Integrais Triplas - Teoria, Exemplos e Aplicações

Integral Tripla Dada uma função de três variáveis fxyz temse ΔVi Δxi Δyi Δzi Onde D é o sólido fechado e limitado no espaço xyz Dividindo ou porcionando o sólido D em n paralelepipedar Sendo que a cada paralelepípedo tomemos o ponto Pi formando o produto fxi yi zi ΔVi f xi yi zi ΔVi i1 n Se n tende para o infinito se e somente se ΔVi tende a zero lim n Sn fx y z dv O cálculo da integral tripla é feito do mesmo modo que a integral simples ou de dupla Ex 0 1 3 1 0 x 2y 43 dx dy dz 0 1 3 1 0 x 2y 43 dx dy dz 0 1 3 1 0 x22 2y x 43 x dy dz 0 1 3 1 0 x22 2y y 43 y dy dz 0 1 3 1 0 22 2y2 432 x22 2yj 43J dy dz 0 1 3 1 0 2 4y 83 12 2y 43 dy dz 0 1 3 2y 43 32 20 dy dz 0 3 22 43 0 32 0 1 43 1 32 1 dz 0 3 x22 43 y 32 y dz 0 ₀³ 1 4z 32 dz ₀³ 12 4z dz 12 z 4z²2 ₀³ 12 z 2z² ₀³ 12 3 2 3² 32 18 392 2 I ₁² ₀³ ₓ₃ˣ z dy dx dz 2 I ₁² ₀³ ₓ₃ˣ z dy dx dz ₁² ₀³ z y ₓ₃ˣ dx dz ₁² ₀³ z x z z x z dx dz ₁² ₀³ 2 z² dx dz ₁² 0 dz ₁² 2 z² z dz 2 z⁴4₁² 2 z⁵5 2 z⁵5 ₁² 2 2⁵5 2 1⁵5 645 25 625 ₁² ₀³ x z z³ x z z³ dx dz ₁² ₀³ 2 z² dx dz ₁² 0 dz ₁² 2 z² z dz 2 z⁴4₁² 2 z⁵5 1400 S Sepulveda Blvd Ste 285 Los Angeles CA 90025 310 4815659 GVK40591 GVK PLASTIC SURGERY COSMETIC DERMATOLOGY 1400 S Sepulveda Blvd Ste 285 Los Angeles CA 90025 310 4815659 GVK40591 Diffusion Models and Views on Representation and Composition We consider whether diffusion models have in implicit representations since assuming explicit latent state is limiting Recently several stagediscriminative models have viewed generative diffusion models as denoising autoencoders DAEs which allows for extracting feature rep resentations useful in other tasks This observation suggests there is an interplay between diffusion and latent representations Further the noisy intermediate latent x t the composition of information from data generation pipeli et al 2022 since the discrimi the reverse noising process Caron ture plays an important role in the interpretablity of the representation Moreover evidence suggests a hierarchical nature in generative diffusion models We highlight key properties of hierarchical representations emergent in diffusion models They have high semantic content and interpretability They provide meaningful compositionality that enables the generation of multiple plausible outputs and closest in distribution outputs using the same representation They show continuous smoothness of the representation manifold allowing meaningful latent interpolations 31 THE REVERSED NOISING PROCESS IS A GENERATIVE MODE We focus on discretetime diffusion models that Ma rkov chains as sequences of T fixed or learned length with noising and denoising processes These models Io define a forward noising process connecting the clean data x 0 at t 0 to a noise variable x T at t T via a Markov chain qx 1T x 0 The goal is to learn a reverse process p x T0 that starts from noise x T and returns a clean sample x 0 by reversing the forward noising process Given Markovian structure for both forward and reverse processes the model can be parameterized using neural networks as follows p x Tx 0T1 px T Π x tx t1 The prior distribution px T is commonly modelled as isotropic Gaussian distributed noise N0 1 although the choice may vary based on data modality To parameterize the reverse denoising transition p x tx t1 approaches vary in parameterization choice but most estimate some variation of the conditional distribution by neural networks θ It is common to use a reparameterization trick to estimate the mean p θ x tx t1 by predicting either the mean or the noise applied learning the data distribution varies The training maximizes the variational lower bound VLB on the negative loglikelihood of the model with an alternative simplified loss based on mean squared error MSE between the predicted noise and true noise used for many diffusion models in practice L MSE EAε εε θ x t t 2 where x t is a noised version of x 0 and ε is sampled noise This training process allows the model to learn the distribution of the data by gradually estimating how to denoise a sample at each intermediate timestep t 32 DIFFUSION MODELS PRODUCE CONTINUOUS SEMILATENT REPRESENTATIONS We argue that the intermediate noisy latents x t as samples from an inferred posterior distribution qx tx 0T carry rich representations of the data at varying fidelities from coarse high t features closer to noise to precise low t features close to x 0 We consider latents x t for t in the range 0 T as residing in a continuous manifold that encodes progressively more information about the data moving from noise at tT least information to clean data at t0 full information In this sense the noisy latent x t can be seen as a semilatent representation that is neither the pure observed data nor an explicitly learned compact latent code but a continuous framing between these extremes The encoding process qx tx 0 allows us to access these meaningful semilatents at arbitrary stages These semilatents can be practical for tasks requiring flexible tradeoffs between computational cost and fidelity as we can stop the generative process at intermediate scales while preserving meaningful representations that capture portions of the data 33 THE REVERSED PROCESS LATENT SPACE IS SEMANTIC AND COMPOS ITIONAL The denoising network in diffusion models learns to reconstruct informative object features from noisy inputs Empirical studies have shown that such intermediate representations learned by the denoising models encode semantically meaningful factors where meaningful features of object parts are disentangled and organized hierarchically These semilatents are also compositional in that intermediate noisy latent represen tations preserve independent components that can be recombined consistently in the denoising process allowing complex image manipulations As a result the semilatents in diffusion models can be seen as a new form of representation that lies between explicit latents and direct pixel space offering a promising framework for compositional visual representations This inherent compositionality is one of the attractive features that makes diffusion models suitable for tasks such as image editing conditional generation and style transfer