,d} is non-zero over the interval t_i to t_{i+d}.

@tracychen Yes, though the question of differentiability is really only important at the internal knots of the spline (you don't need to worry about differentiability at the endpoints, since it will always be differentiable).

@tracychen @donglaix Yes---curvature being zero means only the second-order derivative is zero.

@Qwerty For additional problems on splines, I suggest taking a look at the past finals posted on Piazza (see Spring 2014 & Spring 2016 exams).

I am confused about how the length of the interval is decided by degree?

Does the requirement of differentiable everywhere also include endpoints?

Same question as @donglaix

@Tdog @SlimShday Typically no---the classical gradient descent algorithm does not normalize the magnitude of the gradient. However, there is such a thing as normalized gradient descent, so there may be arguments to normalize.

@SlimShady @crabbage Exactly. In the case of this affine function, sampling the function at "just the right place" refers to choosing the sample x = (a+b)/2.

@BRCO "Random" is perhaps the wrong word here, but the tangents at the endpoints of a Catmull-Rom spline could potentially be user-defined.

@BRCO Yes, h is the grid size here, and h^2 is the area of each grid square. And the units are however you define them (e.g., meters, millimeters, pixels, etc.)

@elenagong @Tdog tau is the time step used here, and h is the grid size in both dimensions.

@Tdog It seems unlikely that these techniques (and graphics in general) will become obsolete. If anything, there's a move towards using graphics to generate more data for training networks.

@A-star The noise in the left image certainly is not desirable, and finding good ways to "smooth" out this noise (lower the variance) is key. At the same time, the objective of physics-based rendering is that the image be a photorealistic representation of the scene, so this needs to be done carefully; simply blurring the image won't necessarily give the desired effect.

Woman with a Parasol. This painting captures the impression that Madam Monet and her son walked through a field during a breezy day, with grass blowing in the day, and a flowing dress. Before the invention of projectors and other similar mediums, this was one artist's way of capturing animations/dynamics. (Hopefully that's justification enough for why this counts. ;-) )

@A-star Maximizing a concave objective function (subject to certain constraints) is effectively the same as minimizing the negated version of that same objective function. Any concave optimization function can be turned into a convex optimization function, and so they are equally easy to solve.

SIGGRAPH Asia 2019 Technical Papers Trailer

Also, check out Ke-Sen Huang's home page for a convenient list of SIGGRAPH and SIGGRAPH Asia papers (among other graphics venues) compiled over the past decade.

Are there subclasses of concave optimizations that are easy to solve as well?

Hi. Can someone point me to more resources on this topic? It still looks confusing to me. Thanks.

How does this count as animation?

This looks a lot like a smoothing effect to me. So I was wondering can we achieve this kind of low variance by first rendering the scene using the high variance setting and then applying specific filter before the "frame" gets displayed onto the screen?

I have never we can apply graphics techniques to hardware.

I wonder how much of this is gonna become obsolete once ML based physics simulations become a thing.

Is tau the timestep?

Is h the size of grid here? I wonder if it has any unit, for example, pixel?

Do we just assign a random tangent value to the end points when using Catmull-Rom?

I guess the x corresponding to the mean f(x)

I don't think we need to? because we have the step size to tune the magnitude, but correct me if I am wrong

https://www.youtube.com/watch?v=ck-r_qmNNG0 Some intuitions behind wave equation.

What is tau representing here? Is it 2pi?

Do we ever normalize the gradient?

Thanks for the links

The elastic man is really fun to play with

Any examples of a double black diamond pde/ what real-world scenario it would correspond to?

@elenagong Well, if you can compute the gradient analytically, then (1) you compute the gradient at a point x_0, and (2) search along the gradient direction d for another point that minimizes your function f(x_0 - \alpha d). (See the next slide.) Otherwise, perhaps you can perform finite differencing at a point x_0 to estimate the gradient.

@Kalecgos Well I suppose you just have to read the paper "A Material Point Method For Snow Simulation". ;-) The equations used to simulate snow physics are listed in Section 4, and involves using "particles (material points) to track mass, momentum and deformation gradient."

Also, I'll point out that the paper mixes both Lagrangian & Eulerian representations to simulate snow, an approach briefly mentioned in this slide.